text:
\begin{document}
\title{Wild recurrent critical points.}

\author{Juan Rivera-Letelier}
\address{Departamento de Matem\'aticas, Universidad Cat\'olica del Norte, Casilla 1280 - Antofagasta, Chile.}
\date{February 2004, revised June 2004.}
\email{rivera-letelier@ucn.cl}
\begin{abstract}
It is conjectured that a rational map whose coefficients are algebraic over $\Q_p$ has no wandering components of the Fatou set.

R.~Benedetto has shown that any counter example to this conjecture must have a wild recurrent critical point.
We provide here the first examples of rational maps whose coefficients are algebraic over $\Q_p$ and that have a (wild) recurrent critical point.
In fact, we show that there is such a rational map in every one parameter family of rational maps that is defined over a finite extension of $\Q_p$ and that has a Misiurewicz bifurcation.
\end{abstract}
\maketitle

\section{Introduction.}
Fix a prime number $p$, let $\Q_p$ be the field of $p$-adic numbers and let $\C_p$ be the completion of an algebraic closure of $\Q_p$.
We consider rational maps whose coefficients in $\C_p$ are algebraic over $\Q_p$, viewed as a dynamical systems acting on the projective line $\pp$.

\subsection{No wandering domains conjecture.}
Fix a rational map $R \in \C_p(z)$ whose coefficients are algebraic over $\Q_p$.
Just as in the complex case, the projective line $\pp$ is partitioned into two sets: the {\it Fatou set} where the dynamics is stable and the {\it Julia set} where the dynamics is (topologically) expanding~\cite{Hs}.
The Fatou set is open and dense in $\pp$ and it is in turn partitioned into three open sets, where the dynamics of $R$ is: {\it contracting}, (eventually) {\it quasi-periodic} and {\it wandering}~\cite{these}.

The situation here is analogous to the complex case, but Sullivan has shown that complex rational maps do not have wandering domains~\cite{Su}.
By analogy to the complex case, it is conjectured that the wandering part of the Fatou set is empty~\cite{Benowandering},~\cite{these}.
In its simplest formulation this conjecture is as follows.

\

{\par\noindent {\bf Conjecture.} {\it Every wandering disk is attracted to an attracting cycle.}}

\

\subsection{Wild recurrent critical points.}
R.~Benedetto has shown that any counter example to the no wandering domains conjecture must have a non-periodic wild recurrent critical point~\cite{Benowandering}.
A {\it critical point} of a rational map $R$ is a point in $\pp$ where the local degree of $R$ is strictly bigger than~1.
If moreover this local degree is a multiple of $p$, then the critical point is called {\it wild}.
A point of $\pp$ is {\it recurrent} under $R$ if it is accumulated by its forward orbit under $R$.

We provide here the first examples of rational maps {\it whose coefficients are algebraic over $\Q_p$}, having a (wild) recurrent critical point that is not periodic.
Such a critical point must belong to the Julia set, so for such rational maps there is a delicate balance between the local contraction of the critical point and the topological expansion of the Julia set.

\

{\par\noindent {\bf Theorem~A.}}
{\it In every one parameter family of rational maps that is defined over a finite extension of $\Q_p$ and that has a Misiurewicz bifurcation, there is a rational map whose coefficients are algebraic over $\Q_p$ and that has a non-periodic recurrent critical point.}

\

A one parameter family of rational maps $R$ has a {\it Misiurewicz bifurcation} at a parameter $t_0$, if there is a critical point that is mapped to a repelling periodic point by some iterate $R^\ell_{t_0}$ of $R_{t_0}$, but for $t \neq t_0$ near $t_0$ this does not happen.
See Section~\ref{Misiurewicz bifurcations} for precise definitions.

We give an algebraic condition that guarantee that the rational map in the theorem has coefficients in a given finite extension of $\Q_p$, see Corollary~2 in Section~\ref{Misiurewicz cascade}.
For a given $d > 1$, we show that the family,
$$
P_t(z) = - p^{-d}z (z - 1)^d + t,
\mbox{ for } t \in \C_p
$$
has a Misiurewicz bifurcation at $t = 0$ and that it satisfies this algebraic condition.
We conclude that there are arbitrarily small parameters $t$ in $\Q_p$, for which the critical point~1 of $P_t$ is recurrent but not periodic.
Note that the local degree of $P_t$ at~1 is equal to $d$, so when $p$ divides $d$ this critical point is wild.

\subsection{Comparison with the complex case.}
In the complex case C.~McMullen has shown that every non-trivial {\it algebraic} family of rational maps has bifurcations, except for some special families of Latt\`es maps~\cite{Mc1}.
Moreover Misiurewicz bifurcations are dense in the bifurcation locus~\cite{Mc2}.
So Misiurewicz bifurcations occur in virtually every algebraic family of complex rational maps.

This statement does not hold in the ultrametric setting considered here.
To begin with, the family of quadratic polynomials $P_c(z) = z^2 + c$, for $c \in \C_p$, is a non-trivial algebraic family of non-Latt\`es rational maps, and yet it does not have Misiurewicz bifurcations.
In fact, when $|4c| \le 1$ all cycles are non-repelling and when $|4c| > 1$ the (unique) finite critical point escapes to infinity, so it cannot be mapped to a repelling periodic point.
Besides parabolic bifurcations, that have a small impact in the overall dynamics, the quadratic family seems to be {\it locally} stable.

\

{\par\noindent {\bf Problem~1.}}
{\it Characterize those algebraic families having a Misiurewicz bifurcation.}

\

A {\it marked critical point} of a one parameter family of rational maps $R$, is a function $c : t \mapsto c_t$, such that for every parameter $t$ the point $c_t$ is a critical point of $R_t$.
A marked critical point is {\it active at a parameter~$t_0$} if on any neighborhood of $t_0$ the family $\{ R^n(c) \}_{n \ge 1}$ is not equicontinuous.
A natural candidate for the {\it bifurcation locus} is the set of those parameters for which there is a (locally) active critical point, compare with Proposition~2.4 of~\cite{Mc2}.

\

{\par\noindent {\bf Problem~2.}}
{\it Consider a one parameter family of rational maps $R$ with a marked critical point that is active at some parameter $t_0$.
Is $t_0$ accumulated by Misiurewicz bifurcations?}

\subsection{Remarks.}
As $\pp$ is totally disconnected (every connected component is reduced to a point), is not natural to consider the decomposition of the Fatou set into its connected components, as it is done in the complex setting.
Moreover the Fatou set is usually equal to $\pp$!
There is however a natural (dynamical) partition of the Fatou set, but it is not easy to describe.
This is done in detail in~\cite{Fatou}, see also~\cite{components} and~\cite{these}.
The situation is much more natural when one considers the action of the rational map on the Berkovich analytic space of $\pp$.

Recently Benedetto has shown examples of (polynomial) rational maps whose coefficients are {\it transcendental over $\Q_p$}, having a wandering domain~\cite{Bewandering}.
Using similar techniques, it can be shown that near every Misiurewicz bifurcation of a wild critical point there are (transcendental) parameters having a wandering domain.

Here we follow in part the axiomatic treatment of Misiurewicz bifurcations in~\cite{Mc2}.

\subsection*{Acknowledgments.}
I'm grateful to J.~Kiwi and R.~Benedetto for useful comments on a preliminary version of this article.
This paper was written during a visit of the author at IMPA and it was partially supported by MECESUP~UCN0202 and PROSUL.

\section{Preliminaries.}
Fix a prime number $p$, let $\Q_p$ be the field of $p$-adic numbers and let $\C_p$ be the completion of an algebraic closure of $\Q_p$.
{\it All finite extensions of $\Q_p$ that we will consider will be subfields of $\C_p$}.
We denote by $| \cdot |$ the norm of $\C_p$, by $\C_p^*$ its multiplicative group and
$$
|\C_p^*|
=
\{ |z| \mid z \in \C_p^* \}
=
\{ p^r > 0 \mid r \in \Q \}.
$$
We also denote $\cO_p = \{ z \in \C_p \mid |z| \le 1 \}$ the ring of integers of $\C_p$ and $\fm_p = \{ z \in \C_p \mid |z| < 1 \}$ the maximal ideal of $\cO_p$.

For $a \in \C_p$ and  $r \in |\C_p^*|$ we call
$$
\{ |z - a| \le r \}
\mbox{ or }
\{ |z - a| < r \}
$$
{\it open} or {\it closed ball of $\C_p$}, respectively.
When $r > 0$ does not belong to $|\C_p^*|$ these sets coincide and they form what we call an {\it irrational ball of $\C_p$}.
By definition a ball of $\C_p$ is irrational if and only if $\diam(B) \not \in |\C_p^*|$, so {\it the diameter of a open or closed ball of $\C_p$ belongs to $|\C_p^*|$}.

We identify the projective line $\pp$ of $\C_p$ with $\C_p \cup \{ \infty \}$.
So, for every subfield $K$ of $\C_p$ the corresponding projective line $\P (K) \subset \pp$ is identified with $K \cup \{ \infty \}$.
The {\it chordal metric} $\Delta$ on $\pp$, is defined in homogeneous coordinates by
$$
\Delta([x, y], [x', y'])
=
\frac{|xy' - x'y|}{\max \{ |x|, |y| \} \cdot \max \{ |x'|, |y'| \}}.
$$
When $z, z' \in \C_p$ we have
$$
\Delta(z, z')
=
\frac{|z - z'|}{\max \{ 1, |z| \} \cdot \max \{ 1, |z'| \} } \le |z - z'|.
$$

\section{Holomorphic functions.}\label{holomorphic functions}
Let $n$ be a positive integer.
A {\it closed} or {\it open polydisk} in $\C_p^n$ is a product $n$ of closed or open balls of $\C_p$, respectively.
When $n = 2$, polydisks are also called {\it bidisks}.

A {\it holomorphic function} defined on a polydisk $\underline{D}$ in $\C_p^n$, is a function given by a convergent power series of the form
$$
(z_1, \ldots, z_n)
\mapsto
\sum_{\nu_1, \ldots, \nu_n \ge 0} c_{\nu_1, \ldots, \nu_n} \cdot (z_1 - \zeta_1)^{\nu_1} \cdot \ldots \cdot (z_n - \zeta_n)^{\nu_n}.
~~\footnote{
Usually holomorphic functions are defined over more general domains.
As our results are of local nature, holomorphic functions defined over polydisks will be enough for our purposes.}
$$
Such a function has a power series expansion at each point $(\zeta_1, \ldots, \zeta_n)$ in $\underline{D}$, that converges in all of $\underline{D}$.
The sum, the product and the composition of holomorphic functions are holomorphic functions.

For a finite extension $K$ of $\Q_p$, we say that a holomorphic function $f$ defined on a polydisk $\underline{D}$ is {\it defined over $K$}, if the power series of $f$ at some point of $\underline{D}$ in $K^n$, has coefficients in $K$.
In that case the power series of $f$ at each point of $\underline{D}$ in $K^n$ has coefficients in $K$.
Clearly a holomorphic function $f$ defined over $K$ is also defined over every finite extension $L$ of $K$.
So the image by $f$ of an element of $L$ belongs to $L$.

The sum, the product and composition of holomorphic functions defined over $K$ are also holomorphic functions defined over $K$.

\subsection{}\label{basic}
The following lemma states some basic properties of holomorphic functions that we will use frequently.
For the proof in the case of the unit polydisk see Section~5.1 of~\cite{BGR}.
The general case follows easily from this case.

\

\lemma
{\it Let $D_1, \ldots, D_n$ be closed balls of $\C_p$ of diameters $r_1, \ldots, r_n$, respectively.
Then, for $(\zeta_1, \ldots, \zeta_n) \in \underline{D} = D_1 \times \ldots \times D_n$, a power series
$$
f(z_1, \ldots, z_n)
=
\sum_{\nu_1, \ldots, \nu_n \ge 0} c_{\nu_1, \ldots, \nu_n} \cdot (z_1 - \zeta_1)^{\nu_1} \cdot \ldots \cdot (z_n - \zeta_n)^{\nu_n}
$$
converges on $\underline{D}$ if and only if
$$
|c_{\nu_1, \ldots, \nu_n}| \cdot r_1^{\nu_1} \cdot \ldots \cdot r_n^{\nu_n} \to 0
\mbox{ as }
\max \{ \nu_1, \ldots, \nu_n \} \to \infty.
$$
In that case
$$
\sup_{\underline{D}} |f|
=
\sup_{\nu_1, \ldots, \nu_n \ge 0} |c_{\nu_1, \ldots, \nu_n}| \cdot r_1^{\nu_1} \cdot \ldots \cdot r_n^{\nu_n}.
$$}

\subsection{Lemma}\label{injective holomorphic functions}(Injective holomorphic functions){\bf .}

{\it Given $r \in |\C_p^*|$ set $D = \{ |z| \le r \}$ and let $f(z) = c_0 + c_1 z + \ldots$ be a power series converging on $D$.
Then the following assertions hold.
\begin{enumerate}
\item[1.]
$f$ is injective on $D$ if and only if for every $k > 1$ we have $|c_k| r^{k - 1} < |c_1|$.
In that case
$$
f(D) = \{ z \in \C_p \mid |z - c_0| \le |c_1| r \}.
$$
\item[2.]
If $f$ is injective, then its inverse function $g : f(D) \to D$ is holomorphic.
\item[3.]
If $f$ is injective and defined over a finite extension $K$ of $\Q_p$, then the inverse of $f$ is defined over $K$ and $f$ induces a bijection between $D \cap K$ and $f(D) \cap K$.
\end{enumerate}}
\proof
Part~1 is an easy consequence of Schwarz' Lemma, as stated in~\cite{these}, Section~1.3.1.
To prove part~2, we reduce to the case $c_0 = 0$ and $c_1 = 1$, so that $f(D) = D$.
Since $f$ converges on $D$ we have $|c_k| r^{k - 1} \to 0$ as $k \to \infty$.
By part~1, the hypothesis that $f$ is injective implies that for $k > 1$ we have $|c_k| r^{k - 1} < 1$.
Therefore $\gamma = \sup \{ |c_k| r^{k - 1} \mid k > 1 \} < 1$.

