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Let $(X,f)$ be a dynamical system with the specification property and $\varphi$ be a continuous function. In this paper, we consider the multifractal irregular set \begin{align*} I_{\varphi}=\left\{x\in X:\lim\limits_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^ix)\ \text{does not exist}\right\} \end{align*} and show that this set is either empty or carries full Bowen upper and lower metric mean dimension with potential., Comment: 23pages. arXiv admin note: substantial text overlap with arXiv:2407.15027

Let $(X,f)$ be a dynamical system with the specification property and $\varphi$ be continuous functions. In this paper, we establish some conditional variational principles for the upper and lower Bowen/packing metric mean dimension with potential of multifractal level set $K_\alpha:=\{x\in X:\lim\limits_{n\to\infty}\dfrac{1}{n}\sum\limits_{i=0}^{n-1}\varphi(f^ix)=\alpha\}.$, Comment: 38pages

In this paper, we first prove the variational principle for amenable packing topological pressure. Then we obtain an inequality concerning amenable packing pressure for factor maps. Finally, we show that the equality about packing topological pressure of the set of generic points when the system satisfies the almost specification property, or $\mu$ is ergodic., Comment: 27 pages

Symbolic dynamical theory plays an important role in the research of amenability with a countable group. Motivated by the deep results of Dougall and Sharp, we study the group extensions for topologically mixing random shifts of finite type. For a countable group $G$, we consider the potential connections between relative Gurevi\v{c} pressure (entropy), the spectral radius of random Perron-Frobenius operator and amenability of $G$. Given $G^{\rm ab}$ by the abelianization of $G$ where $G^{\rm ab}=G/[G,G]$, we consider the random group extensions of random shifts of finite type between $G$ and $G^{\rm ab}$. It can be proved that the relative Gurevi\v{c} entropy of random group $G$ extensions is equal to the relative Gurevi\v{c} entropy of random group $G^{\rm ab}$ extensions if and only if $G$ is amenable. Moreover, we establish the relativized variational principle and discuss the unique equilibrium state for random group $\mathbb{Z}^{d}$ extensions., Comment: 42 pages

We aim to establish Bowen's equations for upper capacity invariance pressure and Pesin-Pitskel invariance pressure of discrete-time control systems. We first introduce a new invariance pressure called induced invariance pressure on partitions that specializes the upper capacity invariance pressure on partitions, and then show that the two types of invariance pressures are related by a Bowen's equation. Besides, to establish Bowen's equation for Pesin-Pitskel invariance pressure on partitions we also introduce a new notion called BS invariance dimension on subsets. Moreover, a variational principle for BS invariance dimension on subsets is established., Comment: 23 pages

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., Comment: arXiv admin note: substantial text overlap with arXiv:2210.16491 by other authors

It is well-known that $\epsilon$-stable sets have a deep connection with the topological entropy of dynamical systems. In the present paper, we investigate the relationships of three types of upper metric mean dimensions with potential between \emph{the blocks of $\epsilon$-stable sets, $\epsilon$-stable sets, the dispersion of preimages of $\epsilon$-stable sets} and the whole phase space. Besides, some chaotic phenomenons are revealed in infinite entropy systems. As an application of main results, we show tail entropy, preimage neighborhood entropy and topological entropy have the same metric mean dimension., Comment: with many changes

Ovadia and Rodriguez-Hertz defined neutralized Bowen open ball as $$B_n(x,e^{-n\epsilon})=\{y\in X: d(T^jx, T^jy)

The present paper contributes to develop metric mean dimension theory of continuous random dynamical systems, which is driven by Tsukamoto's problem [\emph{Adv. Math.} \textbf{361} (2020), 106935, 53 pp.]: For Brody curves of complex dynamical systems, why is mean dimension connected to the certain integral? For continuous random dynamical systems, we introduce the concept of metric mean dimension with potential, and inspired by the Ruelle and Walters' idea of topological pressure determining measure-theoretical entropy, we introduce the concept of measure-theoretical metric mean dimension of probability measures. With the help of separation theorem of convex sets, Stone vector lattice, outer measure theory, and Von Neumann's ergodic theorem, we establish a Ruelle-Walters formula of random metric mean dimension by using functional analysis techniques. This demonstrates the deeper ergodic theoretic phenomena hidden behind of the random metric mean dimension theory.

