arXiv subjects
Publications and source records attributed to Ercai Chen.
We introduce the notion of Feldman-Katok metric mean dimensions in this note. We show metric mean dimensions defined by different metrics coincide under weak tame growth of covering numbers, and establish variational principles for Feldman-Katok metric mean dimensions in terms of FK Katok $ε$-entropy and FK local $ε$-entropy function.
Let $(X,d)$ be a compact metric space, $f:X\rightarrow X$ be a continuous transformation with the specification property. we consider non-dense orbit set $E(z_0)$ and show that for any non-transitive point $z_0\in X$, this set $E(z_0)$ is empty or carries full Bowen upper and lower metric mean dimension.
This paper aims to investigate the thermodynamic formalism of weighted amenable topological pressure for factor maps of amenable group actions. Following the approach of Tsukamoto [\emph{Ergodic Theory Dynam. Syst.} \textbf{43}(2023), 1004-1034.], we introduce the notion of weighted amenable topological pressure for factor maps of amenable group actions, and establish a variational principle for it. As the application of variational principle, we show weighted amenable measure-theoretic entropy can be determined by weighted amenable topological pressure. Equilibrium states of weighted topological pressure are also involved.
In this paper, we study the Feldman-Katok metric in random dynamical systems and establish corresponding fiber topological entropy formula, Brin-Katok local entropy formula and fiber Katok entropy formula by replacing Bowen metric with Feldman-Katok metric. It turns out that the Feldman-Katok metric is also the weakest metric that makes the entropy formulae valid on random dynamical systems.
Let $f_{i},i=1,2$ be continuous bundle random dynamical systems over an ergodic compact metric system $(Ω,\mathcal{F},\mathbb{P},\vartheta)$. Assume that ${\bf a}=(a_{1},a_{2})\in\mathbb{R}^{2}$ with $a_{1}>0$ and $a_{2}\geq0$, $f_{2}$ is a factor of $f_{1}$ with a factor map $Π:Ω\times X_{1}\rightarrowΩ\times X_{2}$. We define the ${\bf a}$-weighted Bowen topological entropy of $h^{\bf a}(ω,f_{1},X_{1})$ of $f_{1}$ with respect to $ω\in Ω$. It is shown that the quality $h^{\bf a}(ω,f_{1},X_{1})$ is measurable in $Ω$, and denoted that $h^{\bf a}(f_{1},Ω\times X_{1})$ is the integration of $h^{\bf a}(ω,f_{1},X_{1})$ against $\mathbb{P}$. We prove the following variational principle: \begin{align*} h^{\bf a}(f_{1},Ω\times X_{1})=\sup\left\{a_{1}h_μ^{(r)}(f_{1})+a_{2}h_{μ\circΠ^{-1}}^{(r)}(f_{2})\right\}, \end{align*} where the supremum is taken over the set of all $μ\in\mathcal{M}_{\mathbb{P}}^{1}(Ω\times X_{1},f_{1})$. In the case of random dynamical systems with an ergodic and compact driving system, this gives an affirmative answer to the question posed by Feng and Huang [Variational principle for weighted topological pressure, J. Math. Pures Appl. 106 (2016), 411-452]. It also generalizes the relativized variational principle for fiber topological entropy, and provides a topological extension of Hausdorff dimension of invariant sets and random measures on the $2$-torus $\mathbb{T}^{2}$. In addition, the Shannon-McMillan-Breiman theorem, Brin-Katok local entropy formula and Katok entropy formula of weighted measure-theoretic entropy for random dynamical systems are also established in this paper.
Firstly, we introduce a new notion called induced upper metric mean dimension with potential, which naturally generalizes the definition of upper metric mean dimension with potential given by Tsukamoto to more general cases, then we establish variational principles for it in terms of upper and lower rate distortion dimensions and show there exists a Bowen's equation between induced upper metric mean dimension with potential and upper metric mean dimension with potential. Secondly, we continue to introduce two new notions, called BS metric mean dimension and Packing BS metric mean dimension on arbitrary subsets, to establish Bowen's equations for Bowen upper metric mean dimension and Packing upper metric mean dimension with potential on subsets. Besides, we also obtain two variational principles for BS metric mean dimension and Packing BS metric mean dimension on subsets. Finally, the special interest about the Bowen upper metric mean dimension of the set of generic points of ergodic measures are also involved.
Firstly, we answer the problem 1 asked by Gutman and $\rm \acute{\ S}$piewak in \cite{gs20}, then we establish a double variational principle for mean dimension in terms of R$\bar{e}$nyi information dimension and show the order of $\sup$ and $\limsup$ (or $\liminf$) of the variational principle for the metric mean dimension in terms of R$\bar{e}$nyi information dimension obtained by Gutman and $\rm \acute{\ S}$piewak can be changed under the marker property. Finally, we attempt to introduce the notion of maximal metric mean dimension measure, which is an analogue of the concept called classical maximal entropy measure related to the topological entropy.
Recently, Kiriki, Nakano and Soma introduced a concept called pointwise emergence as a new quantitative perspective into the study of non-existence of averages for dynamical systems. In the present paper, we consider the set of points with high pointwise emergence for systems with non-uniform structure and prove that this set carries full topological pressure. For the proof of this result, we show that such systems have ergodic measures of arbitrary intermediate pressures.
