text:
\begin{document}
\title{Some Special Cases in the Stability Analysis of Multi-Dimensional Time-Delay Systems Using The Matrix Lambert W function}
\author{Rudy Cepeda-Gomez\thanks{Institute of Automation, University of Rostock, Rostock, Germany. \texttt{rudy.cepeda-gomez@uni-rostock.de}} and Wim Michiels\thanks{Department of Computer Science, KU Leuven, Leuven, Belgium. \texttt{wim.michiels@cs.kuleuven.be}}}
\maketitle

\begin{abstract}
This paper revisits a recently developed methodology based on the matrix Lambert W function for the stability analysis of linear time invariant, time delay systems. By studying a particular, yet  common, second order system, we show that in general there is no one to one correspondence between the branches of the matrix Lambert W function and the characteristic roots of the system. Furthermore, it is shown that under mild conditions only two branches suffice to find the complete spectrum of the system, and that the principal branch can be used to find several roots, and not the dominant root only, as stated in previous works. The results are first presented analytically, and then verified by numerical experiments.
\end{abstract}

\section{Introduction}\label{intro}
In this paper we consider Linear Time Invariant-Time Delay Systems (\textsc{lti-tds}), represented by Delay-Differential Equations (\textsc{ddes}) of the form:
\begin{equation}
\dot{\mathbf{x}}\left(t\right)=\mathbf{Ax}\left(t\right)+\mathbf{Bx}\left(t-\tau\right)
\label{eq:lti-tds}
\end{equation}

The stability analysis and control synthesis of this class of systems is a wide open area of research. The difficulty of this problem arises from the the fact that the delay makes this class of systems infinite dimensional. A nice review of the recent results and challenges int his area can be found in \cite{Sipahi2011}. Several avenues have been followed by different researchers to find solutions to the stability and control questions. Some works study the absolute stability regions with respect to the time delay \cite{Olgac2002,Sipahi2006}, and lead to control strategies that use the time delay as a stabilizing tool \cite{Olgac2005}. Other researchers focus on the numerical computation of the characteristic roots of the system. These works include approaches  based on the discretization of the solution operator\cite{Engelborghs2002,Breda2006,Butcher2011} or its infinitesimal generator \cite{Wu2012,Breda2009}, and methods based on root finding of the characteristic equation \cite{Vhylidal2009}. From here methods have been proposed  to optimize the location of the dominant roots to guarantee a certain performance \cite{Michiels2002,Mondie2003}. Finally, a Krylov method for computing characteristic roots of large-scale problems has been proposed in \cite{Jarlebring2010}.

In the past decade a framework for analyzing \textsc{ddes} based on the Lambert W function has been developed \cite{Asl2003,Yi2009,Yi2010d}. It expands the earlier work \cite{Wright1959}. The main idea of the methodology is to express the solution of a \textsc{dde} as the sum of a series of infinitely many exponential functions. The characteristic roots of the system are found analytically in terms of the Lambert W function. While the problem remains infinite dimensional, a one-to-one correspondence between the characteristic roots of the system and the branches of this multi-valued function is assumed. The stability question is then solved by earmarking the dominant characteristic roots of the system with the branches of the Lambert W function corresponding to $k=0,\,\pm1,\,\ldots,\pm m$, where $m$ is the nullity of the matrix $\mathbf{B}$ in \eqref{eq:lti-tds}. Therefore, only a few branches have to be considered to determine whether a solution is stable or not. Furthermore, the existence of an explicit solution expressed in terms of a power series, allows analyzing structural properties like observability and controllability for systems of \textsc{ddes} \cite{Yi2008}, the development of pole placement techniques for control synthesis \cite{Yi2010a,Yi2010b,Yi2010c,Yi2012b} and other applications like the estimation of decay rates \cite{Duan2012} and spectrum design \cite{Wei2014}.

The basic foundation of his methodology, i.e., the assumption that the principal branch of the Lambert W function defines the stability of the system, is well established for first order systems \cite{Shinozaki2006,Asl2003}. For higher order systems, however, this result has not been extended with the same rigor, and is rather based upon observations \cite{Yi2006a,Yi2006b}.

In this paper, we show that the main assumption does not hold in general. By studying  particular second order delay systems, with a structure that is very common in applications, we show that the full spectrum of a \textsc{lti-tds} of this class can be found using only two branches of the Lambert W function, i.e., there is no one-to-one correspondence between the characteristic roots and the branches of the Lambert W function. This is due to the fact that an important nonlinear equation in the approach does not have a unique solution. Furthermore, we show that the principal branch can be used to find not only the dominant root of the system, but some other roots too.

The organization of the paper is as follows. In section \ref{sec:LambW} we present the Lambert W function and its matrix version, required to work with higher order systems. Section \ref{sec:solLamb} reviews the methodology to solve \textsc{ddes} using the Lambert W function, both in the scalar and vector cases. Section \ref{sec:args} presents the analysis of a second order system and contains the main results. This analysis is illustrated by numerical examples in Section \ref{sec:numer}. Section \ref{sec:odd} presents discussions on another special case for which the results of section \ref{sec:args} cannot be directly applied. Finally, some conclusions of the study are given in section \ref{sec:conclusions}.

In the remainder of the paper, scalar quantities are denoted by italic symbols ($a,\,b,\,\lambda$) whereas vectors and matrices are represented by bold face lowercase ($\mathbf{x}$) and uppercase ($\mathbf{A},\,\mathbf{B}$) letters, respectively. The notation $e^{b}$ is used to represent the exponential function of a scalar and $\exp\left(\mathbf{A}\right)$ represents a matrix exponential function. In a similar way, $W_k\left(z\right)$ represents the $k$-th branch of the Lambert W function of a scalar number, and $\mathbf{W}_k\left(\mathbf{A}\right)$ is the matrix Lambert W function.

\section{The Lambert W Function}
\label{sec:LambW}
The Lambert W function is a function $W(z)$, $\mathbb{C}\mapsto\mathbb{C}$, defined as the solution  to the equation
\begin{equation}
W\left(z\right)e^{W\left(z\right)}=z
\label{eq:lambert_scalar}
\end{equation}
This is a multi-valued function, that is, for a $z\in\mathbb{C}$ there are infinitely many solutions to \eqref{eq:lambert_scalar}. To identify these values a branch number is assigned, and we refer to $W_k\left(z\right)$ as the $k$-th branch of the Lambert W function of $z$. The branch cuts are defined in such a way that each branch has a precisely defined range \cite{Corless1996}. For $z\in\mathbb{R}$, only two of the branches are real valued. The principal branch, $W_0\left(z\right)$ is real for $z\geq -1/e$ and its range is the interval $\left[-1,\,\infty\right)$. The branch $W_{-1}\left(z\right)$ is real for $-1/e\leq z<0$, and its range is $\left(-\infty,\,-1\right]$. These branches are shown in Figure \ref{fig:realbranches}.
\begin{figure}
\centering
\includegraphics[scale=0.55]{realbranches.eps}
\caption{The two real branches of the Lambert W function.}
\label{fig:realbranches}
\end{figure}

A comprehensive study of the history, definition and properties of the Lambert W function is found in \cite{Corless1996}.

The matrix Lambert W function is defined now. Consider a matrix $\mathbf{H}\in\mathbb{C}^{n\times n}$, which has the Jordan canonical decomposition $\mathbf{H}=\mathbf{ZJZ}^{-1}$, with $\mathbf{J}=\mathrm{diag}\left(\mathbf{J}_1\left(\lambda_1\right),\,\mathbf{J}_2\left(\lambda_2\right),\,\ldots,\,\mathbf{J}_p\left(\lambda_p\right)\right)$. Following one of the standard definitions for a function of a matrix \cite{Higham2008}, the Matrix Lambert W function for a Jordan block of size $m$ is defined as:
\begin{equation}
\begin{split}
&\mathbf{W}_k\left(\mathbf{J}_i\right)=\\
&\left[\begin{array}{cccc}
W_k\left(\lambda_i\right)&W_k'\left(\lambda_i\right)&\cdots&\frac{1}{\left(m-1\right)!}W_{k}^{m-1}\left(\lambda_i\right)\\
0&W_k\left(\lambda_i\right)&\cdots&\frac{1}{\left(m-2\right)!}W_{k}^{m-2}\left(\lambda_i\right)\\
\vdots&\vdots&\ddots&\vdots\\
0&0&\cdots&W_k\left(\lambda_i\right)\end{array}\right]
\end{split}
\label{eq:wjordan}
\end{equation}

and the matrix Lambert W function of $\mathbf{H}$ is defined as:
\begin{equation}
\begin{split}
\mathbf{W}_k\left(\mathbf{H}\right)=&\mathbf{Z}\,\mathrm{diag}\left(\mathbf{W}_k\left(\mathbf{J}_1\left(\lambda_1\right)\right),\mathbf{W}_k\left(\mathbf{J}_2\left(\lambda_2\right)\right),\ldots,\right.\\
&\left.\mathbf{W}_k\left(\mathbf{J}_p\left(\lambda_p\right)\right)\right)\mathbf{Z}^{-1}.
\end{split}
\label{eq:wmatrix}
\end{equation}

Every matrix defined by \eqref{eq:wmatrix}, for $k=0,\,\pm 1,\,\pm 2\,\ldots$, is one particular solution to the matrix equation
\begin{equation}
\mathbf{W}_k\left(\mathbf{H}\right)\exp\left(\mathbf{W}_k\left(\mathbf{H}\right)\right)=\mathbf{H}.
\label{eq:wlmatrix}
\end{equation}

The above, standard definition implies that the same branch of the Lambert W function is used in each Jordan block. This is not necessary to have a solution of \eqref{eq:wlmatrix}. Since $W_0\left(0\right)=0$ and $W_k\left(0\right)=\infty$ for $k\neq 0$, we deviate from the standard definition to avoid the infinite value. This special case, called the \emph{hybrid branch} case, is defined in \cite{Yi2009,Yi2010d}. More precisely, there $\mathbf{W}_k\left(\mathbf{H}\right)$ uses the actual value of $k$ for those Jordan blocks with $\lambda\neq 0$ and $k=0$ for those blocks in which $\lambda=0$. This definition is used in the remainder of the paper.

In a similar way, it is assumed that $e^{-1}$ is not eigenvalue of $\mathbf{H}$ corresponding to a Jordan block of dimension higher than 1 when the principal branch is being computed. This is required to overcome the difficulty represented by the fact that $W_0'\left(e^{-1}\right)$ is not defined. This limitation reduces the elegance of the definition of the matrix Lambert W function, but does not affect its usage \cite{Jarlebring2007}.
\section{Solution of Delay-Differential Equations using the Lambert W Function}
\label{sec:solLamb}
\subsection{Scalar Case}
For a scalar, homogeneous \textsc{dde},
\begin{equation}
\dot{x}\left(t\right)=ax\left(t\right)+b\left(t-\tau\right),
\label{eq:scalardde}
\end{equation}
the characteristic equation is
\begin{equation}
s-a-be^{-s\tau}=0.
\label{eq:cescalar}
\end{equation}
The solution to \eqref{eq:cescalar} can be expressed in terms of the Lambert W function following simple steps \cite{Corless1996,Asl2003,Yi2009,Yi2010d}. This solution has the form
\begin{equation}
s_k=\frac{1}{\tau}W_k\left(\tau be^{-a\tau}\right)+a,
\label{eq:solscalar}
\end{equation}
where $k=0,\,\pm 1,\,\pm 2,\,\ldots$ indicates the branch of the Lambert W function to be used. Each one of the infinitely many roots of \eqref{eq:cescalar} corresponds to one of the branches of this function.