Let $f_{i},i=1,2$ be continuous bundle random dynamical systems over an ergodic compact metric system $(\Omega,\mathcal{F},\mathbb{P},\vartheta)$. 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 $\Pi:\Omega\times X_{1}\rightarrow\Omega\times X_{2}$. We define the ${\bf a}$-weighted Bowen topological entropy of $h^{{\bf a}}(\omega,f_{1},X_{1})$ of $f_{1}$ with respect to $\omega\in \Omega$. It is shown that the quality $h^{{\bf a}}(\omega,f_{1},X_{1})$ is measurable in $\Omega$, and denoted that $h^{{\bf a}}(f_{1},\Omega\times X_{1})$ is the integration of $h^{{\bf a}}(\omega,f_{1},X_{1})$ against $\mathbb{P}$. We prove the following variational principle: \begin{align*} h^{{\bf a}}(f_{1},\Omega\times X_{1})=\sup\left\{a_{1}h_{\mu}^{(r)}(f_{1})+a_{2}h_{\mu\circ\Pi^{-1}}^{(r)}(f_{2})\right\}, \end{align*} where the supremum is taken over the set of all $\mu\in\mathcal{M}_{\mathbb{P}}^{1}(\Omega\times X_{1},f_{1})$. 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., Comment: arXiv admin note: text overlap with arXiv:1412.0078 by other authors

Borrowing the idea of topological pressure determining measure-theoretical entropy in topological dynamical systems, we establish a variational principle for upper metric mean dimension with potential in terms of upper measure-theoretical metric mean dimension of invariant measures. Moreover, the notion of equilibrium states is introduced to characterize these measures that attain the supremum of the variational principle., Comment: The main result was covered by the preprint [arXiv:2210.13126] and Theorem A in [Carvalho, Pessil and Varandas, A convex analysis approach to the metric mean dimension: limits of scaled pressures and variational principles , \emph{Adv. Math.} \textbf{436} (2024), Paper No. 109407, 54 pp.]

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., Comment: arXiv admin note: text overlap with arXiv:2203.12251

The present paper aims to investigate the metric mean dimension theory of continuous flows. We introduce the notion of metric mean dimension for continuous flows to characterize the complexity of flows with infinite topological entropy. For continuous flows, we establish variational principles for metric mean dimension in terms of local $\epsilon$-entropy function and Brin-Katok $\epsilon$-entropy; For a class of special flow, called uniformly Lipschitz flow, we establish variational principles for metric mean dimension in terms of Kolmogorov-Sinai $\epsilon$-entropy, Brin-Katok's $\epsilon$-entropy and Katok's $\epsilon$-entropy., Comment: 17 pages

For infinite measure-theoretic entropy systems, we introduce the notion of measure-theoretic metric mean dimension of invariant measures for different types of measure-theoretic $\epsilon$-entropies, and show that measure-theoretic metric mean dimensions of different types of measure-theoretic $\epsilon$-entropies coincide with the packing metric mean dimension of the set of generic points of ergodic measures., Comment: Final version

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., Comment: many typos and mistakes are corrected

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 $\omega$ in a irreducible shift of finite type $\Sigma_A$, the Bowen entropy of the set consisting of all the points that are mean Li-Yorke pairs with $\omega$ 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 $\omega$ in a irreducible shift of finite type $\Sigma_A$, the Bowen entropy of the set consisting of all the points that are Li-Yorke pairs with $\omega$ 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.