Let us now define inductively $b_i \in \C_p$, in such a way that
$$
\sum_{1 \le j \le i} b_j(f(z))^j \equiv z \mod z^{i + 1}.
$$
Set $b_1 = 1$, so that this is satisfied for $i = 1$.
Suppose that this holds for some positive integer $i$ and let $a \in \C_p$ be defined by
$$
\sum_{1 \le j \le i} b_j(f(z))^j \equiv z + a z^{i + 1} \mod z^{i + 2}.
$$
Then $b_{i + 1} = -a$ satisfies the inductive condition.

By definition $b_i$ is a linear combination, with coefficients in $\Z$, of terms of the form
$$
b_j \cdot \prod_{1 \le \ell \le j} c_{k_\ell},
$$
where $j \ge 1$ and $k_1 + \ldots + k_j = i$, with $k_\ell \ge 1$.
So it follows by induction that $b_i \in \Z[c_1, \ldots, c_i]$.
On the other hand,
$$
\left| b_j \cdot \prod_{1 \le \ell \le j} c_{k_\ell} \right| r^i
\le
|b_j|r^j \cdot \min \{ \gamma^j, |c_{k_1}| r^{k_1 - 1}, \ldots, |c_{k_j}| r^{k_j - 1} \}.
$$
It follows by induction on $i$ that $|b_i| r^i < r$, for $i > 1$ and that $|b_i| r^i \to 0$ as $i \to \infty$.
Therefore the power series $z + b_2 z^2 + \ldots$ converges on $D$ to the inverse of $f$.

For part~3, just observe that $b_i \in \Z[c_1, \ldots, c_i]$, so the inverse of $f$ is defined over $K$ whenever $f$ is.
The last assertion of part~3 follows from the fact that holomorphic functions defined over $K$ map points of $K$ to points of $K$.
$\hfill \square$

\subsection{Inverse branches.}\label{inverse branches}
Let $f : D \to \C_p$ be a holomorphic function.
If for a point $z_0 \in D$ we have $f'(z_0) \neq 0$, then $f$ is injective on a small ball containing $z_0$ and therefore $f$ has a local inverse $g$ at $z_0$ (Lemma~\ref{injective holomorphic functions}).
The function $g$ is holomorphic and it is defined over a finite extension $K$ of $\Q_p$, whenever $f$ is defined over $K$ and $z_0 \in K$.

\subsection{Lemma.}\label{Rouche}
{\it Let $f$ be an injective holomorphic function defined on a closed ball $D$ of $\C_p$, such that the ball $f(D)$ contains $0$.
Then for every holomorphic function $g : D \to \C_p$ satisfying
$$
\sup_D |g| < \diam(f(D)),
$$
there is $z_0 \in D$ such that $f(z_0) = g(z_0)$.
Moreover, if $f$ and $g$ are defined over a finite extension $K$ of $\Q_p$, then $z_0 \in K$.}
\proof
After an affine coordinate change on the domain we assume $D = \cO_p$.
Since $f(z) = c_0 + c_1 z + \ldots$ is injective, for every $i > 1$ we have $|c_i| < |c_1|$ and $f(\cO_p) = \{ |z - c_0| \le |c_1| \}$ (Lemma~\ref{injective holomorphic functions}).
So the hypothesis $0 \in f(\cO_p)$ implies $|c_0| \le |c_1|$.

If we set $g(z) = c_0' + c_1'z + \ldots $, then
$$
\sup \{ |c_i'| \mid i \ge 0 \} = \sup_{\cO_p} |g| < \diam(f(\cO_p)) = |c_1|.
$$
So Lemma~\ref{injective holomorphic functions} applied to $f - g$ implies that the holomorphic function $f - g$ is injective on $\cO_p$ and $(f - g)(\cO_p) = \{ |z| \le |c_1| \}$.
So there is $z_0 \in \cO_p$ such that $(f - g)(z_0) = 0$.

If $f$ and $g$ are defined over $K$, then $f - g$ is defined over $K$ and $z_0 \in K$ (part~2 of Lemma~\ref{injective holomorphic functions}).
$\hfill \square$

\subsection{One parameter families of holomorphic functions.}
A {\it one parameter family of holomorphic functions}, is a holomorphic function defined on a bidisk.
Such a function $f : B \times D \to \C_p$ can be written in the form
$$
f_t(z) = c_0(t) + c_1(t) (z  - \zeta) + c_2(t) (z - \zeta)^2 + \ldots,
$$
for each $\zeta \in D$, where each $c_i$ is a holomorphic function defined on $B$ and for each $t \in B$ the function $f_t$ is holomorphic.

When $f$ is defined over a finite extension $K$ of $\Q_p$ and $\zeta \in K$, the functions $c_i$ are defined over $K$.
In that case, for every $t \in K$ the function $f_t$ is defined over $K$.

\subsection{Lemma.}\label{one parameter injectivity}
{\it
Let $f$ be a one parameter family of holomorphic functions defined on a bidisk containing $(0, 0)$, of the form
$$
f_t(z) = c_0(t) + c_1(t) z + \ldots
$$
Given $r, r' \in | \C_p^* |$ set $B = \{ t \in \C_p \mid |t| \le r \}$ and $D = \{ z \in \C_p \mid |z| \le r' \}$.
Assume that $r > 0$ and $r' > 0$ are such that for every $t \in B$, we have
$$
|c_1(t)| = |c_1(0)| > 0, \ \
|c_0(t)| \le |c_1(0)|r'
$$
$$
\mbox{ and for $i > 1$ } \ \
|c_i(t)| (r')^{i - 1} < |c_1(0)|.
$$
Then for every $t \in B$ the holomorphic function $f_t$ maps $D$ bijectively onto
$$
D' = \{ z \in \C_p \mid |z| \le |c_1(0)| r' \}.
$$
Moreover, the inverse functions $(f_t|_D)^{-1}$ form a one parameter family of holomorphic functions $g$, defined on $B \times D'$.

If $f$ is defined over a finite extension $K$ of $\Q_p$, then so is the family~$g$.}

\proof
Replacing $f$ by $f - f(0)$ we assume $c_0 \equiv 0$.
The hypothesis that for every $t \in B$ we have $|c_1(t)| = |c_0(0)| = 1$, implies that the function $t \mapsto c_1(t)^{-1}$ is holomorphic, cf.~\cite{FvP}~Lemme~I.2.5.
So replacing $f$ by $c_1^{-1} f$ we assume $c_1 \equiv 1$.

Then the proof is similar to the proof of part~2 of Lemma~\ref{injective holomorphic functions}.
$\hfill \square$

\subsection{}\label{one parameter inverse branches}
The following lemma is an easy consequence of the previous one.

\

\lemma
{\it
Let $f$ be a one parameter family of holomorphic functions defined on a bidisk $\underline{D}$ and let $(t_0, z_0) \in \underline{D}$ be such that $f_{t_0}'(z_0) \neq 0$.
Then for every small ball $D$ containing $z_0$ there are balls $B$ and $D'$ of $\C_p$, such that $t_0 \in B$ and such that for every $t \in B$ the function $f_t$ induces a bijection between $D$ and $D'$.
Moreover, the inverse functions $(f_t|_D)^{-1}$ form a one parameter family of holomorphic functions $g$ defined on $B \times D'$.

If $f$ is defined over a finite extension $K$ of $\Q_p$ and $t_0, z_0 \in K$, then so is the family $g$.}

\subsection{Lemma.}\label{near critical inverse branches}
{\it
Let $\delta \in |\C_p^*|$ satisfying $\delta \le |p|^2$ and consider a one parameter family of holomorphic functions of the form
$$
f_t(z) = c_d(t) z^d + c_{d + 1}(t) z^{d + 1} + \ldots,
\mbox{ with } c_d(0) \neq 0,
$$
defined for $t$ and $z$ close to $0$.
Then for every $t$ and every $\xi$ close to $0$, the function $f_t$ maps the ball
$$
\{z \in \C_p \mid |z - \xi| \le \delta |\xi| \}
$$
bijectively onto the ball
$$
\{ w \in \C_p \mid |w - c_d(0) \xi^d| \le \delta |d \xi^d| \cdot |c_d(0)| \}.
$$
Moreover, for a fixed $\xi$ the corresponding inverse maps form a one parameter family of holomorphic functions.

If $f$ is defined over a finite extension $K$ of $\Q_p$ and if $\xi \in K$, then this family of inverse maps is defined over $K$.}

\proof
Let $B$ be a ball containing $0$ such that for every $t \in B$ we have $|c_d(t) - c_d(0)| \le \delta |d| \cdot |c_d(0)|$.
Then $|c_d(t)| = |c_d(0)|$ and
$$
\{ w \in \C_p \mid |w - c_d(t) \xi^d| \le \delta |d \xi^d| \cdot |c_d(0)| \}
$$
$$
=
\{ w \in \C_p \mid |w - c_d(0) \xi^d| \le \delta |d \xi^d| \cdot |c_d(0)| \}.
$$
So the function $t \mapsto c_d(t)^{-1}$ is holomorphic on $B$ and, replacing $f$ by $c_d^{-1} f$, we may assume that $c_d(t) \equiv 1$.

Let $\eta \in \C_p$ be of norm equal to $\delta$ and let $r' > 0$ be small enough so that for $j > d$ we have $\sup_B |c_j(t)| (r')^{j - d} < \delta |pd|$.
Set $D = \{ |z| \le r' \}$ and fix $\xi \in D$.
Consider the one parameter family of holomorphic functions $h$, defined on $B \times \cO_p$ by
$$
h_t(z) = \xi^{-d} f_t(\xi (1 + \eta z))  - 1 = b_0(t) + b_1(t) z + \ldots
$$
Observe that, as $|\eta| = \delta \le |p|^2$, the coefficients of the polynomial $(1 + \eta z)^d - 1 - \eta d z$ have norm at most $\delta |pd|$.
Therefore, for every $t \in B$ we have
$$
|b_1(t) - \eta d| \le \delta |pd|
\ \  \mbox{ and for $j \neq 1$ } \ \
|b_j(t)| \le \delta |pd|.
$$
So $h$ satisfies the hypothesis of Lemma~\ref{one parameter injectivity} with $r' = 1$.
The assertions follow easily from the conclusions of this lemma.
$\hfill \square$

\section{Dynamics of rational maps.}
We consider a rational map $R$ with coefficients in $\C_p$ as a dynamical system acting on $\pp$.
We refer the reader to~\cite{Hs} and~\cite{these} for background on dynamics of rational maps.
For a positive integer $n$ we denote by $R^n$ the $n$-th iterate of $R$ and $R^0$ will denote the identity map.

Rational maps are locally holomorphic functions, in the sense that they are given by a convergent power series on a neighborhood of each point of $\pp$, in suitable coordinates.
This follows from the fact every rational map has a power series expansion on a neighborhood of every point of $\C_p$ that is not a pole.

\subsection{Periodic points.}\label{periodic points}
A point $z_0 \in \pp$ is {\it fixed} by a rational map $R \in \C_p(z)$, if $R(z_0) = z_0$.
The point $z_0$ is {\it periodic} by $R$ if for some integer $n \ge 1$ the point $z_0$ is fixed by $R^n$.
When $n$ is the least integer with this property, we say that $n$ is the {\it period} of $z_0$.
In that case the derivative $\lambda = (R^n)'(z_0)$, in a coordinate such that $z_0 \neq \infty$, is invariant under coordinate changes and its is called the {\it multiplier} of $z_0$.
We say that $z_0$ is a {\it repelling, indifferent} or {\it attracting periodic point}, according to $|\lambda| > 1$, $|\lambda| = 1$ or $|\lambda| < 1$, respectively.

\

\lemma
{\it Suppose that $a \in \C_p$ is a repelling fixed point of a rational map $R$ and set $\lambda = R'(a)$.
Denote by $g$ the local inverse of $R$ at $a$.
Then there is a ball $D$ containing $a$, such that $g(D) \subset D$ and such that the sequence of functions $\{ \lambda^i(g^i - a) \}_{i \ge 1}$ converges uniformly on $D$ to a holomorphic function $\zeta$.
Moreover $\zeta$ is locally injective at $a$.}

\

By definition the function $\zeta$ satisfies the functional equation $\zeta \circ g = \lambda^{-1} \zeta$, so that $\zeta$ conjugates $g$ to the map $z \mapsto \lambda^{-1} z$ on a neighborhood of $a$.

\

\proof
After an affine coordinate change assume that $a = 0$ and that $g$ is defined on $\cO_p$.
Then $g$ is of the form
$$
g(z) = \lambda^{-1} z(1 + c_1 z + c_2 z^2 + \ldots ),
$$
with $\lim_{i \to \infty} |c_i| = 0$.
Set $C = \sup_{i \ge 1} |c_i|$ and note that for every $z \in \cO_p$ we have $|\lambda g(z) - z| \le C|z|^2$.
So if we set $D = \{ |z| < C^{-1} \}$, then for every $z \in D$ we have $|g(z)| = |\lambda|^{-1}|z| < |z|$.
So $g(D) \subset D$ and it follows by induction that for every $i \ge 1$ we have $|g^i(z)| = |\lambda|^{-i} |z|$.