It has been proven \cite{Shinozaki2006} that among all  solutions in \eqref{eq:solscalar}, the one that corresponds to the principal branch, $k=0$, always has the largest real part and, therefore, introduces the dominant mode to the solution of the equation. To study the stability of the solution to a one dimensional \textsc{dde} as \eqref{eq:scalardde}, it is necessary and sufficient to find only the solution of \eqref{eq:cescalar} in \eqref{eq:solscalar} corresponding to $k=0$.
\subsection{Higher Order case}
Consider now a higher order \textsc{DDE} described by:
\begin{equation}
\dot{\mathbf{x}}\left(t\right)=\mathbf{Ax}\left(t\right)+\mathbf{Bx}\left(t-\tau\right)
\label{eq:ltitds}
\end{equation}
with $\mathbf{x}\left(t\right)\in\mathbb{R}^{n}$, $\mathbf{A},\,\mathbf{B}\in\mathbb{R}^{n\times n}$ and $\tau>0$.

The following steps, introduced in \cite{Asl2003} and extended in \cite{Yi2009,Yi2010d,Yi2006a,Yi2006b}, aim at computing characteristic roots using the matrix Lambert W function. The proposed method is based on finding a solution of the equation
\begin{equation}
\mathbf{S}-\mathbf{A}-\mathbf{B}\exp\left(-\mathbf{S}\tau\right)=\mathbf{0},
\label{eq:char}
\end{equation}
where $\mathbf{S}\in\mathbb{C}^{n\times n}$. A matrix $\mathbf{Q}\in\mathbb{C}^{n\times n}$ is introduced, such that
\begin{equation}
\tau\left(\mathbf{S}-\mathbf{A}\right)\exp\left(\left(\mathbf{S}-\mathbf{A}\right)\tau\right)=\tau\mathbf{B}\mathbf{Q}
\label{eq:4}
\end{equation}
is satisfied. Let us define $\mathbf{M}:=\tau\mathbf{B}\mathbf{Q}$. Then, from \eqref{eq:4} and \eqref{eq:wlmatrix},
\begin{equation}
\mathbf{S}_k=\frac{1}{\tau}\mathbf{W}_k\left(\mathbf{M}\right)+\mathbf{A},
\label{eq:5}
\end{equation}
with $k\in\mathbb{Z}$, is a solution of \eqref{eq:4}. By substituting \eqref{eq:5} into \eqref{eq:char} the following expression is obtained,
\begin{equation}
\mathbf{W}_k\left(\mathbf{M}\right)\exp\left(\mathbf{W}_k\left(\mathbf{M}\right)+\mathbf{A}\tau\right)-\tau\mathbf{B}=\mathbf{0}.
\end{equation}
Therefore, the steps to compute characteristic roots are given by the following algorithm \cite{Yi2009,Yi2010d}.
\begin{algo}\label{algo}
Repeat for $k=0,\,\pm 1,\,\pm 2,\,\ldots$:
\begin{enumerate}
\item Solve the nonlinear equation
\begin{equation}
\mathbf{W}_k\left(\mathbf{M}_k\right)\exp\left(\mathbf{W}_k\left(\mathbf{M}_k\right)+\mathbf{A}\tau\right)-\tau\mathbf{B}=\mathbf{0},
\label{eq:findQ}
\end{equation}
for $\mathbf{M}_k$\ ( $=\tau \mathbf{B}\mathbf{Q}_k$).
\item Compute $\mathbf{S}_k$ corresponding to $\mathbf{M}_k$ as
\begin{equation}
\mathbf{S}_k=\frac{1}{\tau} \mathbf{W}_k(\mathbf{M}_k)+\mathbf{A}.
\end{equation}
\item Compute the eigenvalues of $\mathbf{S}_k$.
\end{enumerate}
\end{algo}

For system \eqref{eq:ltitds} to be stable, all characteristic roots  must have negative real parts. Calculating the solution for all the branches is not possible. To work around this difficulty, the proposers of this methodology assume that for any branch $k$ of the matrix Lambert W function, there is a unique solution $\mathbf{M}_k$ to \eqref{eq:findQ} and a corresponding $\mathbf{S}_k$ matrix. This assumption, based on observations from many examples, leads to a stronger conjecture: when the rank of $\mathbf{B}$ in \eqref{eq:lti-tds} is at least $n-1$, i.e., $\mathbf{B}$ does not have a repeated zero eigenvalue, the characteristic roots with largest real part correspond to the $\mathbf{S}_0$ matrix, found using the principal branch of the matrix Lambert W function in Algorithm~\ref{algo}. This conjecture is formally stated in \cite{Yi2009} and it is the basis for several derivative works \cite{Yi2010a,Yi2010b,Yi2010c,Yi2012a,Yi2012b,Duan2012,Wei2014}.

In the following section, we show that this conjecture does not hold in general. By studying a particular, yet very common, control system, we can show that (\ref{eq:findQ}) may have multiple solutions and there is no one to one correspondence between the branches of the matrix Lambert W function and the characteristic roots of the system. Furthermore, we show that in particular cases it is possible to find \emph{all} characteristic roots using only \emph{two} branches, those corresponding to $k=0$ and $k=-1$. This contradicts the conjecture regarding stability.

\section{A Common Special Case in Second Order Systems}
\label{sec:args}
We consider a single input, second order system under time delayed state feedback. This type of control system is ubiquitous in applications. Without loss of generality, we assume that the system matrix is in companion form. This system is represented by \eqref{eq:ltitds} with the following $\mathbf{A}$ and $\mathbf{B}$ matrices:
\begin{equation}
\mathbf{A}=\left[\begin{array}{cc}0&1\\a_{21}&a_{22}\end{array}\right]\quad\mathbf{B}=\left[\begin{array}{cc}0&0\\b_{21}&b_{22}\end{array}\right].
\label{eq:sys2nd}
\end{equation}
Since $\mathbf{B}$ in \eqref{eq:sys2nd} has nullity 1, the current theory predicts that the solutions to \eqref{eq:findQ} corresponding to $k=0$ and $k=\pm1$ generate only its dominant roots.

From the structure of $\mathbf{B}$, we can see that $\mathbf{M}_k=\tau\mathbf{B}\mathbf{Q}_k$, for any given $\mathbf{Q}_k$, has the form:
\begin{equation}
\mathbf{M}_k=\left[\begin{array}{cc}0&0\\m_{21}&m_{22}\end{array}\right].
\label{eq:m}
\end{equation}
Applying the definition of the matrix Lambert W function, using the hybrid branch case because $\mathbf{M}_k$  has one eigenvalue equal to zero, we get for $m_{22}\neq 0$:
\begin{equation}
\mathbf{W}_k\left(\mathbf{M}_k\right)=\left[\begin{array}{cc}0&0\\\frac{m_{21}}{m_{22}}W_k\left(m_{22}\right)&W_k\left(m_{22}\right)\end{array}\right],
\label{eq:wm}
\end{equation}
and from here we obtain
\begin{equation}
\begin{split}
\mathbf{S}_k&=\frac{1}{\tau}\mathbf{W}_k\left(\mathbf{M}_k\right)+\mathbf{A}\\
&=\left[\begin{array}{cc}0&1\\\frac{m_{21}}{\tau m_{22}}W_k\left(m_{22}\right)+a_{21}&\frac{1}{\tau}W_k\left(m_{22}\right)+a_{22}\end{array}\right].
\end{split}
\label{eq:s}
\end{equation}
In case $m_{22}=0,\ m_{21}\neq 0$, a simple computation yields
\begin{equation}
\mathbf{S}_k=\left[\begin{array}{cc} 0 &1\\ \frac{m_{21}}{\tau}+a_{21} & a_{22}\end{array}\right],
\end{equation}
where we used $W_0^{\prime}(0)=1$. We are now ready to state the main results of the paper.

\begin{proposition} \label{prop1}
Let $A$ and $B$ be given by (\ref{eq:sys2nd}). Let $\{\lambda,\,\bar{\lambda}\}$ be any pair of complex conjugate characteristic roots of the system defined by \eqref{eq:sys2nd}. Assume their multiplicity is one. Then for either $k=0$ or $k=-1$ there exists a real solution of (\ref{eq:findQ}), such that, if this solution and corresponding value of $k$ are selected in the first step of Algorithm~\ref{algo}, the characteristic roots $\lambda$ and $\bar{\lambda}$ are found in the last step of the algorithm.
\end{proposition}

\noindent \textbf{Proof.} The key idea is to perform the steps of Algorithm~\ref{algo} in \emph{reverse} order, in the course of which $k$ is selected.

From the pair $(\lambda,\bar{\lambda})$ we first construct a real matrix $\mathbf{S}_k$, of which they are the eigenvalues, namely
\begin{equation}
\mathbf{S}_k=\left[\begin{array}{cc}0&1\\-\left|\lambda\right|^2&2\Re\left(\lambda\right)\end{array}\right].
\label{eq:skproof}
\end{equation}

Subsequently, we construct $\mathbf{M}_k$ from (\ref{eq:skproof}), where we make distinction between two cases.

\noindent \emph{Case 1: $2\Re(\lambda)\neq a_{22}$.}
Comparing \eqref{eq:s} and \eqref{eq:skproof}, we can take $\mathbf{M}_k$ of the form (\ref{eq:m}), where $m_{21}\in\mathbb{R}$ and $m_{22}\in\mathbb{R}$ are chosen such that the following equations are satisfied:
\begin{align}
W_k\left(m_{22}\right)&=\tau\left(2\Re\left(\lambda\right)-a_{22}\right),\label{eq:wk}\\
m_{21}&=-\frac{m_{22}\left(\left|\lambda\right|^2+a_{21}\right)}{2\Re\left(\lambda\right)-a_{22}}.
\end{align}
Such a choice  is always possible for either $k=0$ or $k=-1$, which then fix $k$.  Equation \eqref{eq:wk} namely implies that $W_k\left(m_{22}\right)$ must be a real number. As mentioned earlier, the definition of the branch cuts of the Lambert W function makes this function to have two real branches: $k=0$, the principal branch, and $k=-1$, and the union of the ranges of $W_0$ and $W_{-1}$ for real valued arguments includes $\mathbb{R}$, see Figure~\ref{fig:realbranches}. Furthermore, in the case considered, $m_{22}$ computed from (\ref{eq:wk}) is different from zero, justifying the use of (\ref{eq:s}).

