T-entropy and Variational Principle for the spectral radius of transfer and weighted shift operators

局部加权拓扑压

Local weighted topological pressure

本文提出局部加权拓扑压概念，将 Feng 与 Huang 关于加权拓扑压及其变分原理的主要结果推广至局部化情形。

Variational principles on subsets of non-autonomous dynamical systems: topological pressure and topological entropy

This paper discusses the variational principles on subsets for topological pressure and topological entropy of non-autonomous dynamical systems. We define the Pesin-Pitskel topological pressure (weighted topological pressure) and the Bowen topological entropy (weighted Bowen topological entropy) for any subset. Also, we define the measure-theoretic pressure and the measure-theoretic lower entropy for all Borel probability measures. Then, we prove variational principles for topological pressure (topological entropy) which links the Pesin-Pitskel topological pressure (weighted topological pressu…

Variational principle for weighted topological pressure

Let $π:X\to Y$ be a factor map, where $(X,T)$ and $(Y,S)$ are topological dynamical systems. Let ${\bf a}=(a_1,a_2)\in {\Bbb R}^2$ with $a_1>0$ and $a_2\geq 0$, and $f\in C(X)$. The ${\bf a}$-weighted topological pressure of $f$, denoted by $P^{\bf a}(X, f)$, is defined by resembling the Hausdorff dimension of subsets of self-affine carpets. We prove the following variational principle: $$ P^{\bf a}(X, f)=\sup\left\{a_1h_μ(T)+a_2h_{μ\circπ^{-1}}(S)+\int f \;dμ\right\}, $$ where the supremum is taken over the $T$-invariant measures on $X$. It not only generalizes the variational principle of cl…

再加权的加权拓扑压

Weighted topological pressure revisited

本文推广 Tsukamoto 的方法，在高维情形重新定义加权拓扑熵与拓扑压并证明变分原理，从而可初等计算自仿射海绵和某些 sofic 集的 Hausdorff 维数。

加权拓扑熵的两种定义之比较

Comparison of two approaches to weighted topological entropy

本文比较了加权拓扑熵的Feng--Huang与覆盖定义，证明在非不变子集上Feng--Huang熵不超过覆盖熵，且证明不依赖测度论，为相对加权变分原理提供新视角。

顺从群作用的高维加权压力变分原理

Variational principle of higher dimension weighted pressure for amenable group actions

本文将顺从群作用的加权拓扑压力推广到任意高维因子链，并证明其等于加权测度熵与加权积分之和在不变测度上的上确界。

Variational principle for neutralized Bowen topological entropy on subsets of non-autonomous dynamical systems

Ovadia and Rodriguez-Hertz (Neutralized local entropy, arXiv:2302.10874) defined the neutralized Bowen open ball for an autonomous dynamical system on a compact metric space. Replacing the usual Bowen open ball with neutralized Bowen open ball, we introduce the notions of neutralized Bowen topological entropy of subsets, neutralized weighted Bowen topological entropy of subsets, lower neutralized Brin-Katok's local entropy of Borel probability measures, and neutralized Katok's entropy of Borel probability measures for a non-autonomous dynamical system on a compact metric space. Then, we establ…

New approach to weighted topological entropy and pressure

Motivated by fractal geometry of self-affine carpets and sponges, Feng--Huang (2016) introduced weighted topological entropy and pressure for factor maps between dynamical systems, and proved variational principles for them. We introduce a new approach to this theory. Our new definitions of weighted topological entropy and pressure are very different from the original definitions of Feng--Huang. The equivalence of the two definitions seems highly nontrivial. Their equivalence can be seen as a generalization of the dimension formula for the Bedford--McMullen carpet in purely topological terms.

Weighted Topological Entropy of 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^{…

非自治迭代函数系统的拓扑压力变分原理

A Variational Principle for the Topological Pressure of Non-autonomous Iterated Function Systems on Subsets

本文提出非自治迭代函数系统的Pesin-Pitskel拓扑压力定义，并建立其与测度论压力的变分原理。