Set $\zeta_i = \lambda^i g^i$, so that for every $z \in D$ we have $|\zeta_i(z)| = |z|$ and
$$
|\zeta_{i + 1}(z) - \zeta_i(z)| \le C|\lambda|^{i}|g^i(z)|^2 = (C|z|^2)|\lambda|^{-i}.
\hfill \square
$$

\subsection{Misiurewicz maps and homoclinic orbits.}\label{Misiurewicz and homoclinic}
A {\it rational map with marked critical point} is a pair $(R, c)$ consisting of a rational map $R$ and a critical point $c$ of $R$.
When $R$ has coefficients in a finite extension $K$ of $\Q_p$ and $c \in K$, we say that $(R, c)$ {\it is defined over} $K$.

A rational map with marked critical point $(R, c)$ is {\it Misiurewicz}, if the critical point $c$ is mapped to a repelling periodic point of $R$ under iteration.
In this case, replacing $R$ by an iterate if necessary, we assume that the critical point $c$ is mapped to a repelling fixed point of $R$ under iteration.
Fix such $(R, c)$ and let $\ell \ge 1$ be the minimal integer for which $a = R^\ell(c)$ is a repelling fixed point.
A {\it bi-infinite orbit} is a sequence $\{ z_i \}_{i \in \Z}$ such that for every $i \in \Z$ we have $R(z_i) = z_{i + 1}$.
A {\it homoclinic orbit} is a bi-infinite orbit $\{ z_i \}_{i \in \Z}$ for which
$$
\lim_{|i| \to \infty} z_i = a.
$$
As $a$ is a repelling fixed point, this implies that for some $m \in \Z$ we have $z_m = a$.
Moreover, if we denote by $g$ the local inverse of $R$ at $a$, then for some $n \in \Z$ the point $z_n$ belongs to the domain of definition of $g$ and for $j \ge 0$ we have $z_{n - j} = g^j(z_n)$.

\subsection{Ramification and critical homoclinic orbits.}\label{critical homoclinic}
A point $z$ is an {\it unramified preimage} of a point $w$, if for some positive integer $n \ge 1$ we have $R^n(z) = w$ and if the local degree of $R^n$ at $z$ is equal to~1.
A point is {\it unramified} if it has infinitely many unramified preimages.

Let $(R, c), a, \ell, \ldots$ be as in \S\ref{Misiurewicz and homoclinic} above.
We say that a homoclinic orbit $\{ z_i \}_{i \in \Z}$ is {\it critical}, if $z_0 = c$ and if for every $j < 0$ the point $z_j$ is an unramified preimage of $c$.
In this case we have $z_j = a$ for every $j \ge \ell$.

\

\proposition
{\it Let $(R, c)$ be a Misiurewicz rational map with marked critical point, defined over a finite extension $K$ of $\Q_p$.
If the critical point $c$ is unramified, then there is a critical homoclinic orbit contained in a finite extension of $K$.}
\proof
Let $\ell \ge 1$ be the least integer such that $R^\ell(c)$ is a repelling periodic point of $R$.
Replacing $R$ by an iterate if necessary we assume that $a = R^\ell(c)$ is a fixed point of $R$.
Denote $g$ a local inverse of $R$ at $a$.

As $c$ is not periodic under $R$, the sets $R^{-n}(c)$ for $n \ge 1$ are pairwise disjoint.
It follows that for large $n \ge 1$ and every $w' \in R^{-n}(c)$, all preimages of $w'$ are unramified.
As $c$ is unramified, there is such $w'$ that is in addition an unramified preimage of $c$.

By the eventually onto property, there is a point $w$ in the domain of definition of $g$ and an integer $m$ such that $R^m(w) = w'$~\cite{Hs} (note that $w'$ is not an exceptional point, as $R^n(w') = a$ is a repelling fixed point.)
By the choice of $w'$, the point $w$ is an unramified preimage of $c$.
So the backward orbit $\{ g^i(w) \}_{i \ge 0}$ and the forward orbit of $w$ form a critical homoclinic orbit.

As $(R, c)$ is defined over $K$ and $w \in R^{-(m + n)}(c)$, it follows that $w$ and its forward orbit belong to a finite extension $L$ of $K$.
As $a = R^\ell(c) \in K$ it follows that $g$ is defined over $K$, and hence over $L$.
So the backward orbit $\{ g^i(w) \}_{i \ge 0}$ is contained $L$.
$\hfill \square$

\subsection{One parameter families of rational maps.}
A {\it one parameter family of rational maps} defined over a ball $B$ of $\C_p$, is a function
$$
R : B \times \pp \to \pp
$$
such that for each parameter $t \in B$ we obtain a rational map $R_t : \pp \to \pp$ {\it whose degree is independent of} $t$ and whose coefficients are holomorphic functions defined on $B$.
We say that a one parameter family of rational maps $R$ {\it is defined over} a finite extension $K$ of $\Q_p$, if the coefficients of $R_t$ are defined over $K$, as holomorphic functions.

\

The following are natural operations to obtain new families from a given one.
\begin{enumerate}
\item[1.]{\it Coordinate change.}
Replace $R_t$ by $\varphi_t \circ R_t \circ \varphi_t^{-1}$, where $\varphi : B \times \pp \to \pp$ is a one parameter family of M\"oebious transformations (rational maps of degree~1).
\item[2.]{\it Iteration.}
For a given $n \ge 1$, replace $R_t$ by $R_t^n$.
\item[3.] {\it Base change.}
Replace $R_t$ by $R_{\Phi(t)}$, where $\Phi : B \to B$ is a non-constant holomorphic function.
\end{enumerate}
Note that these operations preserve families defined over a finite extension $K$ of $\Q_p$, when $\varphi$ and $\Phi$ are defined over $K$.

\subsection*{Marked critical points.}
A {\it one parameter family of rational maps with marked critical point} $(R, c)$ is a one parameter family of rational maps $R : B \times \pp \to \pp$, together with a holomorphic function $c : B \to \pp$ such that $c_t$ is a critical point of $R_t$, for every $t \in B$.
We say that $(R, c)$ is defined over a finite extension $K$ of $\Q_p$, when both $R$ and $c$ are defined over $K$.

\subsection{}
The following lemma shows that a one parameter family of rational maps is locally a one parameter family of holomorphic functions.
So the considerations of Section~\ref{holomorphic functions} apply to families of rational maps.

\

\lemma
{\it Let $R$ be a one parameter family of rational maps defined over a ball $B$.
Then for every $(t_0, z_0) \in B \times \C_p$ such that $z_0$ is not a pole of $R_{t_0}$, the restriction of $R$ to a sufficiently small bidisk containing $(t_0, z_0)$ is a one parameter family of holomorphic functions.}
\proof
After a coordinate change assume $z_0 = R_0(z_0) = 0$.
As by assumption the degree of $R_t$ does not depend on $t$, we can write $R_t$ as the quotient of polynomials $R_t = P_t / Q_t$, in such a way that the coefficients of $P_t$ and of $Q_t$ are holomorphic functions of $t$ and such that for each $t \in B$ the polynomials $P_t$ and $Q_t$ do not have common factors.
Write $Q_t(z) = c_0(t) + c_1(t) z + \ldots + c_d(t) z^d$.
As by assumption $P_0$ and $Q_0$ do not have common factors and $R_0(0) = 0$, we have $P_0(0) = 0$ and $c_0(0) \neq 0$.
Shrinking $B$ if necessary, assume that the function $t \mapsto c_0(t)^{-1}$ is well defined and holomorphic on $B$.
Then, replacing $P$ by $c_0 P$ and $Q$ by $c_0^{-1}Q$ if necessary, we assume $c_0 \equiv 1$.

Let $h$ be the holomorphic function $1 - Q$ and let $\underline{D}$ be a sufficiently small polydisk so that $\sup_{\underline{D}} |h| < 1$.
Then, on $\underline{D}$ we have that $R = P(1 + h + h^2 + \ldots)$ is a holomorphic function.
$\hfill \square$

\subsection{Holomorphic dependence.}\label{holomorphic dependence}
Let $R$ be a one parameter family of holomorphic functions defined over a ball $B$.
Let $t_0 \in B$ and $a_0 \in \C_p$ be such that $a_0$ is a repelling fixed point of $R_{t_0}$.
Then there is a holomorphic function $a$, defined on a ball $B'$ containing $t_0$, such that for every $t \in B'$ the point $a_t$ is a repelling fixed point of $R_t$.
The function $\lambda_t = R_t'(a_t)$ is holomorphic.

Let $g$ be a one parameter family of holomorphic functions defined on a bidisk containing $(t_0, a_0)$, such that for every $t$ the holomorphic function $g_t$ is a local inverse of $R_t$ at $a_t$ (Lemma~\ref{one parameter inverse branches}).
Then we can show in a similar way as in Lemma~\ref{periodic points}, that the sequence of holomorphic functions $\{ \lambda^i (g^i - a) \}_{i \ge 0}$ converges uniformly to a one parameter family of holomorphic functions $\zeta$.

When $R$ is defined over a finite extension $K$ of $\Q_p$ and $t_0, a_0 \in K$, then $a$, $\lambda$, $g$ and $\zeta$ are all defined over $K$.

\section{Misiurewicz bifurcations.}\label{Misiurewicz bifurcations}
Let $(R, c)$ be a one parameter family of rational maps with marked critical point.
We say that $(R, c)$ has a {\it Misiurewicz bifurcation at $t = t_0$}, if the following conditions are satisfied.
\begin{enumerate}
\item[(M1)]
There is an integer $\ell \ge 1$ such that $R_{t_0}^\ell(c_{t_0})$ is a repelling fixed point of
$R_{t_0}$.
\item[(M2)]
$c_{t_0}$ is unramified for $R_{t_0}$.
\item[(M3)]
$R_t^\ell(c_t)$ is not a fixed point of $R_t$, for some parameter $t$.
\item[(M4)]
$\deg_{R_t^\ell}(c_t) = d$ for every $t$ close to $t_0$.
\end{enumerate}

For $t$ close to $t_0$, denote by $a_t$ the fixed point of $R_t$ that is the continuation of the repelling fixed point $R_{t_0}^\ell(c_{t_0})$ of $R_{t_0}$.
We say that the Misiurewicz bifurcation of $(R, c)$ at $t = t_0$ is {\it transversal}, if the function $t \mapsto R_t^{\ell}(c_t) - a_t$ has a non-zero derivative at $t = t_0$, in a coordinate such that $a_{t_0} \neq \infty$.

\subsection{Admissible homoclinic orbits.}
Let $(R, c)$ be a one parameter family of rational maps with marked critical point, having a Misiurewicz bifurcation at $t = 0$.
Denote by $\ell$ the least integer for which $a_0 = R_0^\ell(c_0)$ is a repelling fixed point of $R_0$.
Assume $a_0 \neq \infty$ and set $\lambda_0 = R_0'(a_0)$.

For a homoclinic orbit $\{ z_i \}_{i \in \Z}$ the limit
$$
\zeta = \lim_{i \to \infty} \lambda_0^i(z_{- i} - a_0)
$$
exists and it is non-zero, cf. Lemma~\ref{periodic points}.
We call $\zeta$ {\it the asymptotic position} of the homoclinic orbit $\{ z_i \}_{i \in \Z}$.
Note that a coordinate change $\varphi$ affects $\zeta$ by a factor of $\varphi_0'(a_0)$.

Set $d = \deg_{R_0^\ell}(c_0)$ and let $\rho \in \C_p^*$ be defined by
$$
R_0^\ell (z) = a_0 + \rho (z - c_0)^d + \ldots
$$
For small $t$ denote by $a_t$ the fixed point of $R_t$ that is the continuation of $a_0$.
As $(R, c)$ has a Misiurewicz bifurcation at $t = 0$, it follows that the function $t \mapsto R_t^\ell(c_t) - a_t$ vanishes at $0$ and that it is non-constant.
So, there is a positive integer $d' \ge 1$ and $\rho' \in \C_p^*$ such that
$$
R_t^\ell(c_t) - a_t = \rho' t^{d'} + \ldots
$$

When $(R, c)$ is defined over a finite extension $K$ of $\Q_p$, then $a_0 = R_0^\ell(c_0), \lambda_0, \rho$ and $\rho'$ all belong to $K$.
If moreover the homoclinic orbit $\{ z_i \}_{i \in \Z}$ is contained in $K$, then $\zeta$ belongs to $K^*$.

\

\definition
{\it
Let $(R, c), \lambda_0, \ldots$ be as above and suppose that $(R, c)$ is defined over a finite extension $K$ of $\Q_p$.
A $K$-{\bf admissible homoclinic orbit} is a critical homoclinic orbit $\{ z_j \}_{j \in \Z}$ contained in $K$, for which there is an integer $m_0$ such that
$$
- \lambda_0^{m_0} \zeta / \rho \in K^d
\ \ \mbox{ and } \ \
\lambda_0^{m_0} \zeta / \rho' \in K^{d'}.
$$}

\

\remark \

\begin{enumerate}
\item[1.]
The $K$-admissibility condition is invariant under coordinate changes and base
changes defined over $K$.
In fact, a coordinate change $\varphi$ affects the first element by a
factor of $(\varphi_0'(c_0))^d$ and leaves the second invariant, while a base change
$\Phi$ leaves the first invariant and affects the second by a factor of
$(\Phi'(0))^{d'}$.
\item[2.]
Every critical homoclinic orbit can be made admissible by a finite extension of the base field.
\item[3.]
When the Misiurewicz bifurcation $(R, c)$ is transversal, that is when $d' = 1$, the second condition of $K$-admissibility is automatically
satisfied.
\end{enumerate}

\subsection{Misiurewicz cascades.}\label{Misiurewicz cascade} \
In this section we reduce the proof of Theorem~A to the following proposition.
The proof of this proposition is deferred to the next section.

