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Register a new userOn variational principles for metric mean dimension
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Ruxi Shi
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In this note, we show several variational principles for metric mean dimension. First we prove a variational principles in terms of Shapiras entropy related to finite open covers. Second we establish a variational principle in terms of Katoks entropy. Finally using these two variational principles we develop a variational principle in terms of Brin-Katok local entropy.
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In this article, we introduce a notion of relative mean metric dimension with potential for a factor map $pi: (X,d, T)to (Y, S)$ between two topological dynamical systems. To link it with ergodic theory, we establish four variational principles in terms of metric entropy of partitions, Shapiras entropy, Katoks entropy and Brin-Katok local entropy respectively. Some results on local entropy with respect to a fixed open cover are obtained in the relative case. We also answer an open question raised by Shi cite{Shi} partially for a very well-partitionable compact metric space, and in general we obtain a variational inequality involving box dimension of the space. Corresponding inner variational principles given an invariant measure of $(Y,S)$ are also investigated.
We introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces. These generalize the Gromov-Lindenstrauss-Weiss mean dimensions for actions of countable amenable groups, and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy.
We introduce some notions of conditional mean dimension for a factor map between two topological dynamical systems and discuss their properties. With the help of these notions, we obtain an inequality to estimate the mean dimension of an extension system. The conditional mean dimension for $G$-extensions are computed. We also exhibit some applications in the dynamical embedding problems.
We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group $Gamma$, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced $Gamma$-action on the Pontryagin dual of M, the mean rank of M, and the von Neumann-Luck rank of M all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin-Schnirelmnn theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups, algebraic actions with zero mean dimension are inverse limits of finite entropy actions.
Associated to a IFS one can consider a continuous map $hat{sigma} : [0,1]times Sigma to [0,1]times Sigma$, defined by $hat{sigma}(x,w)=(tau_{X_{1}(w)}(x), sigma(w))$ were $Sigma={0,1, ..., d-1}^{mathbb{N}}$, $sigma: Sigma to Sigma$ is given by$sigma(w_{1},w_{2},w_{3},...)=(w_{2},w_{3},w_{4}...)$ and $X_{k} : Sigma to {0,1, ..., n-1}$ is the projection on the coordinate $k$. A $rho$-weighted system, $rho geq 0$, is a weighted system $([0,1], tau_{i}, u_{i})$ such that there exists a positive bounded function $h : [0,1] to mathbb{R}$ and probability $ u $ on $[0,1]$ satisfying $ P_{u}(h)=rho h, quad P_{u}^{*}( u)=rho u$. A probability $hat{ u}$ on $[0,1]times Sigma$ is called holonomic for $hat{sigma}$ if $ int g circ hat{sigma} dhat{ u}= int g dhat{ u}, forall g in C([0,1])$. We denote the set of holonomic probabilities by ${cal H}$. Via disintegration, holonomic probabilities $hat{ u}$ on $[0,1]times Sigma$ are naturally associated to a $rho$-weighted system. More precisely, there exist a probability $ u$ on $[0,1]$ and $u_i, iin{0, 1,2,..,d-1}$ on $[0,1]$, such that is $P_{u}^*( u)= u$. We consider holonomic ergodic probabilities. For a holonomic probability we define entropy. Finally, we analyze the problem: given $phi in mathbb{B}^{+}$, find the solution of the maximization pressure problem $$p(phi)=$$
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