In the present paper, we use the generalized multifractal framework introduced by Olsen to study the Bowen entropy and packing entropy of historic sets with typical weights over aperiodic and irreducible shifts of finite type. Following those results and a transfer from almost everywhere to everywhere, we show that for each point $ω$ in a irreducible shift of finite type $Σ_A$, the Bowen entropy of the set consisting of all the points that are mean Li-Yorke pairs with $ω$ is $0$, and its packing entropy is full. This result is beyond the ergodic theory. Also, by the transfer from almost everywhere to everywhere, we show that for each point $ω$ in a irreducible shift of finite type $Σ_A$, the Bowen entropy of the set consisting of all the points that are Li-Yorke pairs with $ω$ is full. This result is also beyond the ergodic theory.
In this paper, we mainly elucidate a close relationship between the topological entropy and mean dimension theory for actions of polynomial growth groups. We show that metric mean dimension and mean Hausdorff dimension of subshifts with respect to the lower rank subgroup are equal to its topological entropy multiplied by the growth rate of the subgroup. Meanwhile, we also prove the above result holds for the rate distortion dimension of subshifts with respect to the lower rank subgroup and measure entropy. Furthermore, some relevant examples are indicated.
In this paper, we consider which kind of invariant measure on hereditary subshifts is not Gibbs measure. For the hereditary closure of a subshift $(X,S)$, we prove that in some situation, the invariant measure $ν*B_{p,1-p}$ can not be a Gibbs measure where $ν$ is an invariant measure on $(X,S)$. As an application, we show that for some $\B$-free subshifts, the unique equilibrium state $ν_η*B_{p,1-p}$ is not Gibbs measure.
Let $(X,G,Φ)$ be a dynamical system, where $X$ is compact Hausdorff space, and $G$ is a countable discrete group. We investigate shadowing property and mixing between subshifts and general dynamical systems. For the shadowing property, fix some finite subset $S\subset G$. We prove that if $X$ is totally disconnected, then $Φ$ has $S$-shadowing property if and only if $(X,G,Φ)$ is conjugate to an inverse limit of a sequence of shifts of finite type which satisfies Mittag-Leffler condition. Also, suppose that $X$ is metric space (may be not totally disconnected), we prove that if $Φ$ has $S$-shadowing property, then $(X,G,Φ)$ is a factor of an inverse limit of a sequence of shifts of finite type by a factor map which almost lifts pseudo-orbit for $S$. On the other hand, let property $P$ be one of the following property: transitivity, minimal, totally transitivity, weakly mixing, mixing, and specification property. We prove that if $X$ is totally disconnected, then $Φ$ has property $P$ if and only if $(X,G,Φ)$ is conjugate to an inverse limit of an inverse system that consists of subshifts with property $P$ which satisfies Mittag-Leffler condition. Also, for the case of metric space (may be not totally disconnected), if property $P$ is not minimal or specification property, we prove that $Φ$ has property $P$ if and only if $(X,G,Φ)$ is a factor of an inverse limit of a sequence of subshifts with property $P$ which satisfies Mittag-Leffler condition.
In this paper, we introduce mean dimension quantities with sub-additive potentials. We define mean dimension with sub-additive potentials and mean metric dimension with sub-additive potentials, and establish a double variational principle for sub-additive potentials.
In this paper, we prove variational principles between metric mean dimension and rate distortion function for countable discrete amenable group actions which extend recently results by Lindenstrauss and Tsukamoto.
A topological dynamical system $(X,f)$ induces two natural systems, one is on the probability measure spaces and other one is on the hyperspace. We introduce a concept for these two spaces, which is called entropy order, and prove that it coincides with topological entropy of $(X,f)$. We also consider the entropy order of an invariant measure and a variational principle is established.
Katok's entropy formula is an important formula in entropy theory. This paper is devoted to establishing Katok's entropy formula of unstable metric entropy which is the entropy caused by the unstable part of partially hyperbolic systems.
Let $σ_{K}: \sum_{K}\rightarrow \sum_{K}$ be a shift map. For an interval $[p,q]\subset[0,1]$, let $D_{σ_{K}}([p,q])$ denote the set of pairs for which the density spectrum of the $ε$-approach time set equals $[p,q]$ when $ε$ is small and $E_{σ_{K}}([p,q])$ the set of pairs for which the density spectrum of the $ε$-approach time set converges to $[p,q]$ when $ε\rightarrow 0^+$. Then $\dim_{H} D_{σ_{K}}([p,q])=\dim_{H} E_{σ_{K}}([p,q])=2-q$. Moreover, $\mathscr{H}^{2-q}(E_{σ_{K}}([p,q]))=1$ when $q=0$ and $\mathscr{H}^{2-q}(E_{σ_{K}}([p,q]))=+\infty$ when $q>0$. Meanwhile, $\mathscr{H}^{2-q}(D_{σ_{K}}([p,q]))=+\infty$ when $q=1$ and $\mathscr{H}^{2-q}(D_{σ_{K}}([p,q]))=0$ when $q<1$.
Let $(X, ϕ)$ be a compact metric flow without fixed points. We will be concerned with the entropy of flows which takes into consideration all possible reparametrizations of the flows. In this paper, by establishing the Brin-Katok's entropy formula for flows without fixed points in non-ergodic case, we prove the following result: for an ergodic $ϕ$-invariant measure $μ$, $$ h_{top}^{B}(ϕ, G_μ(ϕ))=h_μ(ϕ_{1}),$$ where $G_μ(ϕ)$ is the set of generic points for $μ$ and $h_{top}^{B}(ϕ, G_μ(ϕ))$ is the Bowen entropy on $G_μ(ϕ)$. This extends the classical result of Bowen in 1973 to fixed-point free flows. Moreover, we show that the Bowen entropy can be determined via the local entropies of measures.