\

{\par\noindent {\bf Proposition~A.}}
{\it Let $(R, c)$ be a one parameter family of rational maps with marked critical point, having a Misiurewicz bifurcation at $t = 0$.
Suppose that $(R, c)$ is defined over a finite extension $K$ of $\Q_p$ and that there exists a $K$-admissible homoclinic orbit.

Then there is a sequence of parameters $\{ t_r \}$ in $K$ converging to $0$, such that for every $r$, $(R, c)$ has a transversal Misiurewicz bifurcation at $t = t_r$ that admits a $K$-admissible homoclinic orbit.
Moreover, if we denote by $\ell_r \ge 1$ the least integer such that $R^{\ell_r}(c_{t_r})$ is a repelling fixed point of $R_{t_r}$, then}
$$
\lim_{r \to \infty} \min_{1 \le i \le \ell_r} \Delta(c_{t_r}, R^i_{t_r}(c_{t_r})) = 0.
$$

\

The following corollary is immediate consequence of the proposition.

\

\noindent
{\bf Corollary 1.} (Transversal Misiurewicz Bifurcations){\bf .}
{\it Let $(R, c)$ be a one parameter family of rational maps defined over a finite extension $K$ of $\Q_p$.
Among the parameters in $K$ for which $(R, c)$ has a Misiurewicz bifurcation admitting a $K$-admissible homoclinic orbit, those having a transversal Misiurewicz bifurcation are dense.}

\

\subsection*{Proof of Theorem~A}
Let $(R, c)$ and $K$ be as in the proposition.
Then there is a critical homoclinic orbit contained in a finite extension $L$ of $K$ (Proposition~\ref{critical homoclinic}).
Taking $L$ larger if necessary, this critical homoclinic orbit is $L$-admissible.
Then Theorem~A is an immediate consequence of the following corollary of the proposition.

\

\noindent
{\bf Corollary 2.}
{\it Let $(R, c)$ be a Misiurewicz bifurcation at $t = 0$, defined over $K$.
If there exists a $K$-admissible homoclinic orbit, then there are arbitrarily small parameters $t$ in $K$ for which the critical point $c_t$ of $R_t$ is recurrent, but not periodic.}
\proof
Let $m_0$ be the least positive integer such that $a_0 = R_0^{m_0}(c_0)$ is a repelling fixed point of $R_0$.
For small $t$ denote by $a_t$ the repelling fixed point of $R_t$ that is the continuation of $a_0$.

Set $s^0 = 0$.
From the proposition we can define by induction a sequence of parameters $\{ s_n \}_{n \ge 1}$ in $K$, converging to some parameter $t_0 \in K$, such that for every $n \ge 1$ the following properties hold.
\begin{enumerate}
\item[1.]
 $(R, c)$ has a Misiurewicz bifurcation at $t = s_n$ that admits a $K$-admissible homoclinic orbit.
\item[2.]
If we denote $m_n$ the least integer such that $R_{s_n}^{m_n}(c_{s_n}) = a_{s_n}$, then
$$
\min_{1 \le i \le m_n} \Delta(R_{s_n}^i(c_{s_n}), c_{s_n}) \le 2^{-n}.
$$
\item[3.]
$s_n$ is close enough to $s_{n - 1}$ so that for every $1 \le i \le m_{n - 1}$ we have
$$
\Delta(R_{s_n}^i(c_{s_n}), R_{s_{n - 1}}^i(c_{s_{n - 1}})) \le 2^{-n}.
$$
\end{enumerate}

Property~2 implies that for every $n \ge 1$ we have
$$
\min_{1 \le i \le m_n} \Delta(R_{t_0}^i(c_{t_0}), c_{t_0}) \le 2^{-n},
$$
so that the critical point $c_{t_0}$ of $R_{t_0}$ is recurrent.
Property~3 implies that for every $n \ge 1$ we have
$$
\Delta(R_{t_0}^{\ell_n}(c_{t_0}), a_{t_0}) \le 2^{-n},
$$
so that the forward orbit of $c_{t_0}$ under $R_{t_0}$ accumulates on $a_{t_0}$.
As $a_{t_0}$ is a repelling fixed point of $R_{t_0}$, it follows that the critical point $c_{t_0}$ is not periodic.
$\hfill \square$

\subsection{Recurrent critical points in $\Q_p$.}
Given a prime number $p$ and an integer $d > 1$, consider the one parameter family of polynomials
$$
P_t(z) = - p^{-d} z(z - 1)^d + t.
$$
As for every parameter $t$ the local degree of $P_t$ at $1$ is equal to $d > 1$, it follows that $(P, 1)$ is a one parameter family of polynomials with marked critical point, that is defined over $\Q_p$.
We will verify that this family has a Misiurewicz bifurcation at $t = 0$ and that there is a $\Q_p$-admissible homoclinic orbit.
Corollary~2 then implies that there are arbitrarily small parameters $t$ in $\Q_p$ for which the critical point $1$ of $P_t$ is recurrent, but not periodic.

Let us now verify that $(P, 1)$ has a Misiurewicz bifurcation at $t = 0$.
\begin{enumerate}
\item[(M1)]
$P_0(1) = 0$ is a fixed point of $P_0$ whose multiplier $\lambda_0 = -(-1)^d p^{-d}$ satisfies $|\lambda_0| = |p|^{-d} > 1$.
So property (M1) is satisfied with $\ell = 1$.
\item[(M2)]
Note that $P_0$ is injective on $\fm_p = \{ |z| < 1 \}$ and $P_0(\fm_p) = \{ |z| < |p|^{-d} \}$.
So, if we denote by $g$ the inverse of $P_0|_{\fm_p}$, then $\{ g^i(1) \}_{i \ge 1}$ is an infinite sequence of unramified preimages of $1$.
This proves that~1 is unramified for $P_0$.
\item[(M3)]
As $P_t(0) = t$, for $t \neq 0$ the point $P_t(1) = 0$ is not fixed by $P_t$.
\item[(M4)]
As $\ell = 1$, for every $t$ we have $\deg_{P_t^\ell}(1) = d$.
\end{enumerate}
Next observe that this Misiurewicz bifurcation is transversal.
In fact, for small $t$ denote by $a_t$ the repelling fixed point of $P_t$ that is the continuation of $a_0 = 0$.
From the equation $a_t = - p^{-d}a_t(a_t - 1)^d + t$ we deduce that $|a_t| = |p^d t|$ and that the derivative of $t \mapsto P_t(1) - a_t$ at $t = 0$ is different from zero.

We will show now that the homoclinic orbit formed by the backward orbit $\{ g^i(1) \}_{i \ge 1}$ and the forward orbit of~1 under $P_0$ is $\Q_p$-admissible.
As our Misiurewicz bifurcation is transversal, we only need to verify the first condition of $\Q_p$-admissibility.

Since $P_0$ and $g$ are defined over $\Q_p$, this homoclinic orbit is contained in $\Q_p$.
Denote by
$$
\zeta = \lim_{i \to \infty} \lambda_0^i g^i(1),
$$
its asymptotic position.
As
$$
P_0(z) = -p^{-d}(z - 1)^d - p^{-d} (z - 1)^{d + 1},
$$
we have $\rho = -p^{-d}$.
We will show that $\zeta \in (\Q_p)^d$, which implies that $- \zeta / \rho = p^d \zeta \in (\Q_p)^d$ and that the first condition of $\Q_p$-admissibility is satisfied with $m_0 = 0$.
Set $\zeta_0 = 1$ and for $i \ge 1$ set $\zeta_i = \lambda_0^i g^i(1)$, so that $\zeta = \lim_{i \to \infty} \zeta_i$.
From the identity $P_0(g(z)) = z$ we have $\lambda_0 g(z)/z = (1 - g(z))^{-d} \in (\Q_p^*)^d$, whenever $z \in \Q_p$ satisfies $|z| \in (0, |p|^{-d})$.
It follows that for $i \ge 0$ we have $\zeta_{i + 1}/\zeta_i \in (\Q_p)^d$ and that $\zeta_{i + 1} / \zeta_i \to 1$ as $i \to \infty$.
Therefore
$$
\zeta = \prod_{i \ge 0} \left( \zeta_{i + 1}/\zeta_i \right) \in \Q_p^d.
$$

\section{Proof of Proposition~A.}
Denote by $\ell \ge 1$ the least integer for which $a_0 = R_0^\ell(c_0)$ is a repelling fixed point of $R_0$.
For small $t$ denote by $a_t$ the repelling fixed point of $R_t$ that is the continuation of $a_0$.
Since every non-constant rational map has at least one non-repelling fixed point~\cite{Be1}, after a coordinate change we assume that $\infty$ is a non-repelling fixed point of $R_0$.
So for every small~$t$ we have $a_t \neq \infty$ and $c_t \neq \infty$.

Denote by $\lambda_t = (R_t^\ell)'(a_t)$ the multiplier of $a_t$ and denote by $g_t$ the local inverse of $R_t$ at $a_t$.
We assume that the $g_t$ are all defined on a ball of $\C_p$ containing $a_0$, that does not depend on $t$ (Section~\ref{one parameter inverse branches}).

\noindent
{\bf A.}
By hypothesis there is a $K$-admissible homoclinic orbit $\{ z_i \}_{i \in \Z}$ of $(R_0, c_0)$; denote by $\zeta$ its asymptotic position.
Let $\rho, \rho' \in K^*$, $d = \deg_{R_t^\ell}(c_t) > 1$ and $d' \ge 1$ be as in the admissibility condition.
Then there are $m_0 \in \Z$ and $\xi, \xi' \in K^*$ such that
$$
\rho \xi^d = - \lambda_0^{m_0} \zeta
\ \mbox{ and } \
\rho' (\xi')^{d'} = \lambda_0^{m_0} \zeta.
$$

For a given integer $r$ set $m = r dd' - m_0$,
$$
B' = \{ z \in \C_p \mid |z - \lambda_0^{-rd} \xi'| \le |p^2d| \cdot |\lambda_0^{-rd} \xi'| \}
$$
and for $t \in B'$,
$$
B_t = \{ z \in \C_p \mid |z - \lambda_0^{-rd'} \xi - c_t| \le |p^2d'| \cdot |\lambda_0^{-rd'} \xi| \}.
$$
Lemma~\ref{near critical inverse branches} applied to $\delta = |p^2d|$ and to $t \mapsto R_t^\ell(c_t) - a_t$, implies that when $r$ is large enough the function $t \mapsto R_t^\ell(c_t) - a_t$ maps $B'$ bijectively onto
$$
\{ w \in \C_p \mid |w - \lambda_0^{-m} \zeta| \le |p^2dd'| \cdot |\lambda_0^{- m} \zeta| \}.
$$
Moreover, Lemma~\ref{near critical inverse branches} applied to $\delta = |p^2d'|$ and to the family $z \mapsto R_t^\ell(z + c_t) - R_t^\ell(c_t)$, implies that when $r$ is large enough for every $t \in B'$ the function $R_t^\ell$ maps $B_t$ bijectively onto
$$
\{ w \in \C_p \mid |w - R_t^\ell(c_t) + \lambda_0^{- m} \zeta| \le |p^2dd'| \cdot |\lambda_0^{- m} \zeta| \}
$$
$$
=
\{ w \in \C_p \mid |w - a_t| \le |p^2dd'| \cdot |\lambda_0^{- m} \zeta| \}.
$$

\noindent
{\bf B.}
As the homoclinic orbit $\{ z_i \}_{i \in \Z}$ is critical, we have $z_0 = c_0$ and for every $j > 0$ the point $z_{-j}$ is an unramified preimage of $c_0$.
Moreover, for some $n > 0$ the point $z_{- n}$ belongs to the domain of $g_0$ and for $j \ge 0$ we have $z_{- (n + j)} = g_0^j(z_{- n})$.
Since $\infty$ is a non-repelling fixed point of $R_0$, the homoclinic orbit $\{ z_i \}_{i \in \Z}$ is contained in $\C_p$.

By Lemma~\ref{one parameter inverse branches} it follows that, for every $0 \le i \le n$ and every sufficiently small ball $\cD$ containing $c_0$, there is a ball $\cB$ containing $t_0$ and a one parameter family of holomorphic functions $h^i$ defined on $\cB \times \cD$, such that for every $t \in \cB$ the holomorphic function $h_t^i$ is a local inverse of $R_t$ such that $h^i_0(c_0) = z_{- i}$.
Reducing $\cD$ if necessary, we assume that for every $0 \le i \le n$ and $t \in \cB$ the set $h^i_t(\cD)$ is contained in the domain of $g_t$.
Then for $j > 0$ the function $h_t^{n + j} = g_0^j \circ h_t^n$ is a holomorphic function defined on $\cB \times \cD$.
So for every $j > 1$ the function $h^j$ is a holomorphic function defined on $\cD \times \cB$ and for every $t \in \cB$ the holomorphic function $h^j_t$ is a local inverse of $R_t^j$ such that $h_0^j(c_0) = z_{ -j}$.

It follows that for every small $t$ the critical point $c_t$ of $R_t$ is unramified, as $\{ h_t^i(c_t) \}_{i \ge 0}$ is an infinite sequence of unramified preimages of $c_t$.

