论文标题
霍德（Hölder
Hölder continuous maps on the interval with positive metric mean dimension
论文作者
论文摘要
用有限的拓扑维度修复紧凑的度量空间$ x $。令$ c^{0}（x）$为$ x $和$ h^α（x）$连续地图的空间，$α$-Hölder连续地图上的$ x $上的$ x $，以$α\ in（0,1]。$ $ $ $ $ $ $ h^{1}（1}（x）$是$ x $ x $ x $ x $ x y是$ x}的空间。 h^β（x）\子集h^α（x）\ subset c^{0}（x），\ quad \ text {其中} 0 <α<β<1。$$，众所周知，如果$ ϕ \ in H^{1}（1}（x）$），则$ ϕ $ nitrim nitric Meementr Is fil $ niment y niment in niment in niment on niment。然后，$ c^{0}（x）$包含一个残差子集，其元素具有正值度的均值。
Fix a compact metric space $X$ with finite topological dimension. Let $C^{0}(X)$ be the space of continuous maps on $X$ and $ H^α(X)$ the space of $α$-Hölder continuous maps on $X$, for $α\in (0,1].$ $H^{1}(X)$ is the space of Lipschitz continuous maps on $X$. We have $$H^{1}(X)\subset H^β(X) \subset H^α(X) \subset C^{0}(X),\quad\text{ where }0<α<β<1.$$ It is well-known that if $ϕ\in H^{1}(X)$, then $ϕ$ has metric mean dimension equal to zero. On the other hand, if $X$ is a finite dimensional compact manifold, then $C^{0}(X)$ contains a residual subset whose elements have positive metric mean dimension. In this work we will prove that, for any $α\in (0,1)$, there exists $ϕ\in H^α([0,1]) $ with positive metric mean dimension.