\noindent \emph{Case 2: $2\Re(\lambda)=a_{22}$.} We can freely choose $k\in\{0,-1\}$ and take
\begin{equation}
\mathbf{M}_k=\left[\begin{array}{cc}0 & 0 \\ -\tau(a_{21}+|\lambda|^2) & 0\end{array}\right].
\end{equation}

Finally, it remains to show that the constructed pair $(k,\mathbf{M}_k)$ is a solution of (\ref{eq:findQ}). Let $\mathbf{v}$, respectively $\bar{\mathbf{v}}$, be the eigenvector corresponding to $\lambda$, respectively $\bar\lambda$. If the characteristic root $\lambda$ is simple, then the pair $(\mathbf{V},\boldsymbol{\Lambda})$ is an invariant pair of (\ref{eq:ltitds}), where
\begin{equation}
\mathbf{V}=[\mathbf{v}\ \bar{\mathbf{v}}],\ \ \boldsymbol{\Lambda}=\mathrm{diag}(\lambda,\bar\lambda).
\end{equation}
As a consequence it satisfies
\begin{equation}\label{invariant1}
\mathbf{V} \boldsymbol{\Lambda}-\mathbf{A}-\mathbf{B} \mathbf{V} \exp(-\boldsymbol{\Lambda})=0,
\end{equation}
see \cite{Beyn2011}.
The eigenvalue decomposition of $\mathbf{S}_k$ in (\ref{eq:skproof}) takes the form
\begin{equation}
\mathbf{S}_k=\mathbf{V} \boldsymbol{\Lambda} \mathbf{V}^{-1},
\end{equation}
from which we have
\begin{equation}
\exp(-\mathbf{S}_k)=\mathbf{V} \exp(-\boldsymbol{\Lambda}) \mathbf{V}^{-1}.
\end{equation}
It follows that
\begin{equation}
\mathbf{V}\boldsymbol{\Lambda}= \mathbf{S}_k \mathbf{V},\ \ \mathbf{V} \exp(-\boldsymbol{\Lambda})=\exp(-\mathbf{S}_k) \mathbf{V}.
\end{equation}
Substituting the latter in (\ref{invariant1}) yields, as $V$ is invertible,
\begin{equation}
\mathbf{S}_k-\mathbf{A}-\mathbf{B}\exp(-\mathbf{S}_k)=0.
\end{equation}
Finally, replacing $\mathbf{S}_k$ by
\begin{equation}
\frac{1}{\tau}\mathbf{W}_k\left(\mathbf{M}_k\right)+\mathbf{A}
\end{equation}
results in (\ref{eq:findQ}).
\hfill $\Box$

\begin{proposition} \label{prop2}
Let $A$ and $B$ be given by (\ref{eq:sys2nd}). Let $\lambda_1\neq \lambda_2$ be two real, simple characteristic roots of the system defined by \eqref{eq:sys2nd}. Then for either $k=0$ or $k=-1$ there exists a real solution of (\ref{eq:findQ}), such that, if this solution and corresponding value of $k$ are selected in the first step of Algorithm~\ref{algo}, the characteristic roots $\lambda_1$ and $\lambda_2$ are found in the last step of the algorithm.
\end{proposition}

\noindent\textbf{Proof.}\  The proof is completely analogous to the proof of Proposition~\ref{prop1}. The differences are that we start by defining
\begin{equation}
\mathbf{S}_k=\left[\begin{array}{cc} 0 &1 \\-\lambda_1\lambda_2 & \lambda_1+\lambda_2 \end{array}\right],
\end{equation}
and that the two cases to be considered are $\lambda_1+\lambda_2=a_{22}$ and $\lambda_1+\lambda_2\neq a_{22}$. \hfill $\Box$

From Propositions \ref{prop1}-\ref{prop2} the following Corollary can be derived.
\begin{corol}\label{cor1}
Let $A$ and $B$ be given by (\ref{eq:sys2nd}).  If all characteristic roots of \eqref{eq:sys2nd} are simple and if the number of real characteristic roots  is different from one, then \emph{all} characteristic roots can be found using only two branches of the matrix Lambert W function in Algorithm~\ref{algo}, namely $k=0$ and $k=-1$. Moreover, one can restrict to the real solutions of (\ref{eq:findQ}).
\end{corol}

\textbf{Remark} The previous analysis shows that, with proper initial conditions, all the characteristic roots of system l\eqref{eq:sys2nd} can be found using the branches corresponding to $k=0$ and $k=-1$. However, higher branch numbers can be used to find different pairs of roots following the same reverse engineering approach, without any particular structure. This is demonstrated in the following section.

\section{Numerical Examples}
\label{sec:numer}
We consider a system defined by matrices:
\begin{equation}
\mathbf{A}=\left[\begin{array}{rr}0&1\\-5&-1\end{array}\right]\quad\mathbf{B}=\left[\begin{array}{rr}0&0\\-3&-0.6\end{array}\right],
\label{eq:examp}
\end{equation}
with a value $\tau=5$ for the time delay.

When the \emph{LambertDDE} toolbox \cite{Yi2012a} was used to calculate the characteristic roots of the system, the software was not able to find a solution of (\ref{eq:findQ}) for any value of $k$ with the default settings. For the numerical solution, the toolbox uses the matrix $\exp\left(-\mathbf{A}\tau\right)$ as an initial estimation of $\mathbf{Q}_k$. In this case, that value turns out not to be in the region of attraction of a solution of \eqref{eq:findQ}.

In order to obtain an \emph{a posteriori} guess for $\mathbf{Q}_k$, we reverse-engineered the solution, as in the proof of Proposition~\ref{prop1}. First, we use the QPmR algorithm \cite{Vhylidal2009} to find the characteristic roots of the system in a region close to the origin of the complex plane. The roots found are shown in Figure \ref{fig:roots}.
\begin{figure}[!tb]
\centering
\includegraphics[scale=0.55]{roots.eps}
\caption{Characteristic roots of the system under study. The roots represented as blue squares can be found using the principal branch of the matrix Lambert W function, whereas those in black circles are found using the branch corresponding to $k=-1$.}
\label{fig:roots}
\end{figure}

The dominant roots of the system are $\lambda=0.0377\pm j1.7911$. Corresponding to these roots, we create the following $\mathbf{S}$ matrix:
\begin{equation}
\mathbf{S}=\left[\begin{array}{cc}0&1\\-3.2096&0.0753\end{array}\right]
\label{eq:s0}
\end{equation}
From \eqref{eq:s}, we have that
\begin{equation}
\mathbf{W}_k\left(\mathbf{M}\right)=\tau(\mathbf{S}-\mathbf{A})=\left[\begin{array}{cc}0&0\\8.9521&5.3766\end{array}\right].
\label{eq:w0}
\end{equation}
This shows that $W\left(m_{22}\right)\in\left[-1,\,\infty\right)$, which is the range of the principal branch of the Lambert W function. There is, therefore, a matrix $\mathbf{M}$ for which \eqref{eq:w0} is satisfied for $k=0$, and that matrix is
\begin{equation}
\mathbf{M}_0=\left[\begin{array}{cc}0&0\\1.9361&1.1628\end{array}\right]\times10^3.
\label{eq:M0}
\end{equation}
Since $\mathbf{B}$ and $\mathbf{M}$ are singular, there are an infinite number of $\mathbf{Q}_0$ matrices that satisfy $\mathbf{M}_0=\tau\mathbf{B}\mathbf{Q}_0$, for \eqref{eq:M0}. One of such matrices is
\begin{equation}
\mathbf{Q}_0=\left[\begin{array}{cc}1&1\\-650.3812&-392.6121\end{array}\right]
\label{eq:Q0}
\end{equation}
When this $\mathbf{Q}_0$ is used as starting value in the \emph{LambertDDE} toolbox for $k=0$, the numerical solution of \eqref{eq:findQ}, corresponding to the dominant roots, is found at the first iteration, as expected. Furthermore, if the matrix is slightly perturbed, the method still converges to the same solution after a few iterations.

Now, let us consider a non-dominant pair of roots: $\lambda=-0.4113\pm j6.4803$. Following a similar reasoning, we obtain the following matrices:
\begin{equation}
\begin{split}
\mathbf{S}&=\left[\begin{array}{cc}0&1\\-42.1633&-0.8226\end{array}\right],\\
\mathbf{W}_k\left(\mathbf{M}\right)&=\left[\begin{array}{cc}0&0\\-185.8166&0.8868\end{array}\right].
\end{split}
\label{eq:nondom}
\end{equation}
For this case, we again have $W\left(m_{22}\right)\in\left[-1,\,\infty\right)$. This implies that this pair of roots can also be found using the principal branch of the matrix Lambert W function. Indeed, if a matrix close to this one,
\begin{equation}
\mathbf{Q}_0=\left[\begin{array}{cc}1&1\\145.3412&-5.7175\end{array}\right],
\label{eq:Q01}
\end{equation}
which was created as in the previous case, is used as initial condition, the numerical routine within the \emph{LambertDDE} toolbox converges to the solution \emph{using the principal branch}.

In fact, we have observed that using suitable initial conditions, the 11 pairs of roots presented as blue squares in Figure \ref{fig:roots} can be found using the principal branch of the matrix Lambert W function.

If we consider now the pair of roots $\lambda=-0.6169\pm j14.0734$, the corresponding $\mathbf{S}$ and $\mathbf{W}_k\left(\mathbf{M}_k\right)$ are
\begin{equation}
\begin{split}
\mathbf{S}&=\left[\begin{array}{cc}0&1\\-198.4405&-1.2338\end{array}\right],\\
\mathbf{W}_k\left(\mathbf{M}\right)&=\left[\begin{array}{cc}0&0\\-967.2027&-1.1692\end{array}\right].
\end{split}
\label{eq:km1}
\end{equation}
In this case, $W\left(m_{22}\right)\in\left(-\infty,-1\right]$, which is the range of the branch indexed by $k=-1$. Using this branch, we obtain a $\mathbf{Q}_{-1}$ matrix,
\begin{equation}
\mathbf{Q}_{-1}=\left[\begin{array}{cc}1&1\\95.1384&-4.8789\end{array}\right],
\label{eq:Qm1}
\end{equation}
which is a solution to \eqref{eq:findQ} for $k=-1$. Selecting an initial condition close to this matrix guarantees convergence to this solution.

This procedure can be repeated for all the roots marked as black squares in Figure \ref{fig:roots}, as well as for roots further to the left of the complex plane, always using the branch corresponding to $k=-1$. This example shows how the whole spectrum of a system with the structure given in \eqref{eq:sys2nd} can be calculated using only two branches of the matrix Lambert W function and properly selected initial conditions for the solution of the nonlinear equation \eqref{eq:findQ}.

To show how higher branch numbers can also be used to find eigenvalues, let us consider a non conjugate eigenvalue pair, with $\lambda_1=-0.0204+j2.7705$ and $\lambda_2=-0.4658+j7.7500$. Notice how one of these eigenvalues was found using $k=0$, whereas the other was found using $k=-1$ in the previous exercise. With this eigenvalues the following matrix is created:
\begin{equation}
\mathbf{W}_k\left(\mathbf{M}\right)=\left[\begin{array}{cc}0&0\\132.3092+j07.2411&2.5693+j52.6026\end{array}\right]
\end{equation}
which has $W_k\left(m_{22}\right)$ in the range of the 9-th branch of the Lambert W function \cite{Corless1996}. Therefore this pair of eigenvalues can be found using $k=9$ and an appropriate initial condition in \eqref{eq:findQ}. Additionally, we have observed that keeping the same $\lambda_2$ and using the complex conjugate of $\lambda_1$, the matrix created is in the range of the branch indexed by $k=4$. This emphasizes the lack of an structured correspondence between the eigenvalues of \eqref{eq:sys2nd} and the branches of the Matrix Lambert W function.

\section{Odd Number of Real Characteristic Roots}
\label{sec:odd}
The previous discussions considered systems for which all the characteristic roots can be paired in such a way that a real $\mathbf{S}$ matrix is produced. This is the case when the complex roots come in conjugate pairs and the number of roots on the real axis is even. When there is an odd number on real roots, this is not possible. In this section, we undertake further discussions on this topic.

Consider the system with
\begin{equation}
\mathbf{A}=\left[\begin{array}{rc}0&1\\-1&0\end{array}\right],\
\mathbf{B}=\left[\begin{array}{cc}0&0\\1&0\end{array}\right],\
\tau=1,
\label{eq:sys2}
\end{equation}
whose characteristic equation is given by
\begin{equation}\label{eq:charvb}
\lambda^2+1-e^{-\lambda}=0.
\end{equation}

From \eqref{eq:charvb} it can be seen that there is only one real characteristic root, at the origin and with multiplicity one. Furthermore, this is the \emph{rightmost} root, as can be seen in Figure~\ref{fig:roots2}.
\begin{figure}[!tb]
\centering
\includegraphics[scale=0.55]{roots2.eps}
\caption{Characteristic roots of the system in\eqref{eq:sys2}}
\label{fig:roots2}
\end{figure}

For this particular system $\mathbf{B}$ has a repeated zero eigenvalue. According to the observations in \cite{Yi2009,Yi2010d}, in this case the dominant root should be found using the principal branch or the branches $k=\pm1$.

