相关论文: Double variational principle for mean dimension
This paper contributes to the mean dimension theory of dynamical systems. We introduce a new concept called mean dimension with potential and develop a variational principle for it. This is a mean dimension analogue of the theory of…
In this paper, we introduce mean dimension and rate distortion dimension for $\mathbb{Z}^{k}$-actions dynamical system $(\mathcal{X},\mathbb{Z}^k,T)$. Suppose $(\mathcal{X},\mathbb{Z}^k,T)$ has the marker property. Taking these two…
The purpose of this paper is to point out a new connection between information theory and dynamical systems. In the information theory side, we consider rate distortion theory, which studies lossy data compression of stochastic processes…
We develop a variational principle for mean dimension with potential of $\mathbb{R}^d$-actions. We prove that mean dimension with potential is bounded from above by the supremum of the sum of rate distortion dimension and a potential term.…
We prove a variational principle for the metric mean dimension analog to the one in [LT]. Instead of using the rate distortion function we use the function $h_\mu(\epsilon,T,\delta)$ that is closely related to the entropy $h_\mu(T)$ of…
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…
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…
We study variational principles for metric mean dimension. First we prove that in the variational principle of Lindenstrauss and Tsukamoto it suffices to take supremum over ergodic measures. Second we derive a variational principle for…
Metric mean dimension is a dynamical counterpart of the box dimension in fractal geometry to characterize the topological complexity of infinite entropy systems. The classical variational principle states that topological entropy equals the…
We establish three variational principles for the upper metric mean dimension with potential of level sets of continuous maps in terms of the entropy of partitions and Katok's entropy of the underlying system. Our results hold for dynamical…
We prove a variational principle for the upper and lower metric mean dimension of level sets \[ \left\{x\in X: \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^{j}(x))=\alpha\right\} \] associated to continuous potentials $\varphi:X\to…
This paper investigates the relationship between quantization of measures and metric mean dimension of topological dynamical systems. We introduce the concept of mean quantization dimension for invariant probability measures and establish a…
Rate distortion dimension describes the theoretical limit of lossy data compression methods as the distortion bound goes to zero. It was originally introduced in the context of information theory, and recently it was discovered that it has…
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…
It is well-known that the relativized variational principle established by Bogenschutz and Kifer connects the fiber topological entropy and fiber measure-theoretic entropy. In context of random dynamical systems, metric mean dimension was…
In this paper, we prove that for a topological dynamical system with positive mean topological dimension and marker property, it has factors of arbitrary small mean topological dimension and zero relative mean topological dimension which…
In this note, we show several variational principles for metric mean dimension. First we prove a variational principles in terms of Shapira's entropy related to finite open covers. Second we establish a variational principle in terms of…
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…
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…
Generalized dimensions of multifractal measures are usually seen as static objects, related to the scaling properties of suitable partition functions, or moments of measures of cells. When these measures are invariant for the flow of a…