\noindent
{\bf C.}
The proof is organized as follows.
In part~D we define for every $t \in B' = B'(r)$, an orbit $\{ z_i(t) \}_{\ell \le i \le \ell_r}$ ending at $z_{\ell_r}(t) = a_t$.
In part~E we show that when $r$ is sufficiently large there is a parameter $t_r \in B'$ in $K$ such that $R_{t_r}^\ell(c_{t_r}) = z_\ell(t_r)$, so that
$$
R_{t_r}^{\ell_r}(c_{t_r}) = z_{\ell_r}(t_r) = a_{t_r}
$$
and so that $(R, c)$ has a Misiurewicz bifurcation at $t = t_r$.
In part~F we show this Misiurewicz bifurcation is transversal.
In part~G we show that the critical homoclinic orbit (for $t = t_r$) determined by the infinite backward orbit $\{ h_{t_r}^i(c_{t_r}) \}_{i \ge 0}$, is $K$-admissible.
The last assertion of the proposition is easy to verify, this is done in part~E.

\noindent
{\bf D.}
Let $\hr > r$ be such that $|\lambda_0|^{-(\hr - r)dd'} < |p^2dd'|$ and set $\hm = \hr dd' - m_0$.
Given an positive integer $k$ set $\ell_r = (k - 1)(\hm + \ell) + m + 2 \ell$.

We assume $r$ large enough so that for every small parameter $t$ we have $B_t \subset \cD$.
For $t \in B'$ define $w_t^1, \ldots, w_t^k \in B_t$ inductively as follows.
Let $w_t^1$ be the unique inverse image of $a_t$ by $R_t^\ell$ in $B_t$.
Let $1 < j \le k$ be given and assume that $w^{j - 1}_t$ is already defined.
When $r$ is sufficiently large we have
$$
|h_t^{\hm}(w^{j - 1}_t) - a_t|
=
|\lambda_0|^{-\hm} |\zeta|
<
|p^2dd'| \cdot |\lambda_0|^{- m}|\zeta|.
$$
So $h_t^{\hm}(w_t^{j - 1}) \in R_t^\ell(B_t)$ has a unique preimage by $R_t^\ell$ in $B_t$.
We let $w_t^j$ be this preimage.

Set $z_\ell(t) = h_t^m(w_t^k)$ and for $\ell < i \le \ell_r$ set $z_i(t) = R_t^{i - \ell}(z_1(t))$.
Then, for $0 \le j \le k - 1$ we have
$$
z_{j(\hm + \ell) + m + \ell}(t) = w_t^{k - j},
$$
and therefore $z_{\ell_r}(t) = a_t$.

It follows from Lemma~\ref{near critical inverse branches} and from the fact that the functions $h^{\nu}$ are holomorphic and defined over $K$, that the functions $w^j$ and $z_i$ are holomorphic functions defined over $K$.

\noindent
{\bf E.}
Recall that $m = rdd' - m_0$ depends on $r$.
Assume $r$ large enough so that for every $w$ close to $c_t$ and every small~$t$ we have
$$
\left| \zeta - \lambda_0^m(h_t^m(w) - a_t) \right|
\le
|p^3 dd'| \cdot |\zeta|
$$
(Section~\ref{holomorphic dependence}).
Since for every $t \in B' \subset \{ |z| \le |\lambda_0|^{- rd}|\xi'| \}$ we have $B_t \subset \{ |z| \le |\lambda_0|^{- rd'}|\xi| \}$, it follows that for large $r$ the function
$$
g(t)
=
\lambda_0^{- m} \zeta - (h_t^m(w_t^k) - a_t),
$$
satisfies $|g| \le |p^3 dd'| \cdot |\lambda_0|^{- m}|\zeta|$ on $B'$.

On the other hand, by part~A the function $f(t) = R_t^\ell(c_t) - a_t - \lambda_0^{- m} \zeta$ maps the ball $B'$ univalently onto the ball
$$
\{ w \in \C_p \mid |w| \le |pdd'| \cdot |\lambda_0|^{- m} |\zeta| \}.
$$
So $\sup_{B'} |g| < \diam(f(B'))$ and Lemma~\ref{Rouche} implies then that there is parameter $t_r \in B'$ such that $f(t_r) = g(t_r)$, so that $R_{t_r}^\ell(c_{t_r}) = h_{t_r}^m(w_{t_r}^k) = z_\ell(t_r)$.
Since $(R, c)$ and the functions $a$ and $h^m(w^k)$ are defined over $K$ and $\lambda_0^{-m} \zeta \in K$, it follows that $f$ and $g$ are defined over $K$ and $t_r \in K$ (cf. Lemma~\ref{Rouche}).

Let us verify now that $(R, c)$ has a Misiurewicz bifurcation at $t = t_r$.
By definition of $t_r$ we have $R_{t_r}^{\ell_r}(c_{t_r}) = z_{\ell_r}(t_r) = a_{t_r}$.
As remarked in part~B, for every $t \in B'$ the critical point $c_t$ is unramified for $R_t$.
By construction, for every $t \in B'$ none of the points $z_i(t)$, for $\ell \le i \le \ell_r$, is a critical point of $R_t$, so $\deg_{R^{\ell_r}}(c_t) = \deg_{R^{\ell}} (c_t) = d$.
It remains to verify condition (M3).
Suppose by contradiction that for every $t \in B'$ we have $R_t^{\ell_r}(c_t) = a_t$.
This implies that the functions $R_t^\ell(c_t)$ and $z_\ell(t)$, and hence $f$ and $g$, coincide on $B'$.
But we saw above that $g(B')$ is strictly smaller than $f(B')$.

To verify the last assertion of the proposition, observe that $R_{t_r}^{m + \ell}(c_{t_r}) = w_{t_r}^k \in B_{t_r}$, so that
$$
\Delta(c_{t_r}, R_{t_r}^{m + \ell}(c_{t_r})) \le |\lambda_0|^{- rd}|\xi| \to 0
\mbox{ as }
r \to \infty.
$$

\noindent
{\bf F.}
We prove now that for large $r$ the Misiurewicz bifurcation $(R, c)$ at $t = t_r$ is transversal.
As $\infty$ is a non-repelling fixed point of $R_0$,  it follows that for small $t$ the forward orbit of $c$ under $R_{t_r}$ does not contain $\infty$.
So $\partial_t R_t$ is well defined at each of these points.

After an affine change of coordinates we assume that $a = a_t$ and $c = c_t$ do not depend on $t$.
So for all $t$ we have $R_t(a) = a$ and $\partial_t R_t (a) = 0$.
Moreover there is a constant $C > 0$ such that for every point $z$ close to $a$ we have $|\partial_t R_t(z)| \le C |z - a|$.
Enlarging $C$ if necessary we assume that for every $w \in \cD$ near $c$ and every positive integer $i$ we have
\begin{equation}\label{transversality 1}
\left| \partial_t (R_t)|_{t = t_r} (h_{t_r}^i(w)) \right| \le C |\lambda_0|^{-i}
\ \mbox{ and } \
|(R_{t_r}^i)'(h_{t_r}^i(w))| \ge C^{-1} |\lambda_0|^i.
\end{equation}

\noindent
$\bullet$
For $0 \le j \le k - 1$ set $N_j = m + \ell + j(\hm + \ell)$, so that $\ell_r = N_{k - 1} + \ell$.
Moreover for every $0 \le j \le k - 1$ we have $R_{t_r}^{N_j}(c_{t_r}) = w_{t_r}^{k - j}$ and for $1 \le j \le k - 1$ we have $R_{t_r}^{N_{j - 1} + \ell}(c_t) = h_{t_r}^{\hm}(w^{k - j})$.

For $0 \le j \le k - 1$ set $D_j = \partial_t \left( R_t^{N_j + \ell}(c_t) \right)|_{t = t_r}$.
Note that $D_{k - 1} = \partial_t (R_t^{\ell_r}(c_t) - a)$, so we need to prove that $D_{k - 1} \neq 0$.

For $0 \le j \le k - 1$ we have
$$
D_j = A_j + B_j + C_j,
$$
where $A_j = (\partial_t R_t^\ell)|_{t = t_r}(w_{t_r}^{k - j})$, and when $j \neq 0$,
$$
B_j = (R_{t_r}^\ell)'(w_{t_r}^{k - j}) \cdot \partial_t (R_t^{\hm})|_{t = t_r}(h_{t_r}^{\hm}(w_{t_r}^{k - j})),
$$
$$
C_j = (R_{t_r}^{\hm + \ell})'(h_{t_r}^{\hm}(w_{t_r}^{k - j})) \cdot \partial_t \left(R_t^{N_{j - 1} + \ell}(c_t) \right)|_{t = t_r}.
$$
When $j = 0$,
$$
B_0 = (R_{t_r}^\ell)'(w_{t_r}^k) \cdot \partial_t (R_t^{m})|_{t = t_r}(h_{t_r}^{m}(w_{t_r}^k)),
$$
$$
C_0 = (R_{t_r}^{m + \ell})'(h_{t_r}^m(w_{t_r}^{k})) \cdot \partial_t \left(R_t^{\ell}(c_t) \right)|_{t = t_r}.
$$

We will show by induction in $j$ that $|D_j| = |C_j|$.
By induction in $j$ this implies that
$$
|D_j|
=
\left| (R_{t_r}^{N_j})'(R_{t_r}^\ell(c_{t_r})) \cdot \partial_t \left(R_t^{\ell}(c_t) \right)|_{t = t_r} \right|.
$$
But this last quantity is non-zero, since the first factor is non-zero by construction and $\left| \partial_t \left(R_t^{\ell}(c_t) \right)|_{t = t_r} \right| = |\rho'd'(t_r)^{d' - 1}| \neq 0$.
So this claim implies that $D_{k - 1} \neq 0$, as wanted.

\noindent
$\bullet$
Observe that for $t$ close to $0$ and $w$ close to $c$ we have
$$
R_t^\ell(w) = a_0(t) + a_d(t)(w - c)^d + \ldots,
$$
with $a_0(t) = a + \rho't^{d'} + \ldots$ and $a_d(0) = \rho$.
Therefore, enlarging the constant $C$ if necessary, we have
$$
\left |\partial_t (R_t^\ell)|_{t = t_r} (w) \right|
\le
C \max\{ |t_r|^{d' - 1}, |w - c|^d \}.
$$
As $|t_r| = c_0' |\lambda_0|^{-m/d'}$ and $|w_{t_r}^{k - j} - c| = c_0|\lambda_0|^{-m/d}$ for constants $c_0, c_0' > 0$ independent of $j$ and $r$, it follows that when $r$ is large enough we have
$$
|A_j| \le C (c_0')^{d' - 1} |\lambda_0|^{-m(1 - 1/d')}.
$$

\noindent
$\bullet$
For large $r$ we have
$$
|(R_{t_r}^\ell)'(w_{t_r}^{k - j})| = |\rho (w_{t_r}^{k - j} - c)^{d - 1}| = c_1 |\lambda_0|^{-m(1 - 1/d)},
$$
where $c_1 = |\rho|c_0^{d - 1}$.

Fix $0 \le j \le k - 1$ and let $n = \hm$ if $j \neq 0$ and $n = m$ if $j = 0$.
Then
$$
\partial_t \left( R_t^{n} \right)|_{t = t_r} (h_{t_r}^n(w_{t_r}^{k - j}))
=
\sum_{1 \le i \le n}
(R_t^{i - 1})'(h_{t_r}^{i - 1}(w_{t_r}^{k - j}))
\cdot
\partial_t (R_t)|_{t = t_r} (h_{t_r}^{i}(w_{t_r}^{k - j})).
$$
The inequalities ($\ref{transversality 1}$) then imply
$$
\left| \partial_t \left( R_t^{n} \right)|_{t = t_r} (h_{t_r}^n(w_{t_r}^{k - j}))  \right|
\le C^2|\lambda_{0}|^{-1} \le C^2.
$$

So $|B_j| \le C^2c_1 |\lambda_0|^{-m(1 - 1/d)}$.

\noindent
$\bullet$
We have
$$
C_0 = (R_{t_r}^m)'(h_{t_r}^m(w^k)) \cdot (R_{t_r}^\ell)'(w^k) \cdot \partial_t (R_t^\ell(c_t))|_{t = t_r},
$$
so $|C_0| \ge C^{-1}|\lambda_0|^m \cdot c_1 |\lambda_0|^{-m(d - 1)/d} \cdot |\rho'd'| |t_r|^{d' - 1} = c_2 |\lambda_0|^{- m (1 - 1/d - 1/d')}$, where $c_2 = C^{-1} c_1 |\rho'd'|(c_0')^{d' - 1}$.

We assume $r$ large enough so that
\begin{equation}\label{transversality 2}
c_2|\lambda_0|^{- m(1 - 1/d - 1/d')} > \max\{ C(c_0')^{d' - 1} |\lambda_0|^{- m(1 - 1/d') }, C^2c_1 |\lambda_0|^{-m(1 - 1/d)} \}.
\end{equation}
This implies $|C_0| > \max \{ |A_0|, |B_0| \}$ and $|D_0| = |C_0|$.

For $1 \le j \le k - 1$ we have
$$
C_j = (R_{t_r}^{\hm})'(h_{t_r}^{\hm}(w_{t_r}^{k - j})) \cdot (R_{t_r}^\ell)'(w_{t_r}^{k - j}) \cdot D_{j - 1},
$$
so for large $r$ we have $|C_j| \ge C^{-1}|\lambda_0|^{\hm} c_1 |\lambda_0|^{-m(1 - 1/d)} |C_{j - 1}| \ge |C_{j - 1}|$.
It follows by induction in $j$ that $|C_j| \ge |C_0|$.
Then inequality ($\ref{transversality 2}$) implies that $|C_j| > \max \{ |A_j|, |B_j| \}$ and $|D_j| = |C_j|$.