Using the ``reverse-engineering" approach for this example, we observe that all the complex conjugate eigenvalues can be obtained using $k=-1$ and a suitable initial condition. However, the dominant root at the origin cannot be found using the principal branch of the Lambert W function. The logic behind Propositions~\ref{prop1}-\ref{prop2} is based on constructing real $\mathbf{S}_k$, which cannot be done with this single real eigenvalue. A formal proof of this statement is given by contradiction. Assume that there is a real solution $\mathbf{M}_k$, and corresponding real $\mathbf{S}_k$, defined by (\ref{eq:5}), having eigenvalue $\lambda=0$. By (\ref{eq:5}) and (\ref{eq:findQ}) this implies that $\mathbf{S}_k$ satisfies (\ref{eq:char}). This implies on its turn that $(I,\mathbf{S})$ is an invariant pair of (\ref{eq:ltitds}), i.e.,~the eigenspace corresponding to the real characteristic roots is two-dimensional.

If we relax the conditions and allow $\mathbf{S}$ to be complex in the method presented earlier, we can match the characteristic root at the origin with any complex eigenvalue of the form $\lambda=a+jb$. This leads to matrices of the form:
\begin{align}
\mathbf{S}&=\left[\begin{array}{cc}0&1\\0&-a-jb\end{array}\right]\\
\mathbf{W}\left(\mathbf{M}\right)&=\left[\begin{array}{cc}0&0\\\tau&-\tau\left(a+jb\right)\end{array}\right]\label{eq:wsingle}.
\end{align}
The matrix given in \eqref{eq:wsingle} can be found using a value of $k$ such that $-j\tau b$ belongs to the range of the $k-$th branch of the Lambert W function. Therefore, for this particular system, the dominant root can be found using \emph{any} branch of the matrix Lambert W function, if a suitable initial condition is provided.

\section{Concluding Remarks}
\label{sec:conclusions}
The methodology for the stability analysis and control synthesis of \textsc{lti-tds} presented in \cite{Yi2009,Yi2010d}, based on the matrix Lambert W function, assumes that there is a one to one correspondence between the characteristic roots of the system and the branches of the matrix Lambert W function. It also assumes that this correspondence is such that using the principal branch always leads to the dominant roots of the system. This paper illustrates that such a correspondence does not exist in all the cases, and that for a particular, albeit very common structure for the system, the branches corresponding to $k=0$ and $k=-1$ can be used to find all the characteristic roots of the system. It is also illustrated that the correspondence cannot always be restored when selecting the particular initial condition $\exp(-\mathbf{A}\tau)$ in solving \eqref{eq:findQ}.

In our examples matrix $\mathbf{B}$ had reduced rank. An interesting path of research is to find out whether the correspondence between characteristic roots and branches of the Lambert W function, holding for scalar systems, can be extended to higher-order systems, provided \emph{additional} structural conditions on the system are assumed.

The MATLAB code used to create the examples can be requested via email to the authors or can be downloaded from \texttt{https://db.tt/mSI3VwbO}.

\section*{Acknowledgments}
This paper is the result of a research visit that R. Cepeda-Gomez made to the Computer Science department at KU Leuven, funded by a Coimbra Group Scholarship for Young Professors and Researchers of Latin America. This scholarship was awarded while he was a lecturer of the Mechatronic Engineering Department at Universidad Santo Tom\'as in Bucaramanga, Colombia.

The work of W. Michiels work is supported by the Programme of Interuniversity Attraction Poles of the Belgian
Federal Science Policy Office (IAP P6-DYSCO), by OPTEC, the Optimization in Engineering Center
of KU Leuven, and by the project G.0712.11N of the Research Foundation-Flanders (FWO).

The authors thank the comments received from the anonymous reviewers, which helped to improve the presentation of the paper, specially in the second example.

\bibliographystyle{plain}
\bibliography{lambert}
\end{document}


TITLE: In an isothermal process how is the heat change non zero?
QUESTION [2 upvotes]: The temperature change in an isothermal process is zero. As a result (if the system is made of an ideal gas) the change in internal energy must also be zero. I don't want to consult the equations. I'm trying to understand this with logic.
Temperature change accounts for heat flow. If there's no change in temperature how can we say the heat supplied to or the heat released from the system is non zero?
My second question : Let's consider a simple example of state change (like the melting of ice) at constant temperature, heat is supplied to the system and the system is in thermal equilibrium, how is work being done here?

REPLY [1 votes]: If there's no change in temperature how can we say the heat supplied
  to or the heat released from the system is non zero?

Heat and temperature are not the same thing. Heat is a mode of energy transfer whereas the latter is a measure of average kinetic energy. Heat flow may or may not change the temperature. If the heat add in to the internal kinetic energy of the system you will observe a change in temperature but if something else happens while, heat is being transferred, which robs the system of its internal kinetic energy, you will observe no change in temperature.
For instance, consider a closed system with moveable piston. If there is no net change in temperature and you wish to find about heat transfer to (or from) the system, you have to look at the work done by (or on) system. According to the first law of thermodynamics,
$$\Delta U=q-P\Delta V$$
For an isothermal process,
$q=P\Delta V$
We have a given condition that the temperature of the system is constant. So if something happens in the system which increases its $U$, something else must happen which could nullify this increment, so that the net effect of such a process brings no change in the internal energy of the system.
Heat supplied or work done on the system increases the internal energy and heat released or work done by the system decreases the internal energy. So if heat is supplied to the system, work has to be done $by$ the system, if the temperature has to remain constant. And these two energy terms operate opposite to each other.

State change (like the melting of ice )at constant temperature, heat
  is supplied to the system and the system is in thermal equilibrium,
  how is work being done here?

$\Delta U$ is zero at constant temperature only for ideal gases. Internal energy contains both kinetic energy and potential energy terms, but since ideal gases do not interact with each other, the potential energy term is removed.
However, in the real world potential energy comes into play. The heat supplied externally adds in to the internal potential energy of ice. No change occurs in the internal kinetic energy of the system, which implies that the temperature remains constant $(\Delta T=0)$.


TITLE: Prove that multiplication by an integer $a$ that is relatively prime to $n$ defines a bijection from $\mathbb{Z}_n-\{0\}$ to itself
QUESTION [0 upvotes]: If gcd$(a,n)=1$, then multiplication by $a$ defines a bijection from $\mathbb{Z}_n-\{0\}$ to itself.
My working:
If $n=p$ a prime, then we can use the Fermat's Little Theorem. If $n$ is not prime in general, I tried the following:
$x\equiv0,1,2,\ldots,n-1$ (mod $n$), then $ax\equiv0,a,2a,\ldots,(n-1)a$ (mod $n$).
Since gcd$(a,n)=1$, we can cancel $a$ on both sides to get back $x\equiv0,1,2,\ldots,n-1$ (mod $n$).
My doubts:
I think my working is wrong. I know what we need to do is to show that all the elements after multiplied by $a$ have different remainders modulo $n$. But I am not sure what theorems to use, I only know some basic elementary number theory theorems. Can we solve this problem by using any simple methods?
Thanks for the help!

REPLY [2 votes]: Sketch:

Step $1$: Prove that the map $x\mapsto ax$ is injective.  Suppose that $ax\equiv ay\pmod n$, then $ax=ay+kn$, so $a(x-y)=kn$.  Therefore, $n\mid a(x-y)$.  Since $a$ is relatively prime to $n$, $n\mid x-y$ and $x\equiv y\pmod n$.
Step $2$: Prove that the map $x\mapsto ax$ is surjective.  Let $z\in(\mathbb{Z}/n)^\times$  Since $a$ and $n$ are relatively prime, there exist $x$ and $y$ so that $ax+ny=1$.  Consider $xz$: under this map, $zx$ maps to $axz=(1-ny)z=z-nyz\equiv z\pmod n$.  Since $z$ is nonzero, $xz$ must also have been nonzero.

One could also use that $(\mathbb{Z}/n)^\times$ is a finite set and injectivity or many other techniques.  Also, $(\mathbb{Z}/n)^\times$ is just another way to write the set that you describe.


TITLE: How to evaluate $\int \frac {x^3} {1+x^6} dx $?
QUESTION [1 upvotes]: How to evaluate $\int \dfrac {x^3} {1+x^6} dx $ ? I am completely at a loss , please help , thanks in advance .

REPLY [5 votes]: HINT:
Set $x^2=y$
$$\int\dfrac{x^3}{1+x^6}dx=\int\dfrac y{1+y^3}\cdot\dfrac{dy}2$$
Now use  partial fraction decomposition $$\dfrac y{1+y^3}=\dfrac A{1+y}+\dfrac{By+C}{1-y+y^2}$$


text:
\begin{document}

\title{Minimal model-universal flows for locally compact Polish groups}
\author{Colin Jahel and Andy Zucker}

\maketitle
\begin{abstract}
Let $G$ be a locally compact Polish group. A metrizable $G$-flow $Y$ is called model-universal if by considering the various invariant probability measures on $Y$, we can recover every free action of $G$ on a standard Lebesgue space up to isomorphism. Weiss has shown that for countable $G$, there exists a minimal, model-universal flow. In this paper, we extend this result to all locally compact Polish groups.
\let\thefootnote\relax\footnote{2020 Mathematics Subject Classification. Primary: 37B05. Secondary: 22D40, 28D15.}
\let\thefootnote\relax\footnote{A.\ Zucker was supported by NSF Grant no.\ DMS 1803489. C.\ Jahel and A.\ Zucker were partially supported by the ANR project AGRUME (ANR-17-CE40-0026).}
\end{abstract}

\section{Introduction}

Topological dynamics can be understood as an attempt to describe the actions of topological groups on compact spaces. Part of this process consists of understanding which objects are generic or universal among the actions of a given group. A famous instance of this is the universal minimal flow, whose existence was proven by Ellis (\cite{Ellis}). This is a minimal flow which maps onto any other minimal flow; by understanding the properties of this one object, we can better understand the collection of all minimal flows. In this paper, we prove the existence of a minimal flow which is universal in a different sense, in that it contains a copy of any measured free action. Similarly, this ``universal minimal model" can help shed light on the dynamical properties of a given locally compact group.

Let $G$ be a locally compact non-compact Polish group. A \emph{$G$-flow} is a continuous $G$-action on a compact space. A $G$-flow is said to be \emph{minimal} if every orbit is dense. If $Y$ is a $G$-flow, then $P_G(Y)$ denotes the $G$-invariant regular Borel probability measures on $Y$.

By a \emph{$G$-system}, we will mean a Borel $G$-action on a standard Lebesgue space $(X, \mu)$ which preserves $\mu$. We say that a $G$-system $(X, \mu)$ is \emph{free} if the set
$$\mathrm{Free}(X):= \{x\in X: \forall g\in (G\setminus \{1_G\})\,\, gx\neq x\}$$
has measure $1$. Because $G$ is locally compact, this set is Borel. Therefore when dealing with free $G$-systems, we will often just assume that $X = \mathrm{Free}(X)$.

If $(X, \mu)$ and $(Y, \nu)$ are $G$-systems, we say that $Y$ is a \emph{factor} of $(X, \mu)$ if there is a Borel, $G$-invariant subset $X'\subseteq X$ with $\mu(X') = 1$ and a Borel, $G$-equivariant map $f\colon X'\to Y$ with $\nu = f_*\mu$.

