The Topological Entropy Conjecture
The Topological Entropy Conjecture
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拓扑熵猜想
DOI：
10.3390/math9040296
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发表时间：
2021
期刊：
影响因子：
2.4
通讯作者：
Lvlin Luo
中科院分区：
文献类型：
--
作者：
Lvlin Luo
For a compact Hausdorff space $X$,.let $J$ be the ordered set associated with the set of all finite open covers of $X$.such that there exists $n_J$,.where $n_J$ is the dimension of $X$ associated with $\partial$..Therefore, we have $\check{H}_p(X;\mathbb{Z})$,.where $0\leq p\leq n=n_J$..For a continuous self-map $f$ on $X$,.let $\alpha\in J$ be an open cover of $X$ and $L_{f}(\alpha)=\{L_{f}(U)|U\in\alpha\}$..Then, there exists an open fiber cover $\dot L_f(\alpha)$ of $X^f$ induced by $L_{f}(\alpha)$..In this paper, we define %\v{C}ech homology group $\check{H}_{i}(X;\mathbb{Z})$ and.a topological fiber entropy $ent_L(f)$.as the supremum of $ent(f,\dot L_f(\alpha))$ through all finite open covers of $X^f=\{L_f (U); U \subset X\}$,.where $L_f(U)$ is the f-fiber of $U$, that is the set of images $f^n(U)$ and preimages $f^{-n}(U)$ for $n\in\mathbb{N}$..Then, we prove the conjecture $\log \rho\leq ent_L(f)$ for $f$ being a continuous self-map on a given compact Hausdorff space $X$, where $\rho$ is the maximum absolute eigenvalue of $f_*$,.which is the linear transformation associated with $f$ on the \v{C}ech homology group $\check{H}_{*}(X;\mathbb{Z})=\bigoplus\limits_{i=0}^{n}{\check{H}_{i}(X;\mathbb{Z})}.$
For a compact Hausdorff space $X$,.let $J$ be the ordered set associated with the set of all finite open covers of $X$.such that there exists $n_J$,.where $n_J$ is the dimension of $X$ associated with $\partial$..Therefore, we have $\check{H}_p(X;\mathbb{Z})$,.where $0\leq p\leq n=n_J$..For a continuous self-map $f$ on $X$,.let $\alpha\in J$ be an open cover of $X$ and $L_{f}(\alpha)=\{L_{f}(U)|U\in\alpha\}$..Then, there exists an open fiber cover $\dot L_f(\alpha)$ of $X^f$ induced by $L_{f}(\alpha)$..In this paper, we define %\v{C}ech homology group $\check{H}_{i}(X;\mathbb{Z})$ and.a topological fiber entropy $ent_L(f)$.as the supremum of $ent(f,\dot L_f(\alpha))$ through all finite open covers of $X^f=\{L_f (U); U \subset X\}$,.where $L_f(U)$ is the f-fiber of $U$, that is the set of images $f^n(U)$ and preimages $f^{-n}(U)$ for $n\in\mathbb{N}$..Then, we prove the conjecture $\log \rho\leq ent_L(f)$ for $f$ being a continuous self-map on a given compact Hausdorff space $X$, where $\rho$ is the maximum absolute eigenvalue of $f_*$,.which is the linear transformation associated with $f$ on the \v{C}ech homology group $\check{H}_{*}(X;\mathbb{Z})=\bigoplus\limits_{i=0}^{n}{\check{H}_{i}(X;\mathbb{Z})}.$