\noindent
{\bf G.}
Now we will prove that, if $k = d$ and $r$ is large enough, then the critical homoclinic orbit, for $t = t_r$, determined by the infinite backward orbit $\{ h_{t_r}^i(c_{t_r}) \}_{i \ge 0}$ is $K$-admissible.
Denote by $\zeta_r$ the asymptotic position of this homoclinic orbit.

By part~F the Misiurewicz bifurcation $(R, c)$ at $t = t_r$ is transversal, so we only need to verify the first condition of $K$-admissibility.
We fix the parameter $t = t_r$ and omit all the indexes corresponding to parameters.
We will use several times that $(K^*)^d$ is open in $K^*$, so that any element in $K$ close to~1 belongs to $K^d$.

Define $\rho_r, \trho_r \in K^*$ by
$$
R^{\ell_r}(z) = a + \rho_r(z - c)^d + \ldots
\mbox{ and }
R^\ell(z) = R^\ell(c) + \trho_r(z - c)^d + \ldots
$$
We have $\rho_r = (R^{\ell_r - \ell})'(R^\ell(c)) \cdot \trho_r$.
As $\lambda = \lambda_{t_r} \to \lambda_0$, $\zeta_r \to \zeta$ and $\trho_r \to \rho$ as $r \to \infty$, it is enough to prove that
$$
(R^{\ell_r - \ell})'(R^\ell(c_t)) = \rho_r/\trho_r \in K^d.
$$
By construction $R^\ell(c) = h^m(w^k)$.
Moreover, recall that
$$
\ell_r - \ell = (k - 1)(\hm + \ell) + m + \ell
\ \mbox{ and } \
\hm = \hr dd' - m_0 = m + (\hr - r) dd'.
$$
Therefore,
\begin{eqnarray*}
(R^{\ell_r - \ell})'(R^\ell(c))
& = &
(R^{\ell_r - \ell})'(h^m(w^k)) \\
& = &
(R^{k(\hm + \ell)})'(h^{\hm}(w^k)) \cdot \left( (R^{(\hr - r)dd'})'(h^{\hm}(w^k)) \right)^{-1}.
\end{eqnarray*}
We will verify that each of the two factors of the last term belong to $K^d$, thus completing the proof of the proposition.
Observe that, as $t_r \in K$, we have that $c, a, \lambda, w^j, h^i(w^j), \ldots$ all belong to $K$.

\noindent
$\bullet$
For the second factor, observe that when $r$, and hence $m$, is large enough, for every $m < i \le \hm$ the point $h^i(w^k)$ is close enough to the fixed point $a$, so that $R'(h^i(w^k))/\lambda \in K^d$.
Therefore
$$
(R^{(\hr - r)dd'})'(h^{\hm}(w^k))
=
\lambda^{(\hr - r)dd'} \prod_{m < i \le \hm} (R'(h^i(w^k))/\lambda)
$$
belongs to $K^d$, as it is the product of elements in $K^d$.

\noindent
$\bullet$
For the first factor, observe that
$$
(R^{k(\hm + \ell)})'(h^{\hm}(w^k))
=
\prod_{1 \le j \le k} (R^{\hm + \ell})'(h^{\hm}(w^j)).
$$

Since for every $w \in \cD$ we have $h^i(w) \to a$ as $i \to \infty$, it follows that if $w \in K$ is close enough to $c$, then for every positive integer $s$ we have
$$
R'(h^s(w)) / R'(h^s(c)) \in K^d.
$$
So if $r$ is sufficiently large then for every $1 \le j \le k$ we have
$$
(R^{\hm})'(h^{\hm}(w^j)) / (R^{\hm})'(h^{\hm}(c)) \in K^d.
$$

On the other hand, for $1 \le j \le k - 1$ we have
$$
|h^{\hm}(w^j) - a| / |R^\ell(c) - a| = |\lambda_0|^{-(\hr - r)dd'}.
$$
Since $R^\ell(w^1) = a$ and for $1 \le j \le k - 1$ we have $R^\ell(w^{j + 1}) = h^{\hm}(w^j)$, it follows that, if $\hr$ is large enough compared to $r$, the quotient
$$
(R^\ell)'(w^{j + 1}) / (R^\ell)'(w^1)
$$
is close enough to~1, to belong to $K^d$.
Setting $\eta = (R^{\hm})'(h^{\hm}(c)) \cdot (R^\ell)'(w^1)$, we thus have $(R^{\hm + \ell})'(h^{\hm}(w^j)) \eta^{-1} \in K^d$.
When $k = d$ it follows that
$$
\prod_{1 \le j \le k} (R^{\hm + \ell})'(h^{\hm}(w^j))
=
\eta^d \cdot \prod_{1 \le j \le d} \left( (R^{\hm + \ell})'(h^{\hm}(w^j)) \eta^{-1} \right)
\in K^d.
\hfill \square
$$

\bibliographystyle{plain}

config: arxiv


TITLE: Determine whether a series expansion will be a Taylor or Laurent series?
QUESTION [0 upvotes]: Let $f(z)= \frac {1 } {z^3+z^2 } $ and consider its series expansions around the point $z=1 $.
Clearly there will be three such, on 1) $|z-1 |< 1 $, 2) $1<|z-1 |<2 $, 3) $|z-1 |>2$.
Now my question: how can I tell which of these will be Taylor series or which will be Laurent series? Obviously one way would be to partial fraction the expression and then express the fractions as series using the fact that $\frac {1 } {1-z } =\sum _{n=0 } ^{\infty } z^n, |z |<1$, and then see if the series contains negative exponents. But is there an easier way to determine this without doing this work?
Thanks in advance!

REPLY [2 votes]: If the Laurent series of $f$ in the annulus $r < \lvert z-a\rvert < R$ is actually a Taylor series (i.e., it contains no negative powers of $z-a$), then the series converges (at least) on the disk $\lvert z-a\rvert < R$, and therefore $f$ can have no singularities in that disk.
Thus in the given example, the Laurent series will be a Taylor series only for the disk $\lvert z-1\rvert < 1$, since $f$ has poles in $0$ and $-1$.


text:
\begin{document}

\title{Cyclic Homology of $DG$ Coalgebras and a K\"{u}nneth Formula}
\author{Masoud Khalkhali\thanks{\; Author's research is partially supported
by NSERC of Canada.  Presented in a conference in honor of Professors
Peter Fillmore and Heydar Radjavi, Halifax, Nova Scotia, June 1998.}}

\maketitle

In this paper we extend the cyclic homology functor, and in particular
the periodic cyclic homology, to the category of $DG$ (= differential
graded) coalgebras.  We are partly motivated by the question of products
and coproducts in periodic cyclic homology of algebras.  As an
application, we will show how one can start from the classical shuffle
map in homological algebra and algebraic topology, interpreted in [HMS]
as a morphism of $DG$ coalgebras, and build a theory of products and
coproducts in (periodic, negative, etc.) cyclic homology.  Along these
lines we also recover a K\"{u}nneth isomorphism
\[ \widehat{S}:CC_\ast (A_1)\widehat{\otimes}CC_\ast
(A_2)\overset{\sim}{\longrightarrow}CC_\ast (A_1\otimes A_2)\; , \]
relating the periodic cyclic complexes of unital $(DG)$ algebras over a
field of characteristic zero.  Here $CC_\ast$ denotes the periodic
cyclic complex and $\widehat{\otimes}$ denotes the topologically
completed tensor product of complexes.  We note that a homotopy
equivalent result, with a different choice for $CC_\ast$ and
$\widehat{S}$, can be
found in the works of Cuntz and Quillen [CQ$_2$], Puschnigg [P] and more
recently Bauval [B].  Our method is based on ideas of Cuntz and Quillen
in cyclic homology (cf. [CQ$_1$] and reference therein) and in a sense
is dual to them.

An interesting problem suggests itself.  It would be interesting to see
how one can use a similar approach to define the cyclic cohomology of
$(DG)$ Hopf algebras and compare it with the definition of periodic
cyclic cohomology of Hopf algebras as recently defined by Connes and
Moscovici [CM].

I am much indebted to Joachim Cuntz for his comments on a first draft of
this paper and ensuing discussions which greatly improved my understanding
of the K\"{u}nneth formula and the importance of the topological tensor
product.  I would also like to thank A. Bauval for communicating her
results and for very informative discussions.  I am also grateful to J.
Williams for typing the manuscript.

\section{Bar and cobar constructions and the shuffle map}
The present form of the bar and cobar constructions first appeared in
a paper by Husemoller, Moore and Stasheff [HMS] under the name of
{\em algebraic classifying space} and {\em loop space constructions}.
The reason for these names is that these constructions provide models
for singular chains on the classifying space of groups and singular cochains
on the (based) loop space of simply connected spaces.  In the same paper the
{\em shuffle map} is introduced as a morphism of $DG$ coalgebras.  This is
crucial for applications to periodic cyclic homology.  The bar construction,
denoted here by $B$, is a functor from the category of $DG$ algebras to the
category of $DG$ coalgebras.  It has a left adjoint, namely the cobar
construction, denoted here by $B^c$, from the category of $DG$ coalgebras to
$DG$ algebras.  Let us recall their definitions.

Let \d A = \oplus_{i\geq 0}A_i$ be a positively graded $DG$ algebra
over a ground ring $k$.  We assume its differential $d$ has degree $-1$.
Let $A[1]$ denote the suspension of $A$ defined by $A[1]_n=A_{n-1}$.
The {\em bar construction} of $A$, denoted by $BA$, is a $DG$ coalgebra whose
underlying graded coalgebra is $BA=T^cA[1]$, the cofree coaugmented
counital coalgebra generated by the graded vector space $A[1]$.  We have
$(BA)_0=k$, $(BA)_1=A_0$ and in general
\[ (BA)_n = \bigoplus^n_{s=0}\;\bigoplus_{i_1+\cdots +i_r+r=s}\;
A_{i_1}\otimes\cdots\otimes A_{i_r}\; . \]
The differential of $BA$ is $d+b^\prime$, where $d$ and $b^\prime$ are
defined by
\[ d(a_1,\dots ,a_n) = \sum^n_{i=1}(-1)^{\varepsilon_i}(a_1,\dots
,da_i,\dots ,a_n)\]
and
\[ b^\prime (a_1,\dots ,a_n) =
\sum^{n-1}_{i=1}(-1)^{\varepsilon_i+|a_i|}(a_1,\dots ,a_ia_{i+1},\dots
,a_n)\; , \]
where
\[ \varepsilon_i = \sum^{i-1}_{j=1}|a_j|+i-1\; . \]
Computation shows that $d^2 = b^{\prime\; ^2} = db^\prime +b^\prime d=0$,
so that $d+b^\prime$ is a differential of degree -1.  It is moreover a
graded coderivation of $BA$.

Morphisms of $DG$ coalgebras into $BA$ are defined via {\em twisting
cochains} and vice versa.  More precisely, let $C$ be a $DG$ coalgebra.  A
degree -1 map $\theta :C\to A$ is called a twisting cochain if
\[ \delta\theta +\theta^2 = 0\; , \]
where, $\delta\theta = [d,\theta ]$ in the Hom complex differential of
$\theta$ and $\theta^2= m\circ (\theta\otimes\theta)\Delta$, where
$\Delta$ is the coproduct of $C$ and $m$ is the multiplication of $A$.
The {\em universal twisting cochain} is the map $\theta :BA\to A$ defined by
$\theta (a) = a$ for $a\in A$ and zero otherwise.  There is a 1-1
correspondence between morphism of $DG$ coalgebras $\widehat{\theta}:C\to
BA$ of degree zero and twisting cochains $\theta :C\to A$.  In this
correspondence, $\theta$ is simply the corestriction of
$\widehat{\theta}$ to $A$.  Conversely, given $\theta$, we have
\begin{equation*}
\widehat{\theta} = \sum^\infty_{n=1}\widehat{\theta}_n\tag{1}
\end{equation*}
where
\[\widehat{\theta}_n = \theta^{\otimes n}\circ\Delta^{(n)}\; , \]
and $\Delta^{(n)}$ is the $n-th$ iteration of the coproduct.

Next, we discuss the {\em cobar construction}.  Let \d C =
\oplus_{i\geq 0}C_i$ be a positively graded $DG$ coalgebra with a
differential $d$ of degree -1.  We assume $C_0 = k$.  Let $C[-1]$ be the
graded space defined by $C[-1]_i=0$ if $i<0$, and $C[-1]_i=C_{i+1}$ for
$i\geq 0$.  The cobar construction of $C$, denoted by $B^cC$, is the
$DG$ algebra whose underlying graded algebra is $B^cC=TC[-1]$, the augmented
unital free algebra generated by $C[-1]$.  We have \d (B^cC)_0 =
\oplus_{n\geq 0}C^{\otimes n}_1$, and
\[ (B^cC)_n = \bigoplus^n_{r=1}\;\bigoplus_{i_1+i_2+\cdots +i_r=n+r}\;
C_{i_1}\otimes\cdots\otimes C_{i_r}\; . \]
The differential of $B^cC$ is $d+b^\prime$, where $d$ and $b^\prime$ are
defined by
\[ d(a_1,\dots , a_n) = \sum (-1)^{\varepsilon_i}(a_1,\dots , da_i,\dots
, a_n) \]
and
\[ b^\prime = \sum
1\otimes\cdots\otimes\tilde{\Delta}\otimes\cdots\otimes 1\; , \]
where $\varepsilon_i$ is the same as before and
$\tilde{\Delta}:C[-1]\longrightarrow C[-1]\otimes C[-1]$ is the unique
morphism of complexes of degree -1 such that
$\tilde{\Delta}\circ s^{-1} = (s^{-1}\otimes s^{-1})\Delta$.  Here
$s:C\longrightarrow C[-1]$ is the desuspension.