Let $(X,G,\Phi)$ 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 $\Phi$ has $S$-shadowing property if and only if $(X,G,\Phi)$ 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 $\Phi$ has $S$-shadowing property, then $(X,G,\Phi)$ 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 $\Phi$ has property $P$ if and only if $(X,G,\Phi)$ 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 $\Phi$ has property $P$ if and only if $(X,G,\Phi)$ is a factor of an inverse limit of a sequence of subshifts with property $P$ which satisfies Mittag-Leffler condition., Comment: 23 pages

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., Comment: Any comments are welcome

The notion of $\Delta$-weakly mixing set is introduced, which shares similar properties of weakly mixing sets. It is shown that if a dynamical system has positive topological entropy, then the collection of $\Delta$-weakly mixing sets is residual in the closure of the collection of entropy sets in the hyperspace. The existence of $\Delta$-weakly mixing sets in a topological dynamical system admitting an ergodic invariant measure which is not measurable distal is obtained. Moreover, Our results generalize several well known results and also answer several open questions., Comment: 26 pages

This article is devoted to the investigation of the topological pressure of generic points for nonuniformly hyperbolic systems via Pesin theory. In particular, our result can be applied to the nonuniformly hyperbolic diffeomorphisms described by Katok and several other classes of diffeomorphisms derived from Anosov systems., Comment: 21 pages. arXiv admin note: text overlap with arXiv:1412.0764; text overlap with arXiv:1110.6091 by other authors

This article is devoted to the study of the historic set of ergodic averages in some nonuniformly hyperbolic systems. In particular, our results hold for the robust classes of multidimensional nonuniformly expanding local diffeomorphisms and Viana maps., Comment: 18 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:1310.2348

This paper is devoted to the study of the topological pressure dimension for almost additive sequences, which is an extension of topological entropy dimension. We investigate fundamental properties of the topological pressure dimension for almost additive sequences. In particular, we study the relationships among different types of topological pressure dimension and identifies an inequality relating them. Also, we show that the topological pressure dimension is always equal to or greater than 1 for certain special almost additive sequence., Comment: 23 pages. arXiv admin note: text overlap with arXiv:1104.2195 by other authors

We propose a new definition of preimage entropy dimension for continuous maps on compact metric spaces, investigate fundamental properties of the preimage entropy dimension, and compare the preimage entropy dimension with the topological entropy dimension. The defined preimage entropy dimension holds various basic properties of topological entropy dimension, for example, the preimage entropy dimension of a subsystem is bounded by that of the original system and topologically conjugated systems have the same preimage entropy dimension. Also, we discuss the relation between the preimage entropy dimension and the preimage entropy., Comment: 13 pages

This article is devoted to the study of the multifractal analysis of ergodic averages in some nonuniformly hyperbolic systems. In particular, our results hold for the robust classes of multidimensional nonuniformly expanding local diffeomorphisms and Viana maps., Comment: 15 pages

Let $(X,d,\mu)$ be a metric measure space satisfying both the geometrically doubling and the upper doubling measure conditions, which is called non-homogeneous metric measure space. In this paper, via a sharp maximal operator, the boundedness of commutators generated by multilinear singular integral with $RBMO(\mu)$ function on non-homogeneous metric measure spaces in $m$-multiple Lebesgue spaces is obtained., Comment: 17 pages

This article is devoted to showing the product theorem for Bowen's topological entropy., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1012.1103 by other authors

This article establishes the variational principle of topological pressure for actions of sofic groupoids., Comment: 23 pages. arXiv admin note: text overlap with arXiv:1210.1992 by other authors

In this article, the historic set is divided into different level sets and we use topological pressure to describe the size of these level sets. We give an application of these results to dimension theory. Especially, we use topological pressure to describe the relative multifractal spectrum of ergodic averages and give a positive answer to the conjecture posed by L. Olsen (J. Math. Pures Appl. {\bf 82} (2003))., Comment: 30 pages

Let $\mu$ be the self-similar measure supported on the self-similar set $K$ with open set condition. In this article, we discuss the packing dimension of the set $\{x\in K: A(\frac{\log\mu(B(x,r))}{\log r})=I\}$ for $I\subseteq\mathbb{R}$, where $A(\frac{\log\mu(B(x,r))}{\log r})$ denotes the set of accumulation points of \frac{\log\mu(B(x,r))}{\log r}$ as $r\searrow0$. Our main result solves the conjecture about packing dimension posed by Olsen and Winter \cite{OlsWin} and generalizes the result in \cite{BaeOlsSni}., Comment: 13 pages

This study establishes the variational principle for local pressure in the sofic context., Comment: 13 pages