论文标题
随机动力学系统的加权拓扑熵
Weighted Topological Entropy of Random Dynamical Systems
论文作者
论文摘要
令$ f_ {i}，i = 1,2 $是连续的束随机动态系统，上面是紧凑型公制系统$（ω，\ mathcal {f}，\ mathbb {p}，\ vartheta）$。假设$ {\ bf a} =（a_ {1}，a_ {2}）\ in \ mathbb {r}^{2} $带有$ a_ {1}> 0 $ a_ {1}> 0 $ and $ a_ {2} \ geq0 $，$ f_ {2} $，$ f_ {2} $是$ f _ $ f _ $ f _ $ f _}; x_ {1} \rightarrowΩ\ times x_ {2} $。我们定义$ h^{\ bf a}的$ {\ bf a} $ - 加权鲍恩拓扑熵（ω，f_ {1}，x_ {1}）$ of $ f_ {1} $ of $ f_ {1} $相对于$ω\ incom incom yinch $。结果表明，质量$ h^{\ bf a}（ω，f_ {1}，x_ {1}）$可在$ω$中测量，并表示$ h^{\ bf a}（f_ {1}，ω\ times x_ {1}）$是$ h^\ bf a}（ω，f_ {1}，x_ {1}）$ fions $ \ mathbb {p} $。我们证明了以下变异原理：\ begin {align*} h^{\ bf a}（f_ {1}，ω\ times x_ {1}）= \ sup \ lest \ {a_ {1}h_μ^{（r）}（f_ {1}）+a_ {2} h_ {2} \ Mathcal {M} _ {\ Mathbb {p}}}^{1}（ω\ times x_ {1}，f_ {1}）$的所有$μ\的集合。对于具有千古和紧凑驱动系统的随机动力学系统，这给了Feng和Huang提出的问题[加权拓扑压力的变化原理，J。Math。 Pures Appl。 106（2016），411-452]。它还概括了纤维拓扑熵的相对变异原理，并提供了不变套件的Hausdorff尺寸和$ 2 $ -TORUS $ \ MATHBB {T}^{2} $的拓扑扩展。此外，本文还建立了Shannon-McMillan-Breiman定理，Brin-katok局部熵公式和加权方法的Katok熵公式，用于随机动力学系统的加权方法理论熵。
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.