If we can find $f$ as above which is also injective, then we say that $(X, \mu)$ and $(Y, \nu)$ are \emph{isomorphic} $G$-systems.  We will denote this $(X, \mu)\cong (Y, \nu)$.

A compact metric $G$-flow $Y$ is \emph{weakly model-universal} if for every free $G$-system $(X, \mu)$, there is $\nu\in P_G(Y)$ with $(Y, \nu)$ a factor of $(X, \mu)$. We say that $Y$ is \emph{model-universal} if for every free $G$-system $(X, \mu)$, there is $\nu\in P_G(Y)$ with $(X, \mu)\cong (Y, \nu)$.

The main theorem of this paper is the construction of a minimal, model-universal flow for every locally compact, non-compact Polish group.
\vspace{2 mm}

\begin{theorem}\label{Thm:Princip}
Let $G$ be a locally compact, non-compact Polish group. Then there exists a minimal model-universal flow for $G$.
\end{theorem}
\vspace{2 mm}

This result extends a result of Weiss \cite{W}, who proved Theorem~\ref{Thm:Princip} in the case of countable groups. Weiss uses the slightly different terminology \emph{universal minimal model} to describe this object. More recently, Zucker \cite{Z} gives a new proof of Weiss's result, and it is this proof that we generalize. We remark that Theorem~\ref{Thm:Princip} cannot extend to all Polish groups. As an example, the group $\aut(\mathbb{Q})$ admits no non-trivial minimal flows, while the shift action on $[0,1]^\mathbb{Q}$ equipped with the product Lebesgue measure is a free $G$-system. It would be interesting to understand the precise class of Polish groups for which Theorem~\ref{Thm:Princip} is true.

The universality of the flow from Theorem~\ref{Thm:Princip} is quite different from that of the universal minimal flow $M(G)$. Indeed, $M(G)$ is universal in the sense that it surjects onto every minimal flow, while we construct a minimal flow into which every free $G$-system can be injected. Another difference is uniqueness; while $M(G)$ is unique up to isomorphism, it is shown in \cite{Z} that when $G$ is countable, there are continuum many minimal model-universal flows up to isomorphism. Unfortunately, the proof of this requires some machinery for countable groups that is not yet known to generalize to locally compact groups. However, we strongly suspect that for any locally compact $G$, minimal model-universal flows are not unique.

Another result of \cite{Z} that we do not address here is whether a minimal, model-universal flow can be free. If $Y$ is a minimal flow and $\nu\in P_G(Y)$ is such that $(Y, \nu)$ is a free $G$-system, then $Y$ must be \emph{essentially free}, meaning that $\mathrm{Free}(Y)\subseteq Y$ is dense $G_\delta$. However, Weiss's construction of a minimal model-universal flow left open the question of whether such an object could be free. The first construction of a free minimal model-universal flow was given by Elek \cite{Elek}, and in \cite{Z}, an easy method of transforming any minimal model-universal flow into a free one is provided. The method is roughly as follows: start with $Z$ a minimal model-universal flow, where we note that the construction from \cite{Z} gives a zero-dimensional flow. Then construct an \emph{almost one-one} extension $\pi\colon Y\to Z$ with $Y$ free and so that for every $z\in \mathrm{Free}(Z)$, $|\pi^{-1}(\{z\})| = 1$. Then $\pi\colon \pi^{-1}(\mathrm{Free}(Z))\to \mathrm{Free}(Z)$ is a $G$-equivariant homeomorphism, and thus the map $\pi^{-1}\colon \mathrm{Free}(Z)\to Y$ will show that $Y$ is also model-universal. Unfortunately, it is essential that $Z$ be zero-dimensional for this to work, and our construction here does not produce zero-dimensional flows (indeed this is impossible when $G$ is connected). A simple example is provided by Antonyan \cite{Ant}; let $G = \mathbb{Z}/2\mathbb{Z}$ and set $Z = [-1, 1]^\omega$. $G$ acts on $Z$ by negating every coordinate. Hence $\mathrm{Free}(Z) = Z\setminus \{0\}$, where $0 = (0,0,0,...)$. However, Antonyan shows that $Z\setminus \{0\}$ does not embed as a $G$-subspace of any free $G$-flow. Therefore any soft method of transforming a minimal model-universal flow into a free one must use a different method to work for all locally compact groups.

One immediate consequence of Theorem~\ref{Thm:Princip} is a result that was suggested by Angel, Kechris and Lyons in \cite{AKL}. Recall that a topological group $G$ is said to be \emph{uniquely ergodic} if all minimal $G$-flows admit exactly one $G$-invariant probability measure.
\vspace{2 mm}

\begin{theorem}\label{Thm:nonUE}
Let $G$ be a locally compact non-compact Polish group.  Then there is a minimal $G$-flow with multiple invariant probability measures. In particular, $G$ is not uniquely ergodic.
\end{theorem}

\begin{proof}
Let $Z$ be a minimal model-universal flow for $G$. It suffices to show that $Z$ admits an invariant probability measure that is not ergodic. Take $(X, \mu)$ a free $G$-system (see \cite{AEG}, Proposition 1.2). Then $G$ acts on $X\times 2$ by acting on the first coordinate. Letting $\delta_{1/2}$ be the $(1/2, 1/2)$-measure on $2$, then $(X\times 2, \mu\times \delta_{1/2})$ is a free, non-ergodic $G$-system. So letting $\nu\in P_G(Z)$ be chosen so that $(Z, \nu)\cong (X\times 2, \mu\times \delta_{1/2})$, we see that $\nu$ is not ergodic.
\end{proof}

\section{Preliminaries}
Fix a non-compact, locally compact Polish group $G$, on which we fix a right-invariant compatible metric $d$. Throughout we assume that $\{g\in G: d(g, 1_G)\leq 1\}$ is compact.
\vspace{2 mm}

\begin{defin}
	\label{Def:Lip}
	If $(X, d_X)$ and $(Y, d_Y)$ are compact metric spaces, $\lip(Y, X)$ denotes the space of Lipschitz functions from $Y$ to $X$ with the topology of pointwise convergence. If $f\in \lip(G, X)$ and $g, h\in G$, we set
	$$(g\cdot f)(h) = f(hg).$$
	This action turns $\lip(G, X)$ into a $G$-flow. Given a subflow $Y\subseteq \lip(G, X)$ and any $B\subseteq G$, we set $$S_B(Y) := \{y|_B: y\in Y\}.$$
	Notice that $S_B(Y)\subseteq \lip(B, X)$ is a compact metric space; when $B\subseteq G$ is pre-compact, we will use the uniform metric
	$$d_B(u, v):= \sup\{d_X(u(g), v(g)): g\in B\}.$$
	If $u\in S_B(Y)$ and $g\in G$, we define $g\cdot u \in S_{Bg^{-1}}(Y)$ via $(g\cdot u)(h) = u(hg)$. If $A\subseteq G$ is another subset, we set
	$$A|B := \{g\in G: A\subseteq Bg^{-1}\}.$$
	In particular, if $g\in A|B$ and $u\in S_B(Y)$, then $(g\cdot u)|_A\in S_A(Y)$.
\end{defin}

\begin{rem}
	If $(X, d_X)$ and $(Y, d_Y)$ are metric spaces, we will equip $X\times Y$ with the metric $d_{X\times Y}((x_0, y_0), (x_1, y_1)) = \max\{d_X(x_0, x_1), d_Y(y_0, y_1)\}$.
\end{rem}
\vspace{2 mm}

We now spend some time understanding minimality in subflows of $\lip(G, X)$.
\vspace{2 mm}

\begin{defin}
	\label{Def:MinimalFragment}
	Let $(X, d_X)$ be a compact metric space. If $\epsilon > 0$ and $S\subseteq X$, we say that $S\subseteq X$ is \emph{$\epsilon$-dense} if for every $x\in X$, there is $s\in S$ with $d_X(x, s)< \epsilon$.

	Now fix a pre-compact $D\subseteq G$ and $\epsilon > 0$. We say that a subflow $Y\subseteq \lip(G, X)$ is \emph{$(D, \epsilon)$-minimal} if for any $y\in Y$, we have that $\{(g\cdot y)|_D: g\in G\}\subseteq S_D(Y)$ is $\epsilon$-dense.
\end{defin}
\vspace{2 mm}

We remark that this notion is monotone; if $Y$ is $(D, \epsilon)$-minimal, then also $Y$ is $(D_0, \epsilon_0)$-minimal for any $D_0\subseteq D$ and $\epsilon_0\geq \epsilon$.
\vspace{2 mm}

\begin{prop}
	\label{Prop:MinimalFragment}
	With notation as in Definition~\ref{Def:MinimalFragment}, the following are equivalent.
	\begin{enumerate}
		\item
		The subflow $Y\subseteq \lip(G, X)$ is $(D, \epsilon)$-minimal.
		\item
		There is a pre-compact open $E\subseteq G$ so that for any $u\in S_E(Y)$, we have that \newline $\{(g\cdot u)|_D: g\in D|E\}\subseteq S_D(Y)$ is $\epsilon$-dense.
	\end{enumerate}
\end{prop}

\begin{proof}
	Item $2$ certainly implies item $1$. Conversely, if Item $2$ fails, let $E_n\subseteq G$ be an exhaustion of $G$ by pre-compact open sets, and find $u_n\in S_{E_n}(Y)$ and $v_n\in S_D(Y)$ with $d_D((g\cdot u_n)|_D, v_n)\geq \epsilon$ for every $g\in D|E_n$. Passing to a subsequence, let $u_n\to y\in Y$ and $v_n\to v\in S_D(Y)$. Towards a contradiction, suppose there were $g\in G$ with $d_D((g\cdot y)|_D, v)< \epsilon$. Then for some $N< \omega$, we have $g\in D|E_n$ for every $n\geq N$. But then we must have $d_D((g\cdot u_n)|_D, v_n)< \epsilon$ for some $n$, a contradiction. Hence $Y$ is not $(D, \epsilon)$-minimal.
\end{proof}
\vspace{2 mm}

\begin{prop}
	\label{Prop:MinimalGDelta}
	With notation as in Definition~\ref{Def:MinimalFragment}, the subflow $Y\subseteq \lip(G, X)$ is minimal iff it is $(D, \epsilon)$-minimal for every pre-compact $D\subseteq G$ and $\epsilon > 0$.
\end{prop}

\begin{proof}
	If $Y$ is minimal, then fix pre-compact $D\subseteq G$, $\epsilon > 0$, and $y\in Y$. If $u\in S_D(Y)$, find $z\in Y$ with $z|_D = u$. Find $g_n\in G$ with $g_n\cdot y\to z$. It follows that we must have $(g_n\cdot y)|_D \to u$, implying that $d_D((g_n\cdot y)|_D, u)< \epsilon$ for some $n< \omega$. Hence $Y$ is $(D, \epsilon)$-minimal.