We have $d^2=b^{\prime^2} = db^\prime +b^\prime d=0$, so that
$d+b^\prime$ is a differential of degree -1.  It is moreover a graded
derivation of $B^cC$.

Let $A_1$ and $A_2$ be unital algebras (no differential, no grading).
The {\em shuffle map} is a morphism of $DG$ coalgebras
\begin{equation*}
S:BA_1\otimes BA_2\longrightarrow B(A_1\otimes A_2)\tag{2}
\end{equation*}
defined as follows.  Let $\theta_i:BA_i\to A_i$, $i=1,2$, be universal
twisting cochains of $A_i$.  Define a twisting cochain
\[ \theta :BA_1\otimes BA_2\to A_1\otimes A_2 \]
by
\[ \theta = \theta_1\otimes\varepsilon\eta
+\varepsilon\eta\otimes\theta_2\; , \]
where $\eta$ is the counit map of $BA_i$ and $\varepsilon$ is the unit
map of $A_i$.  Note that $\theta$ is zero except on linear span of tensors
of the type $a_1\otimes 1$, or $1\otimes a_2$.  Checking the twisting
cochain condition for $\theta$ is easy.  By universal property of the bar
construction we obtain $S$.

A simple computation using (1) gives the following explicit formula for
$S$:
\[ S((a_1,\dots , a_p)\otimes (b_1,\dots b_q)) = \sum_{\sigma\in
S_{p,q}}sgn(\sigma )\sigma (a^\prime_1,\dots ,a^\prime_p,b^\prime_1,\dots
,b^\prime_q)\; , \]
where $a^\prime_i=a_i\otimes 1$ and $b^\prime_i=1\otimes b_i$, and
$S_{p,q}$ is the set of all $(p,q)$ shuffle permutations in the symmetric group
$S_{p+q}$.

We need to know that $S$ is a quasi-isomrophism.  This is trivial for
unital algebras over a field, since it is well known that $b^\prime$
is acyclic for unital algebras and one can use the K\"{u}nneth formula
for tensor product of complexes over a field.  We mention, however, that
$S$ has an explicit homotopy inverse
\[ A:B(A_1\otimes A_2)\to BA_1\otimes BA_2\; , \]
the {\em Alexander-Whitney map} (see, e.g. [M]), which proves that
$S$ is a quasi-isomorphism over any ground ring.  Same proof works when
$A_1$ and $A_2$ are unital $DG$ algebras.  It is important, however, to
note that $A$ is not a morphism of $DG$ coalgebras.  This makes finding
an explicit formula for the inverse of the product map in periodic cyclic
homology a difficult task.  See however [B] where this problem is
successfully solved.  An alternative approach would be to extend $A$ to
an $A_\infty$-morphism of $DG$ coalgebras which in theory one knows to
exist and let it act on cyclic complexes.

\section{Periodic cyclic homology of $DG$ coalgebras}
Periodic cyclic homology of $DG$ algebras was defined by Goodwillie and
others (see [G] and references therein).  One of the main results of [G]
is the fact that if $f:A\to B$ is a morphism of $DG$ algebras which is a
quasi-isomorphism of complexes, then the induced map on periodic cyclic
homology $f_\ast :HP_\ast (A)\to HP_\ast (B)$ is an isomorphism.  We need
the analogue of this result for $DG$ coalgebras.  We also need the fact that
for $DG$ coalgebras of finite cohomological dimension (1 and 2 in our
applications), the periodic cyclic complex is quasi-isomorphic to some
higher order versions of the $X$-complex.

Let $C$ be a $DG$ coalgebra.  In [Kh], we have carefully defined the
$DG$ coalgebra of (noncommutative) differential forms over $C$, and
denoted it by $(\Omega C,d)$.  Its construction is dual to the
corresponding construction for algebras and $DG$ algebras.

We need to adopt some basic definitions and constructions from [CQ$_1$] to
our $DG$ coalgebraic set up.  Let $C$ be a $DG$ coalgebra and let
$(\Omega C,d)$ denote the $DG$ coalgebra of {\em universal
codifferential forms} over $C$.  Let $\eta :C\longrightarrow k$ be the
counit of $C$.  We have $\Omega^nC = C\otimes \overline{C}^{\otimes n}$,
where $\overline{C} = Ker\;\eta$.  Let $b:\Omega^\bullet
C\longrightarrow\Omega^{\bullet +1}C$ be the analogue of the {\em
Hochschild boundary operator} and let $N$ be the {\em number operator}
which multiplies a differential form by its degree.  Let
\[ \Omega^{norm}C = ker\{ (b+dN)^2:\Omega C\longrightarrow\Omega C\}\; .
\]

Equipped with the differential $b+dN$ and with its natural
$\mathbb{Z}/2$ grading, $(\Omega^{norm}C, b+dN)$ can be regarded as a
supercomplex.  There is a decreasing filteration
$\{F^n\Omega^{norm}C\}_{n\geq 2}$ on $\Omega^{norm}C$, where $F^n$
consists of forms of degree at least $n$.  The successive quotient
complexes $\Omega^{norm}C/F^n$ approximate the normalized cyclic
bicomplex for $DG$ coalgebras.  We need only the first two quotients,
denoted by $X(C)$ and $X^2(C)$.  These are the supercomplexes
\[ \begin{array}{rl}
X(C): & \begin{CD} C @<\overset{b}{\longrightarrow}<{d}<
\Omega^1C_\natural\end{CD}\\
X^2(C): & \begin{CD} C\bigoplus\Omega^2C_\natural
@<\overset{b+2d}{\longrightarrow}<{b+d}< \dot{\Omega}^1C,
\end{CD}\end{array} \]
where $\natural$ denotes the cocommutator subspace and $\dot{\Omega}^C =
\Omega^{norm,1}C$.  Note that
$\Omega^1C_\natural\subset\dot{\Omega}^1C$.

We are mostly interested in the total complexes of these bicomplexes which
we denote by $\widehat{X}(C)$ and $\widehat{X}^2(C)$.  Here total means
taking direct products.  When working with $X$-complex of $DG$ algebras,
total means taking direct sums.

We define the {\em periodic cyclic complex} of a $DG$ coalgebra $C$ by
\[ CC_\ast (C) = \widehat{X}(B^cC)\; . \]
We denote the homology of this complex by
$HP_\ast (C)$.  The motivation for this definition is as follows.  In
[Q$_1$], D. Quillen computed $\widehat{X}(BA)$, when $A$ is an algebra, and
showed that it is isomorphic with the total complex of Connes-Tsygan
bicomplex of $A$.  The same proof easily extends to the case of $DG$
algebras.  Our definition of periodic cyclic complex of $DG$ coalgebras
is simply the dual of this definition-theorem for $DG$ algebras.

The following result was first proved by T. Goodwillie for $DG$ algebras
[G].

\begin{proposition}
Let $f:C\to D$ be a morphism of $DG$ coalgebras such that $f$ is a
quasi-isomorphism of complexes.  Then $f_\ast :HP_\ast (C)\to HP_\ast
(D)$ is an isomorphism.
\end{proposition}

Let $C$ be a $DG$ coalgebra.  Note that we have a natural morphism of
complexes
\[ I:\widehat{X}(C)\longrightarrow\widehat{X}^2(C)\; , \]
obtained from the inclusion $C\longrightarrow C\oplus\Omega^2C_\natural$
and $\Omega^1C_\natural\longrightarrow\dot{\Omega}^1C$.  In general,
there is no natural map $\widehat{X}^2(C)\longrightarrow\widehat{X}(C)$.
However, it is shown in [Kh] that if $C = BA$ is the bar construction
then $I$ is a homotopy equivalence and a homotopy inverse
\[ R:\widehat{X}^2(BA)\longrightarrow\widehat{X}(BA) \]
is constructed.  There is a dual statement for $DG$ coalgebras, but we
don't need it in this paper.

Let us call a $DG$ coalgebra of {\em finite cohomological dimension} if
its underlying coalgebra has finite cohomological dimension.  We need to
know that if a $DG$ coalgebra $C$ has finite cohomological
dimension, then its periodic cyclic complex $CC_\ast (C)$ is
quasi-isomorphic to a ``small'' complex.  We need this only for
cohomological dimensions 1 and 2.

\begin{proposition}
Let $C$ be a $DG$ coalgebra.  Then
\begin{enumerate}
\item[1.] The natural map $\widehat{X}(C)\longrightarrow CC_\ast (C)$
is a quasi-isomorphism if $C$ has cohomolgical dimension 1.
\item[2] The natural map $\widehat{X}^2(C)\longrightarrow CC_\ast (C)$
is a quasi-isomorphism if $C$ has cohomological dimension 2.
\end{enumerate}
\end{proposition}

\section{A Proof of the K\"{u}nneth formula}
Let $A_1$ and $A_2$ be unital algebras over a field of characteristic
zero.  In general, $X(A_1)\otimes X(A_2)$ can not be quasi-isomorphic to
$X(A_1\otimes A_2)$.  However, when $A_1$ and $A_2$ are quasifree (in
particular free) algebras then the cohomological dimension of $A_1\otimes
A_2$ is at most 2 and $X(A_1)\otimes X(A_2)$ is quasi-isomorphic to
$X^2(A_1\otimes A_2)$.  This fact is due to Cuntz and Quillen [CQ$_1$].  In
order to obtain explicit formulas, M. Puschnigg constructed a morphism
of complexes [P],
\[ P:X^2(A_1\otimes A_2)\longrightarrow X(A_1)\otimes X(A_2)\; , \]
and showed that $P$ is a quasi-isomorphism when $A_1$ and $A_2$ are
quasi-free.  In fact, an explicit right inverse to $P$ is the map defined in
[CQ$_1$], for arbitrary algebras, $X(A_1)\otimes X(A_2)\longrightarrow
X(A_1\otimes A_2)$ combined with the canonical inclusion
$X(A_1\otimes A_2)\longrightarrow X^2(A_1\otimes A_2)$.

Next we observe that the above constructions are completely functorial
and extend to $DG$ algebras and $DG$ coalgebras.  To be precise, let us
define the {\em completed tensor product} $C\widehat{\otimes} D$ of
infinite product vector spaces \d C = \prod_{i\geq 0} C_i$ and \d D =
\prod_{i\geq 0}D_i$ by
\[
C\widehat{\otimes}D = \prod_{n\geq 0}\bigoplus_{i_+j=n}C_i\otimes D_j\; .
\]
There is an obvious injection
$C\otimes D\longrightarrow C\widehat{\otimes}D$.  Since the bar construction
is free as a coalgebra, dualizing the above map $P$ we obtain a
quasi-isomorphism of complexes
\[ \widehat{X}(BA_1)\widehat{\otimes}\widehat{X}(BA_2)\longrightarrow
\widehat{X}^2(BA_1\otimes BA_2)\; . \]

Consider the sequence of maps
\[ \widehat{X}(BA_1)\widehat{\otimes}\widehat{X}(BA_2)\overset{P}{\longrightarrow}
\widehat{X}^2(BA_1\otimes BA_2)\overset{\overline{S}}{\longrightarrow}
\widehat{X}^2(B(A_1\otimes A_2))\overset{R}{\longrightarrow}
\widehat{X}(B(A_1\otimes A_2))\; , \]
where $\overline{S}$ is induced by the shuffle map
$S:BA_1\otimes BA_2\longrightarrow B(A_1\otimes A_2)$ and $R$
is the retraction introduced earlier.
\begin{theorem}
$P$, $\overline{S}$ and $R$ are quasi-isomorphisms.
\end{theorem}
\begin{proof}
We only have to show that $\overline{S}$ is a quasi-isomorphism.
Consider the diagram
\[ \begin{CD}
\widehat{X}^2(BA_1\otimes BA_2) @<<< \widehat{X}(B^c(BA_1\otimes BA_2) =
CC_\ast (BA_1\otimes BA_2)\\
@VV \overline{S} V @ VV\widetilde{S} V\\
\widehat{X}^2(B(A_1\otimes A_2)) @<<< \widehat{X}(B^cB(A_1\otimes
A_2))=CC_\ast (B(A_1\otimes A_2))
\end{CD} \]
where $\widetilde{S}$ is the map induced from the $DG$ coalgebra map
$S:BA_1\otimes BA_2\longrightarrow B(A_1\otimes A_2)$ on periodic
cyclic complexes.  By proposition 2.1, $\widetilde{S}$ is a
quasi-isomorphism as $S$ is a quasi-isomorphism of complexes.  By
proposition 2.2, the horizontal arrows are also quasi-isomorphisms and hence
$\overline{S}$ is a quasi-isomorphism.
\end{proof}

It follows that the composition
\[ \widehat{S} = R\circ \overline{S}\circ P:CC_\ast
(A_1)\widehat{\otimes}CC_\ast (A_2)\longrightarrow CC_\ast (A_1\otimes A_2) \]
is a quasi-isomorphism of complexes.


TITLE: Parametrized complex equation
QUESTION [1 upvotes]: What are the values that satisfy the following equation:
$$\frac{z+i}{\bar{z}-2}=-2i$$
I have tried substituting $z$ with $a+bi$, and multiplying the left hand side by $\frac{z-2}{z-2}$ but the equations became too complicated.

