Equilibrium states of intermediate entropies

Ergodic measures of intermediate entropies for dynamical systems with the approximate product property

For a dynamical system satisfying the approximate product property and asymptotically entropy expansiveness, we characterize a delicate structrue of the space of invariant measures: The ergodic measures of intermediate entropies and intermediate pressures are generic in certain subspaces. This proves a conjecture of Katok for a broad class of systems and extends a sequence of known results.

非自治动力系统子集的中间拓扑熵

Intermediate topological entropies for subsets of nonautonomous dynamical systems

本文提出非自治动力系统子集的中间拓扑熵概念，通过参数θ调节覆盖长度范围，推导了相关不等式和性质，揭示了熵的连续性和不连续性特征。

斜积同态的中间熵测度

Measures of Intermediate Entropies for Skew Product Diffeomorphisms

本文研究了具有正熵测度$μ$的斜积映射$F$.证明了如果纤维上的映射是具有非零李雅普诺夫指数的$C^{1+α}$同构，则$F$具有遍历的中间熵测度.为了构建这些措施，我们找到一个集上的返回地图是一个斜产品与马蹄沿纤维。我们可以控制平均返回时间，并表明这些措施的最大熵可以任意接近$h_μ(F)$。

Measures of intermediate entropies for star vector fields

We prove that all star vector fields, including Lorenz attractors and multisingular hyperbolic vector fields, admit the intermediate entropy property. To be precise, if $X$ is a star vector field with $h_{top}(X)>0$, then for any $h\in [0,h_{top}(X))$, there exists an ergodic invariant measure $μ$ of $X$ such that $h_μ(X)=h$. Moreover, we show that the topological entropy is lower semi-continuous for star vector fields.

$\mathbb{Z}^{d}$作用的中间熵遍历测度

Ergodic measures of intermediate entropies for $\mathbb{Z}^{d}$-action

本文研究满足近似$\mathbb{Z}^{d}$或$\mathbb{Z}_+^{d}$-积性质且渐近熵扩张的动力系统，证明了其不变测度空间的结构，并证明了具有任意给定中间熵的遍历测度在某些自然子空间中是稠密的，从而验证了Katok关于此类系统存在任意中间熵遍历测度的猜想。

Ubiquity of entropies of intermediate factors

We consider topological dynamical systems $(X,T)$, where $X$ is a compact metrizable space and $T$ denotes an action of a countable amenable group $G$ on $X$ by homeomorphisms. For two such systems $(X,T)$ and $(Y,S)$ and a factor map $π: X \rightarrow Y$, an intermediate factor is a topological dynamical system $(Z,R)$ for which $π$ can be written as a composition of factor maps $ψ: X \rightarrow Z$ and $φ: Z \rightarrow Y$. In this paper we show that for any countable amenable group $G$, for any $G$-subshifts $(X,T)$ and $(Y,S)$, and for any factor map $ π:X \rightarrow Y$, the set of entrop…