	Now assume that $Y$ is $(D, \epsilon)$-minimal for every pre-compact $D\subseteq G$ and $\epsilon > 0$. Let $E_n\subseteq G$ be an exhaustion of $G$ by pre-compact open sets, and let $\epsilon_n\to 0$. Fix $y, z\in Y$. We can then find for each $n< \omega$ some $g_n\in G$ with $d_{E_n}((g_n\cdot y)|_{E_n}, z|_{E_n}) < \epsilon_n$. It follows that $g_n\cdot y\to z$, showing that $Y$ is minimal.
\end{proof}
\vspace{2 mm}

The remainder of this section looks at other important flows closely related to $\lip(G, X)$.
\vspace{2 mm}

\begin{defin}
	\label{Def:ClosedSetsG}
We let $2^G$ denote the space of closed subsets of $G$ with the Fell topology. If $B\subseteq G$, we set $\mathrm{Meets}(B) := \{F\in 2^G: F\cap B \neq \emptyset\}$ and $\mathrm{Avoids}(B) := \{F\in 2^G: F\cap B = \emptyset\}$. By considering the sets $\mathrm{Meets}(U)$ and $\mathrm{Avoids}(C)$ for $U\subseteq G$ open and $C\subseteq G$ compact, we obtain a sub-basis for the Fell topology. The action we take is not quite the obvious one: given $g\in G$ and $F\in 2^G$, we set $g\boldsymbol{\cdot} F = Fg^{-1}$. We do this for the following reason: letting $\iota\colon 2^G\to \lip(G, [0,1])$ denote the map $\iota(S)(g) = d(g, S)$, then $\iota$ is an injective $G$-map.
\end{defin}
\vspace{2 mm}

\begin{defin}
	\label{Def:MaxSpacedSets}
	Fix $D\subseteq G$ any symmetric subset. Then a set $S\subseteq G$ is \emph{$D$-spaced} if $Dg\cap Dh = \emptyset$ for any $g\neq h\in S$. Notice that $S\subseteq G$ is a maximal $D$-spaced set iff $S$ is $D$-spaced and $D^2S = G$.

	Now suppose $D\subseteq G$ is a compact symmetric neighborhood of the identity. We let $G_D\subseteq 2^G$ denote the collection of $S\subseteq G$ which are $\mathrm{Int}(D)$-spaced and with $D^2S = G$. In particular, $G_D$ contains every maximal $D$-spaced set and every maximal $\mathrm{Int}(D)$-spaced set.
\end{defin}

\begin{prop}
	\label{Prop:GDWeak}
	For every $D\subseteq G$ a compact symmetric neighborhood of the identity, the $G$-flow $G_D$ is weakly model-universal.
\end{prop}

\begin{proof}
This fact follows from a theorem of Kechris and a theorem of Slutsky. Suppose $G\times X\to X$ is a standard Borel $G$-space. A \emph{cross-section} is any Borel $C\subseteq X$ so that $G\cdot C = X$.  If $D\subseteq G$ is a compact neighborhood of the identity, we say that a cross-section $C$ is \emph{$D$-lacunary} if whenever $x\neq y\in C$, we have $Dx\cap Dy = \emptyset$. A \emph{maximal $D$-lacunary cross-section} is a $D$-lacunary cross-section $C$ such that for any $x\not\in C$, $C\cup\{x\}$ is not $D$-lacunary. Notice that if $C$ is a maximal $D$-lacunary cross-section, then $D^2C = X$.

Kechris in \cite{Kechris} proves the following.
\vspace{2 mm}

\begin{theorem}
	\label{Thm:KechrisLacunary}
	Let $G$ be a locally compact group and $D$ a compact neighborhood of the identity. Then any standard Borel $G$-space $X$ admits a $D$-lacunary cross section.
\end{theorem}
\vspace{2 mm}

The measurable version of this result is a classical result of Feldman, Hahn, and Moore \cite{FHM}; however, we appeal to the Borel version as we will need the following strengthening due to Slutsky (\cite{Slutsky}, Theorem 2.4).
\vspace{2 mm}

\begin{theorem}
	\label{Thm:SlutskyMaxLacunary}
	Let $G$ be a locally compact group, and let $X$ be a standard Borel $G$-space. Then if $D$ is a compact neighborhood of the identity and $C\subseteq X$ is a $D$-lacunary cross-section of $X$, there is a maximal $D$-lacunary cross section $C'\supseteq C$.
\end{theorem}
\vspace{2 mm}

Now fix $(X, \mu)$ a free $G$-system. Applying Theorem~\ref{Thm:SlutskyMaxLacunary}, let $C\subseteq X$ be a maximal $D$-lacunary cross section. We define a Borel $G$-map $f: X\to G_D$ by setting
$$f(x) = C_x:= \{g\in G: gx\in C\}.$$
As $C$ is a maximal $D$-lacunary cross-section, we have $C_x\in G_D$, and certainly $f$ is $G$-equivariant. To see that $f$ is Borel, suppose $B\subseteq G$ is Borel, and consider $\mathrm{Meets}(B)\subseteq 2^G$. Then $f(x)\in \mathrm{Meets}(B)$ iff $Bx\cap C\neq \emptyset$. The set $Y:= \{(g, x): g\in B \text{ and } gx\in C\}\subseteq G\times X$ is Borel, and $f^{-1}(\mathrm{Meets}(B)) = \pi_X[Y]$. We note that since $C$ is $D$-lacunary, the projection $\pi_X$ is countable-to-one on $Y$, showing that $f^{-1}(\mathrm{Meets}(B))$ is Borel.
\end{proof}

\section{A (non minimal) model-universal flow}

In this section, we prove
\vspace{2 mm}

\begin{theorem} \label{Thm:BetterVarad}
	The $G$-flow $\lip(G, [0,1])^\omega$ is model-universal.
\end{theorem}
\vspace{2 mm}

The proof is an adaptation of the proof of a result of Varadarajan.
\vspace{2 mm}

\begin{theorem}[Varadarajan \cite{V}]
	\label{Thm:Varad}
	Let $G$ be a locally compact Polish group, and let $G\times X\to X$ be a Borel action of $G$ on a standard Borel space $X$. Then there is a compact metric $G$-flow $Y$ and a $G$-equivariant Borel injection $X\hookrightarrow Y$.
\end{theorem}
\vspace{2 mm}

Write $\lip := \lip(G, [0,1])$, and let $\lambda$ denote the left Haar measure on $G$.
\vspace{2 mm}

\begin{defin}
	\label{Def:Convolution}
	Suppose $f\colon X\to [0,1]$ is Borel. We let $f_x\colon G\to [0,1]$ be the Borel function given by $f_x(g) := f(gx)$. If $\phi\colon G\to \mathbb{R}^+$ is in $L^1(G, \lambda)$ with $\|\phi\|_1 \leq 1$, we let $(\phi*f)_x\colon G\to [0,1]$ be defined via
	$$(\phi*f)_x(g) = \int_G \phi(h)f(h^{-1}gx) d\lambda(h)$$
\end{defin}
\vspace{2 mm}

\begin{lemma}
	\label{Lemma:LipschitzConv}
	With notation as in Definition~\ref{Def:Convolution}, suppose $\lambda(\mathrm{Supp}(\phi)) \leq L$ and that $\phi$ is $K$-Lipschitz. Then $(\phi*f)_x$ is $2L\cdot K\cdot \|\phi\|_1$-Lipschitz.
\end{lemma}

\begin{proof}
	Fix $g_0, g_1\in G$. By considering the change of variables $h\to g_ih$, we see that
	$$\left|(\phi*f)_x(g_0) - (\phi*f)_x(g_1)\right| \leq \int_G \left|\phi(g_0h)-\phi(g_1h)\right|f(h^{-1}x)d\lambda(h)$$
	The right hand side is identically zero whenever $h\not\in g_0^{-1}\cdot\mathrm{Supp}(\phi)\cup g_1^{-1}\cdot \mathrm{Supp}(\phi)$. For $h$ inside this set, the integrand is at most $K\cdot \|\phi\|_1\cdot d(g_0, g_1)$.
\end{proof}
\vspace{2 mm}

In particular, suppose $\phi$ is such that $(\phi*f)_x\in \lip$ for every $x\in X$. Then the map $\phi*f\colon X\to \lip$ given by $\phi*f(x) = (\phi*f)_x$ is Borel and $G$-equivariant.

Recall that a sequence $(\phi_n)_n$ from $L^1(G, \lambda)$ is an \emph{approximate identity} if $\phi_n \geq 0$, $\phi_n$ is symmetric, $\|\phi_n\|_1 = 1$, $\mathrm{Supp}(\phi_n)$ is compact, and $\mathrm{Supp}(\phi_n)\to \{1_G\}$.
\vspace{2 mm}

\begin{fact}[\cite{F}, Proposition 2.44]
	\label{Fact:ApproxIdDistinguish}
	Suppose $f\colon [0,1]\to G$ is Borel, and let $(\phi_n)_n$ be an approximate identity. Then for any $x\in X$ and any compact $K\subseteq G$, we have that $\|(\phi_n*f)_x\cdot \chi_K - f_x\cdot \chi_K\|_1\to 0$.
\end{fact}
\vspace{2 mm}

We can now work towards our proof of Theorem~\ref{Thm:BetterVarad}. Let $\{f_k: k< \omega\}$ be a sequence of characteristic functions of Borel subsets of $X$ which generate $X$. Let $(\phi_n)_n$ be an approximate identity with each $\phi_n$ $C_n$-Lipschitz for some $C_n\in \mathbb{R}^+$. Using Lemma~\ref{Lemma:LipschitzConv}, choose constants $c_n > 0$ so that $(c_n\phi_n*f_k)_x\in \lip$ for every $n, k< \omega$ and $x\in X$. We define the map $\gamma\colon X\to \lip^{\omega\times \omega}\cong \lip^\omega$ by setting $\gamma(x)(n, k) = (c_n\phi_n*f_k)_x$. Then $\gamma$ is Borel and $G$-equivariant, and we need only check that it is injective. Suppose $\gamma(x) = \gamma(y)$. Using Fact~\ref{Fact:ApproxIdDistinguish}, this implies that for each $k< \omega$, $(f_k)_x(g) = (f_k)_y(g)$ for $\lambda$-almost every $g\in G$. So for most $g\in G$, this is true for every $k< \omega$. Fix such a $g\in G$. But then $gx = gy$, since the $f_k$ separate points. In particular, also $x = y$.

\section{A minimal model-universal flow}

In this section we prove our main theorem.

\begin{defin}
	Suppose $D\subseteq G$ is a compact symmetric neighborhood of the identity. We call $S_0, S_1\subseteq G$ \emph{$D$-apart} if $DS_0\cap DS_1 = \emptyset$.

	Suppose $(X, d_X)$ is a compact metric space. We call a subflow $Y\subseteq \lip(G, X)$ \emph{$D$-irreducible} if whenever $S_0, S_1\subseteq G$ are $D$-apart and whenever $y_0, y_1\in Y$, there is $z\in Y$ with $z|_{S_i} = y_i|_{S_i}$.
\end{defin}
\vspace{2 mm}

We note that if $d_X$ has diameter $1$ and $D\supseteq \{g\in G: d(g, 1_G) \leq 1\}$, then $\lip(G, X)$ is $D$-irreducible.
\vspace{2 mm}

\begin{defin}
	\label{Def:Theta}
	Suppose $(X, d_X)$ is a compact metric space of diameter $1$, suppose $D\supseteq \{g\in G: d(g, 1_G)\leq 1\}$ is a compact symmetric neighborhood of $1_G$, and let $Y\subseteq \lip(G, X)$ be a $D$-irreducible subflow.