REPLY [1 votes]: $$\frac{z+i}{\bar{z}-2}=-2i$$
Let's multiply by $\bar{z}-2$:
$$z+i=-2i(\bar{z}-2)$$
Let's substitute $z$ for $a+bi$:
$$a+bi+i=-2i(a-bi-2)$$
$$a+bi+i=-2ai+2bi^2+4i$$
$$a+(b+1)i=-2ai-2b+4i$$
$$a+(b+1)i=-2b+(-2a+4)i$$
$$
\left\{
\begin{array}{c}
a=-2b \\
b+1=-2a+4 \\
\end{array}
\right.
$$
$$b+1=-2(-2b)+4$$
$$b+1=4b+4$$
$$3b=-3$$
$$b=-1$$
$$a=2$$
Thus $z=2-i$


TITLE: Relativistic Accelerating Frame (non-inertial frame)
QUESTION [1 upvotes]: One way to resolve the Twin paradox is to count in the effect of acceleration when velocity suddenly changes. I am therefore searching for a relativistic version of "accelerating frame" or non-inertial frame.  Apparently it is much less straightforward than accelerating frame of Galilean relativity.
The key question is: if an observer stays at rest at the origin of an accelerating frame, how much (proper) time will she experience? In the laboratory frame, it certainly should not equal $T/\gamma(u),$ where $\gamma(u) = (1-u^2/c^2)^{-1/2},$ and $u$ is the instantaneous velocity $\frac{dx}{dt},$ because this does not resolve the Twin paradox.
Could anyone explain how the proper time changes with acceleration?

REPLY [1 votes]: You only need a single frame, and an inertial one at that, to calculate the twin paradox exactly. Just focus on the fact that if two events are separated by $dt,dx,dy,dz$ in some inertial frame, then the proper time between them is
$$
d\tau = \sqrt{ dt^2 - (dx^2 + dy^2 + dz^2)/c^2}.
$$
For motion in one spatial dimension this simplifies to
$$
d\tau = \sqrt{ dt^2 - dx^2 /c^2}.
$$
It follows that the total proper time along a worldline $p$ between events $A$ and $B$ is
$$
\tau = \int_{(p),A}^B \sqrt{ dt^2 - dx^2 /c^2} \;
= \int_{(p),A}^B \sqrt{1 - v^2 /c^2} dt.
$$
If we pick a frame where those events are at the same spatial location, then for the worldline going straight from $A$ to $B$ we have $v=0$ all the way, so the proper time for that worldline is
$$
\tau_{v=0} = t(B) - t(A)
$$
For some other worldline between the same two events, one will have $v^2 > 0$ for some or all of the worldline, so then the proper time is smaller. The twin who follows such a worldline will therefore age less, between $A$ and $B$, than the twin whose worldline was straight.
The above method of calculation is perfectly accurate for any worldline at all, including ones which first accelerate away and then turn around and come back. No approximations are introduced, and you do not need to assume a sudden turn-around or anything like that.


TITLE: Example of a field extension of a cyclotomic extension
QUESTION [0 upvotes]: I have for example the cyclotomic field extension $\mathbb{Q}(\zeta_3)$ . And I think it's possible:  $\mathbb{Q}(\sqrt{\zeta_3})$ . Then it will be possible the field extension of the cyclotomic extension:  $\mathbb{Q}(\zeta_3,\sqrt{\zeta_3})$ .
How  $\mathbb{Q}(\zeta_3)$  and  $\mathbb{Q}(\sqrt{\zeta_3})$ are linearly disjoint over $\mathbb{Q}$ , I have to multiply its two respective bases to get the basis of  $K=\mathbb{Q}(\zeta_3,\sqrt{\zeta_3})$ .
Then:  $K=\{1,\zeta_3\}\,\,x\,\, \{1,\sqrt{\zeta_3}\}\,=\,\{1,\,\zeta_3,\,\sqrt{\zeta_3},\,\,\zeta_3\sqrt{\zeta_3}\}$
Is it correct?
Is it truly the basis  $k_1=\mathbb{Q}(\sqrt{\zeta_3})=\{1,\sqrt{\zeta_3}\}$ , how I'm thinking?
Thanks in advance

REPLY [1 votes]: Let $a=\sqrt{\zeta_3}$. Since $a^2=\zeta_3$ we have $a^6=1$ and $a$ is a $6$-th root of unity. However, we have
$$
\Bbb Q(\zeta_6)=\Bbb Q(\zeta_3),
$$
see here: Is $\mathbb Q(\zeta_6)=\mathbb {Q}(\zeta_3)$?
So the degree is indeed $2$.


TITLE: Honest definitions and UDTFS
QUESTION [2 upvotes]: I am trying to understand the Chernikov-Simon theorem on Uniform Definability of Types over Finite Sets see Corollary 6.17 in Pierre Simon's book.
The proof rests on two results. One is the so-called (p,q)-theorem, a deep result in combinatorics. The second ingredient is Theorem 3.13 in Pierre Simon's book:

Theorem  (honest definitions). Let $M\models T$, $A\subseteq M$, $\phi(x;y) \in L$ a formula and $b\in M^{|y|}$ a tuple. Then there is an elementary extension $(M,A)\prec (M',A')$, a formula $\psi(x;z)\in L$ and a tuple $d$ of elements of $A'$ such that $$\phi(A;b)\subseteq \psi(A';d) \subseteq \phi(A';b).$$

I am puzzled because the theorem above seems trivial when $A$ is finite. Though this is exactly the case relevant for Corollary 6.17.

REPLY [2 votes]: You're correct that the statement of Theorem 3.13 is trivial when $A$ is finite. This is just because types over finite sets are trivially definable in any theory: For a finite set $A$ and a formula $\phi(x;y)$, the set $\phi(A;b)$ can be defined by $\psi(A;d)$, using parameters $d$ from $A$, just by taking $\psi$ to be $\bigvee_{i=1}^k (x = d_k)$ when $|\phi(A;b)| = \{d_1,\dots,d_k\}$.
Of course, the key issue here is uniformity: The formula $\psi$ above depended on the size of $\phi(A;b)$. We'd like to be able to pick a formula $\psi(x;z)$ which only depends on $\phi(x;y)$, which works for every finite set $A$ and every tuple of parameters $b$.
And note how Theorem 3.13 gets used in the proof. It's in the middle of a (technical) uniformity result, Lemma 6.16 of A Guide to NIP Theories. Roughly speaking, if we want to achieve uniformity over arbitrary finite sets, compactness implies that we're going to have to "handle" infinite sets too: We introduce a new unary predicate $P$ which names the set $A$, and we can't axiomatize "$P$ is finite". But the honest definitions theorem is strong enough to give us the right kind of leverage on infinite sets in this compactness argument.


TITLE: Quotient space of the torus with two points identified
QUESTION [1 upvotes]: Take the torus $T=S_1 \times S_1$. Choose two points $x,y \in T$ and define a quotient topology by identifying $x$ and $y$. Let $X$ denote the quotient space. Prove that: a) Compute the fundamental group of $X$. b) Prove that $X$ is not homeomorphic to a surface.

REPLY [2 votes]: Here's how I might do it. (In fact, this is just the method suggested in PseudoNeo's comment above, but I wrote this out before I saw that particular comment, so I'm posting it for posterity.)
Consider the torus, and choose your two points $x$ and $y$. The space $X$ in which these two points are identified is homotopy equivalent to the space formed by attaching a copy of $I$ to the torus with endpoints $x$ and $y$, and since the torus is path-connected, this is homotopy equivalent to the wedge sum of a torus and a circle. By van Kampen's Theorem, and the homotopy invariance of $\pi_1$ we now know the fundamental group. Much like Jacob's proof, these homotopy equivalences are very easy to convince yourself of, but perhaps require a little more work to actually prove.
EDIT: My solution to part b) is obviously nonsense, homotopy equivalence doesn't preserve manifold-ness; consider a sphere with an interval attached at the basepoint, for instance. You can show that it's not homeomorphic to a closed compact surface by using van Kampen's theorem, the classification of such surfaces, the fundamental group calculated in part a), and the homeomorphism invariance of $\pi_1$. You can show that it's not even a 2-manifold by looking at Jacob's solution above.


TITLE: A question about thermal conductivity and heat transfer
QUESTION [5 upvotes]: Put a piece of cold pizza on top of a sheet of aluminum foil and then stick it in the oven to heat up. After about 10 minutes, the pizza should be nice and hot—the aluminum foil is the approximately the same temperature. You can pull the aluminum foil out with your fingers, but not the pizza.
I read this on wired.com Someone please explain the reason behind it.

REPLY [9 votes]: The sensation of something feeling "hot" is a function of the rate of heat transfer to the skin and the duration of the transfer. The product of the rate and duration of heat transfer is the amount of heat transferred to the skin.
The rate of heat transfer of the aluminum foil, being a metal, is relatively high because of its high thermal conductivity. But the amount of heat available is low because it is very thin making its mass per unit area and its heat capacity very low, all other things being equal. This is why you can take the foil directly out of the oven without an oven mitt and not get burned. If, on the other hand, you attempted to take the metal rack out of the oven without a mitt you could get burned.
Example:
A widely cited thermal injury study (Alice Stoll et al) determined the theoretical combinations of absorbed energy rate and duration of exposure for thresholds of blister and pain. It was based on experiments conducted on human subjects.
For an absorbed rate of 5 w/cm$^2$ and duration of exposure of 1 s, a theoretical total absorbed energy of 5 J/cm$^2$ is predicted for threshold of blister and 2.5 J/cm$^2$ for threshold of pain.
The amount of thermal energy available to transfer to the finger, bringing the temperature of the aluminum to the temperature of the finger, from a 1 cm$^2$ piece of aluminum foil (approximate surface area of finger tip), 0.02 mm thick (average for foil) at a temperature of 200 C (392 F) is about 0.8 J, making it more than 5 times less than the blister threshold and more than 2.5 times less than the pain threshold.

Just a curious thought, what should we do to get burn by the aluminium
foil? I mean what modification should we make to the foil? And may you
please provide formula/methodology for the calculations?

In order to increase the potential for pain and burn from the aluminum, one would have to sufficiently increase its thickness and/or its temperature.
The formula used to calculate the theoretical amount of heat $Q$ available per cm$^3$ of foil relative to, say, the finger, and assuming the foil does not transfer heat to anything else, was
$$Q=\rho c (T_{Al}-T_{f})$$
Where
$\rho$ = the density of aluminum, taken as 2.71 g/cm$^3$
$c$ = the specific heat of aluminum, taken as 0.872 J/g$^o$C
$\rho c$ = the volumetric heat capacity of aluminum, 2.43 J/cm$^{3}$ $^o$C
$T_{Al}$ = temperature of aluminum, taken as 200$^0$C (upper temperature of a moderately hot oven)
$T_{f}$ = temperature of the skin of finger, taken as 32.5$^o$C.
From the above data the available thermal energy is about 400 J/cm$^3$. Multiplying this by the thickness of the foil, 0.002 cm, gives you 0.8 J/cm$^2$ of foil.
As indicated previously, the theoretical threshold for blister and pain for 1 sec contact is 5 J/cm$^2$ and 2.5 J/cm$^2$. To approach those levels the foil thickness would need to be theoretically increased by a factor of 5 and 2.5 respectively. Alternatively, the oven temperature (= temperature of the foil prior to touch) would have to theoretically be increased to about 870 $^o$C and 450 $^0$C, respectively. These temperatures are well above those attainable in household ovens.
It should be noted that there were a number of assumptions made in the above, most of which err on the side of caution.  Perhaps the most significant is the theoretical thresholds of pain and burn. They were based on experiments on very thin (most vulnerable) skin (the inside surface of forearm and back of finger (middle phalanges)). The skin on the finger pads is thicker making them less vulnerable. So the thresholds for the finger pads would be higher than those used here. This should err on the side of caution.
Hope this helps.


TITLE: Domain and range of a multivariable function
QUESTION [2 upvotes]: I have this exercise $$z={2x\over y+5}$$, and I am supposed to obtain the domain and range. I understand that the domain is all the pair of $(x,y)$ except $y=-5$ , then the exercise said that the range is $z=R$

I dont understand why z accept all the values, suppose that you want to plot the point $(1,-5)$ you wont be able to plot that point because $y=-5$ its not accepted by the domain so it cant output a value for z (range)
Please help me undestand this or how i get the range for rational functions
Thank you!

REPLY [0 votes]: Domain is the set of all inputs a function can take. As you correctly note, the only thing that makes your $f$ undefined is any input with $y=-5$, hence the domain of $f$ is the entire $\mathbb{R}^2$ with the exception of the line $y=-5$, i.e. the set
$$
dom(f) = \{(x,y)|y \ne -5\}.
$$
As for the range, it is the set of all points to which $f$ can map something. Note that if $r \in \mathbb{R}$, then $f(r/2, -4)=r$, so the range of $f$ is the entire real line.
UPDATE
For example, to clarify range, suppose you want to check if $\pi$ is in the range of $f$. Note that
$$
f(\pi/2, -4) = \frac{2 \cdot (\pi/2)}{(-4)+5} = \pi,
$$
so indeed $\pi$ is in the range of $f$. So would be any other real number.

REPLY [0 votes]: Or, let $x=1$, then $z=\dfrac{2}{y+5}$, as a function of $y$, $z$ takes all values except $z=0$.
But, for $x=0$, $y=1$, $z=\dfrac{2\cdot 0}{1+5}=0$.
So it takes all values.