	Fix $E\subseteq G$ another compact symmetric neighborhood of $1_G$ with $D^2\subseteq E$. Fix some $u\in S_E(Y)$. Suppose $F\subseteq G$ is yet another compact symmetric neighborhood of $1_G$ with $E^3\subseteq F$. We define the flow
	$$\Theta(Y, u, F)\subseteq \lip(G, X)$$
	to consist of those functions $f\in \lip(G, X)$ so that all of the following hold:
	\begin{enumerate}
		\item
		There is $T\in G_F$ so that $(g\cdot f)|_E = u$ for each $g\in T$.
		\item
		There is $y\in Y$ with $f(g) = y(g)$ for any $g\not\in E^2T$.
		\item
		For every $g\in G$, we have $(g\cdot f)|_D \in S_D(Y)$.
	\end{enumerate}
\end{defin}
\vspace{2 mm}

If $f_n\in \Theta(Y, u, F)$, where items $1$ and $2$ are witnessed by $T_n\in G_F$ and $y_n\in Y$, respectively, then suppose $f_n\to f\in \lip(G, X)$. To show that $f\in \Theta(Y, u, F)$, we first note that item $3$ is a closed condition. Then pass to a subsequence with $T_n\to T\in G_F$ and $y_n\to y\in Y$. Then $T$ and $y$ will witness that items $1$ and $2$ hold for $f$, showing that $\Theta(Y, u, F)$ is closed. To see that $\Theta(Y, u, F)$ is $G$-invariant, take $f\in \Theta(Y, u, F)$ and $g\in G$. Item $3$ is clear for $g\cdot f$, and if $T\in G_F$ and $y\in Y$ witness items $1$ and $2$ for $f$, then $g\boldsymbol{\cdot} T = Tg^{-1}$ and $g\cdot y$ will be witnesses for $g\cdot f$.

We remark that since $Y$ is $D$-irreducible, the flow $\Theta(Y, u, F)$ is non-empty; the next proposition shows this and more.
\vspace{2 mm}

\begin{prop}
	\label{Prop:Uniformize}
	In the setting of Definition~\ref{Def:Theta}, suppose in addition that $Y$ is weakly model-universal. Then so is $\Theta(Y, u, F)$.
\end{prop}

\begin{proof}
	First consider the restriction map $S_{E^3}(Y)\to S_{E^3\setminus E^2}\times S_E(Y)$. Since $Y$ is $D$-irreducible, this map is surjective. As this is a continuous surjection between compact metric spaces, let $\eta\colon S_{E^3\setminus E^2}(Y)\times S_E(Y) \to S_{E^3}(Y)$ be a Borel section.

	We now define a map $\theta\colon Y\times G_F\to \Theta(Y, u, F)$, where given $y\in Y$ and $T\in G_F$, $\theta(y, T)$ is defined as follows.
	\begin{enumerate}
		\item
		If $g\not\in E^2T$, then $\theta(y, T)(g) = y(g)$.
		\item
		If $g = hk$ for some $h\in E^3$ and $k\in T$, we set $\theta(y, T)(g) = \eta((k\cdot y)|_{E^3\setminus E^2}, u)(h)$.
	\end{enumerate}

Then $\theta$ is a Borel $G$-equivariant map. We check that $\theta(y, T)\in \Theta(Y, u, F)$. By construction, items 1 and 2 from Definition~\ref{Def:Theta} are satisfied. For item 3, consider $g\in G$. Since $D^2\subseteq E$, either $Dg\cap E^2T = \emptyset$ or $Dg\subseteq E^3k$ for some $k\in T$. If $Dg\cap E^2T = \emptyset$, then $(g\cdot \theta(y, T))|_D = (g\cdot y)|_D\in S_D(Y)$. If $D_g\subseteq E^3k$ for some $k\in T$, then
$$\left(g\cdot \theta(y, T)\right)|_D = \left(gk^{-1}\cdot \eta\left((k\cdot y)|_{E^3}, u\right)\right)\bigg|_D \in S_D(Y).$$
Lastly, to see that $\theta(y, T)\in \lip(G, X)$, we note that by assumption $D\supseteq \{g\in G: d(g, 1_G)\leq 1\}$, and we have just seen that item 3 holds.

Since $Y\times G_F$ is weakly model-universal, then so is $\Theta(Y, u, F)$.
\end{proof}
\vspace{2 mm}

We will sometimes want to refer to the $\theta$ constructed here as the various parameters change. In this case, we will refer to the map as $\theta\langle Y, u, F\rangle$.
\vspace{2 mm}

\begin{prop}
	\label{Prop:Irreducible}
	In the setting of Definition~\ref{Def:Theta}, $\Theta(Y, u, F)$ is $F^6$-irreducible.
\end{prop}

\begin{proof}
	Let $f_0, f_1\in \Theta(Y, u, F)$, where the membership of $f_i$ is witnessed by $T_i\in G_F$ and $y_i\in Y$. Suppose $S_0, S_1\subseteq G$ are $F^6$-apart. Set $T = (F^5S_0\cap T_0)\cup (F^5S_1\cap T_1)$. Then $T$ is an $\mathrm{Int}(F)$-spaced set, so let $U\supseteq T$ be a maximal $\mathrm{Int}(F)$-spaced set, and let $V = (U\setminus F(S_0\cup S_1))\cup T$. So in particular, $V\cap FS_i = T_i\cap FS_i$.

	We claim that $V\in G_F$. Since $V\subseteq U$, $V$ is $\mathrm{Int}(F)$-spaced. To see that $F^2V = G$, let $g\in G$. If $g\not\in F^3(S_0\cup S_1)$, then $g\in \mathrm{Int}(F)^2h$ for some $h\in U$, and since $h\not\in F(S_0\cup S_1)$, we have $h\in V$. If $g\in F^3S_i$, then $g\in F^2h$ for some $h\in (T_i\cap F^5S_i) \subseteq T\subseteq V$.

	Since $Y$ is $D$-irreducible and $D\subseteq F$, let $y\in Y$ be chosen with $y|_{F^5S_i} = y_i|_{F^5S_i}$. Using $V$ and $y$, we define the $f\in \Theta(Y, u, F)$ which will satisfy $f|_{S_i} = f_i|_{S_i}$. We set $(g\cdot f)|_E = u$ for each $g\in V$, and we set $f(g) = y(g)$ whenever $g\not\in E^2V$. It remains to define $f$ on $(E^2\setminus E)g$ for $g\in V$. If $g\in V$ and $E^2g\cap S_i \neq \emptyset$, we set $f|_{E^2g} = f_i|_{E^2g}$. If $g\in V$ and $E^2g\cap (S_0\cup S_1) = \emptyset$, then we use the $D$-irreducibility of $Y$ to find any $v_g\in S_{E^3}(Y)$ with $v_g|_{E^3\setminus E^2} = (g\cdot f)|_{E^3\setminus E^2}$ and $v_g|_E = u$, and we set $(g\cdot f)(h) = v_g(h)$ for $h\in E^3$.

	We verify that $f$ is as desired. Since $V\cap FS_i = T_i\cap FS_i$, we have $f|_{S_i} = f_i|_{S_i}$. To check that $f\in \Theta(Y, u, F)$, items $1$ and $2$ are witnessed by $V$ and $y$. For item $3$, let $g\in G$. Then either $Dg\cap E^2V = \emptyset$, in which case $(g\cdot f)|_E = (g\cdot y)|_E\in S_E(Y)$, or $Dg\subseteq E^3h$ for some $h\in V$, in which case $(g\cdot f)|_E = (gh^{-1})\cdot v_h|_E\in S_E(Y)$.
\end{proof}

\vspace{4 mm}

We can now undertake the main construction. We set $\lip = \lip(G, [0,1])$. So note in particular that $\lip^n = \lip(G, [0,1]^n)$, and we can treat $\lip^0$ as the trivial (singleton) $G$-flow. We will inductively construct the following objects:
\begin{itemize}
	\item
	$D_n, E_n, F_n\subseteq G$ compact symmetric neighborhoods of the identity,
	\item
	A subflow $Y_n\subseteq \lip^n$ so that $Y_n\times \lip\subseteq \lip^{n+1}$ is $D_n$-irreducible.
\end{itemize}

The sets $D_n, E_n, F_n\subseteq G$ will have the following properties:

\begin{enumerate}
	\item
	$D_n^2\subseteq E_n$, $E_n^3\subseteq F_n$, $F_n^6\subseteq D_{n+1}$
	\item
	$\bigcup_n D_n = G$
	\item
	Suppose $K_n< \omega$ is such that $S_{D_n}(Y_n\times \lip)$ can be covered by $K_n$-many balls of radius $1/2^n$. Then there is a $D_n^2$-spaced set $\{g_{n,i}: i< K_n\}\subseteq G$ so that $D_ng_{n,i}\subseteq E_n$ for each $i< K_n$.
	\item
	There is a $E_n^4$-spaced set $\{h_{n,i}: i< 2^n\}\subseteq G$ so that $E_n^4h_{n,i}\subseteq \mathrm{Int}(F_n)$ for each $i< 2^n$.
\end{enumerate}

We remark that the construction of the sets $D_n, E_n, F_n\subseteq G$ requires that $G$ be non-compact, especially in regards to items 3 and 4. Start by setting $D_0 = \{g\in G: d(g, 1_G)\leq 1\}$ and $Y_0 = \lip^0$.

Suppose we have defined $Y_0,...,Y_n$; $D_0,...,D_n$; $E_0,...,E_{n-1}$; and $F_0,...,F_{n-1}$. In particular, we have arranged so far that $Y_n\times \lip\subseteq \lip^{n+1}$ is $D_n$-irreducible. Find $E_n\subseteq G$ satisfying items $(1)$ and $(3)$.
\vspace{2 mm}

\begin{lemma}
	\label{Lem:ConstructString}
	There is $u_n\in S_{E_n}(Y_n\times \lip)$ so that $$\{(g\cdot u_n)|_{D_n}\colon g\in D_n|E_n\}\subseteq S_{D_n}(Y_n\times \lip)$$
	is $(1/2^n)$-dense.
\end{lemma}

\begin{proof}
	By assumption, $Y_n\times \lip$ is $D_n$-irreducible. Suppose $K_n< \omega$ is as in item 3, and let $\{f_i: i< K_n\}\subseteq S_{D_n}(Y_n\times \lip)$ be chosen so that every $f\in S_{D_n}(Y_n\times \lip)$ satisfies $d_{D_n}(f, f_i)< 1/2^n$ for some $i< K_n$. Let $\{g_{n,i}: i< K_n\}\subseteq G$ be as guaranteed by item 3. Then $Dg_{n,i}$ and $Dg_{n, j}$ are $D_n$-apart whenever $i\neq j < K_n$. Using $D_n$-irreducibility, we can find $u_n\in S_{E_n}(Y_n\times \lip)$ so that $(g_{n,i}\cdot u_n)|_{D_n} = f_i$ for every $i< K_n$. Then $u_n$ is as desired.
\end{proof}
\vspace{2 mm}

Now find $F_n\subseteq G$ satisfying items $(1)$ and $(4)$, and form
$$Y_{n+1} := \Theta(Y_n\times L_1, u_n, F_n)\subseteq L_{n+1}.$$
Then find $D_{n+1}\subseteq G$ satisfying item $(1)$. By Proposition~\ref{Prop:Irreducible}, $Y_{n+1}\times \lip$ is $D_{n+1}$-irreducible.

We can regard each $Y_n$ as a subflow of $\lip^\omega$ by adding a tail of constant zero functions. Conversely, if $m< n\leq \omega$, we let $\pi^n_m\colon [0,1]^n\to [0,1]^m$ denote projection onto the first $m$ coordinates. So if $Z\subseteq \lip^\omega$ is a subflow, we let $\pi^\omega_n\circ Z$ denote its projection to a subflow of $\lip^n$. We now consider the space $\mathrm{Sub}(\lip^\omega)$ of subflows of $\lip^\omega$ equipped with the Vietoris topology. In this topology, we have $Z_n\to Z$ iff for each compact $D\subseteq G$ and $m< \omega$, we have $S_D(\pi^n_m\circ Z_n)\to^n S_D(\pi^\omega_m\circ Z)$ in the space $K(\lip(D, [0,1]^m))$, the space of compact subsets of $\lip(D, [0,1]^m)$ also equipped with the Vietoris topology.
\vspace{2 mm}

\begin{lemma}
	\label{Lem:ConvergeToMinimal}
	In the space $\mathrm{Sub}(\lip^\omega)$, we have $Y_n\to Z$ for some minimal flow $Z$.
\end{lemma}

\begin{proof}
	For notation, set $S(k, m, n) := S_{D_k}(\pi^n_m\circ Y_n)$. Notice that when $k\leq n$, we have that $S(k, m, n+1)\subseteq S(k, m, n)$ by item $3$ in Definition~\ref{Def:Theta}. So it follows that $S(k, m, n)\to^n S(k, m)$ for some compact $S(k, m)\subseteq \lip(D_k, [0,1]^m)$. Also notice that $\pi^{m+1}_m\circ S(k, m+1) = S(k, m)$. Furthermore, if $\ell \geq k$ and $m< \omega$, let
	$$\rho^l_k\colon \lip(D_\ell, [0,1]^m)\to \lip(D_k, [0,1]^m)$$
	denote the restriction map. We note that whenever $k \leq \ell \leq n$, we have $\rho^l_k[S(\ell, m, n)] = S(k, m, n)$, so also $\rho^l_k[S(\ell, m)] = S(k, m)$. It follows that
	$$Z := \varprojlim_k \varprojlim_m S(k, m)\subseteq \lip^\omega$$
	satisfies $Y_n\to Z$ in $K(\lip^\omega)$. Since $\mathrm{Sub}(\lip^\omega)$ is a closed subspace of $K(\lip^\omega)$, it follows that $Z\in \mathrm{Sub}(\lip^\omega)$.

	To see that $Z$ is minimal, it suffices to argue that $\pi^\omega_m\circ Z$ is minimal for each $m< \omega$. To do this, we use Proposition~\ref{Prop:MinimalGDelta}. Fix $z\in \pi^\omega_m\circ Z$ and $k< \omega$. We will argue that $\{(g\cdot z)|_{D_k}: g\in G\}$ is dense in $S(k, m)$, thus handling all $\epsilon > 0$ simultaneously.

	 Let $n\geq k$. The construction of $Y_{n+1}$ yields (keeping in mind that $F_n^6\subseteq D_{n+1}$) that for any $v\in S(n+1, m, n+1)$, we have that $\{(g\cdot v)|_{D_n}: g\in D_n|D_{n+1}\}\subseteq S(n, m, n)$ is $(1/2^n)$-dense. Since $z|_{D_{n+1}}\in S(n+1, m)$, we have that $\{(g\cdot z)|_{D_n}: g\in D_n|D_{n+1}\}\subseteq S(n, m)$ is $(1/2^n)$-dense. By restricting to $D_k$, we see that $\{(g\cdot z)|_{D_k}: g\in D_n|D_{n+1}\}\subseteq S(k, m)$ is also $(1/2^n)$-dense. Letting $n$ grow, we see that $\{(g\cdot z)|_{D_k}: g\in G\}\subseteq S(k, m)$ is dense as desired.
\end{proof}
\vspace{2 mm}

Set $X = \lip^\omega\times \prod_{n} G_{F_n}$. Weakly model-universal flows are closed under products, and if any member of the product is model-universal, then the product is as well (\cite{Z}, Proposition 6). So $X$ is model-universal. We will often write elements of $X$ as tuples $((f_n)_n, (S_n)_n))$, where $f_n\in \lip$ and $S_n\in G_{F_n}$. We will find a Borel $G$-invariant set $W\subseteq X$ with the following two properties:
\begin{enumerate}
	\item
	For every measure $\mu\in P_G(X)$, $\mu(W) = 1$.
	\item
	There is a Borel, $G$-equivariant injection $\phi\colon W\to Z$.
\end{enumerate}
This will show that $Z$ is model-universal. We set
$$W = \{((f_n)_n, (S_n)_n): \forall k< \omega \,\, \exists m< \omega\,\, \forall n\geq m\,\, (E_k\cap E_n^3S_n = \emptyset)\}.$$
In particular, membership in $W$ only depends on the sets $S_n$. Fix $\mu\in P_G(X)$; we will show that $\mu(W) = 1$. Fix $n< \omega$, and let $\nu\in P_G(G_{F_n})$ be the projection of $\mu$ onto $G_{F_n}$.
\vspace{2 mm}

\begin{lemma}
	\label{Lem:BorelCantelli}
If $k\leq n$, we have
$$\nu(\{S\in G_{F_n}: E_k\cap E_n^3S \neq \emptyset\}) \leq 1/2^n$$
\end{lemma}

\begin{proof}
	Recall that by item $(4)$ of the properties of the sets $D_n, E_n, F_n$, there is an $E_n^4$-spaced set $\{h_{n,i}: i< 2^n\}\subseteq G$ so that $E_n^4h_{n,i}\subseteq \mathrm{Int}(F_n)$ for each $i< 2^n$. Note that we have:
	\begin{align*}
	h_{n,i}^{-1}\boldsymbol{\cdot} \{S\in G_{F_n}: E_k\cap E^3_nS\neq \emptyset\} &= \{Sh_{n,i}\in G_{F_n}: E_k\cap E_n^3S\neq \emptyset\} \\
	&= \{T\in G_{F_n}: E_n^3E_kh_{n,i}\cap T\neq \emptyset\}\\
	&\subseteq \{T\in G_{F_n}: E_n^4h_{n,i}\cap T\neq \emptyset\}.
	\end{align*}
	Note that if $g_i\in E_n^4h_{n,i}\subseteq \mathrm{Int}(F_n)$ and $g_j\in E_n^4h_{n,j}\subseteq \mathrm{Int}(F_n)$, then we have
	$$1_G\in \mathrm{Int}(F_n)g_i\cap \mathrm{Int}(F_n)g_j\neq \emptyset.$$
	It follows that the collection
	$$\{h_{n,i}^{-1}\boldsymbol{\cdot} \{S\in G_{F_n}: E_k\cap E_n^3S\}: i< 2^n\}$$
	is pairwise disjoint. Since $\nu$ is $G$-invariant, we are done.
\end{proof}
\vspace{2 mm}

We can now apply the Borel-Cantelli lemma to conclude that $\mu(W) = 1$.

We now turn to defining $\phi\colon W\to Z$. First let $j\colon \omega\to \omega\times 2$ be an infinite-to-one surjection. We define for each $n< \omega$ a Borel $G$-equivariant map $\phi_n\colon W\to Y_n$ so that $\phi(w) = \lim_n \phi_n(w)$. Start by letting $\phi_0$ denote the only map to $Y_0$. Suppose $\phi_n$ is defined. Let $w = ((f_n)_n, (S_n)_n)\in W$ be given. For notation, we set $f_{(n, 0)} = f_n$ and $f_{(n, 1)} = \iota(S_n)$. We set
$$\theta_n := \theta\langle Y_n\times \lip, u_n, F_n\rangle$$
i.e.\ $\theta_n\colon (Y_n\times \lip)\times G_{F_n}\to Y_{n+1}$ denotes the map defined in the proof of Proposition~\ref{Prop:Uniformize}. We set $$\phi_{n+1}(w) = \theta_n((\phi_n(w), f_{j(n)}), S_n).$$
\vspace{2 mm}

\begin{lemma}
	\label{Lem:PWConvergence}
	For every $w\in W$, the sequence $\phi_n(w)$ is convergent.
\end{lemma}

\begin{proof}
	Fix $w := ((f_n)_n, (S_n)_n)\in W$, and write $\phi_n(w) = (\alpha_{n,m})_{m< \omega}$, with $\alpha_{n,m}\in \lip$. When $m> n$, we set $\alpha_{n,m}$ to the constant zero function. Fix $m< \omega$ and $g\in G$. We will show that as $n\to \infty$, eventually $\alpha_{n,m}(g)$ is constant. Suppose $k< \omega$ is such that $g\in E_k$. Since $w\in W$, we eventually have that $E_n^3S_n\cap E_k = \emptyset$. This implies that $\phi_{n+1}(w)|_{E_k} = (\phi_n(w), f_{j(n)})|_{E_k}$, so in particular $\alpha_{n,m}(g) = \alpha_{n+1, m}(g)$ whenever $n$ is suitably large.
\end{proof}
\vspace{2 mm}

We can now define $\phi\colon W\to Z$ by setting
$$\phi(w) = \lim_n \phi_n(w)$$
Then $\phi$ is Borel and $G$-equivariant. To see that $\phi$ is injective, suppose $w, w'\in W$, where $w = ((f_n)_n, (S_n)_n)$ and $w' = ((f'_n)_n, (S'_n)_n)$. Find $n< \omega$ and $i< 2$ with $f_{(n,i)}\neq f'_{(n,i)}$. In particular, there is some $k< \omega$ with $f_{(n,i)}|_{E_k}\neq f'_{(n,i)}|_{E_k}$. Because $j\colon \omega\to \omega\times 2$ is an infinite-to-one surjection, we can find $N<\omega$ with $j(N) = (n,i)$ which is as large as desired, in particular large enough so that $E_N^3S_N\cap E_k = \emptyset$ and $E_N^3S'_N\cap E_k = \emptyset$. Writing $\phi(w) = (\alpha_n)_n$ and $\phi(w') = (\alpha'_n)_n$, we have that $\alpha_N|_{E_k} = f_{j(N)}|_{E_k}$ and $\alpha'_N|_{E_k} = f'_{j(N)}|_{E_k}$. Hence $\phi(w)\neq \phi(w')$ as desired.

This concludes the proof that $Z$ is a minimal, model-universal $G$-flow.

config: arxiv


TITLE: Properties of finite magmas $(S,\cdot)$ with $\forall(x,y,z)\in S^3, x\cdot(y\cdot z)=y\cdot(x\cdot z)$?
QUESTION [3 upvotes]: I am considering finite magmas $(S,\cdot)$ with $\forall(x,y,z)\in S^3, x\cdot(y\cdot z)=y\cdot(x\cdot z)$. Any finite commutative group is an example of such thing. But in the context (this question on crypto.stackexchange.com), I am not interested in groups; or at least, not in groups with an efficiently computable inverse.
I am wondering if/how the classical magma-to-group classification simplifies for such structures.

REPLY [2 votes]: About quasigroups
The submagma $M_S$ of $S\to S$ generated by left multiplication $\phi_x: y\mapsto x\cdot y$ is commutative. If $S$ is a finite quasigroup, $M_S$ is only composed of permutations (as $\phi_x(x\backslash y)=y$ for all $y$), so it is a (commutative) subgroup of the symmetric group $\mathcal S_n$. In particular for each $x$ there is an $x^{-1}$ such that $x^{-1}\cdot(x\cdot y)=y$.
Every commutative subgroup $H$ of $\mathcal S_n$ ($|H|\ge 3$) can be associated to a quasigroup without identity in this way: pick a permutation $\pi$ of the $|H|$ elements of $H$ distinct from the identity such that $\pi(1)=1$. Then the operation $x\cdot y=\pi^{-1}(x\circ\pi(y))$ defines a quasigroup structure, with $x\backslash y=x^{-1}\cdot y$ and $x/y=\pi(x)\circ\pi(y)^{-1}$. Obviously $1\cdot x=x\ne x\cdot 1$.
In fact this is exactly the set of quasigroups $Q$ without identity such that $x\mapsto \phi_x$ is injective, since necessarily $Q=M_Q$ and $\pi(x)=x\cdot 1$.
If $H$ has an element of order greater than 2, we can choose $\pi(x)=x^{-1}$ and $\cdot$ boils down to left division in $H$: $x\cdot y=x^{-1}\circ y$. So $C_3$ equipped with left division is a simple example of a quasigroup satisfying the condition.