Variational principles for topological entropies of subsets
By De-Jun Feng, Wen Huang. 2010
174 citations recorded by the source corpus.
Let $(X,T)$ be a topological dynamical system. We define the measure-theoretical lower and upper entropies $\underline{h}_\mu(T)$, $\bar{h}_\mu(T)$ for any $\mu\in M(X)$, where $M(X)$ denotes the collection of all Borel probability measures on $X$. For any non-empty compact subset $K$ of $X$, we show that $$\htop^B(T, K)= \sup \{\underline{h}_\mu(T): \mu\in M(X),\; \mu(K)=1\}, $$ $$\htop^P(T, K)= \sup \{\bar{h}_\mu(T): \mu\in M(X),\; \mu(K)=1\}. $$ where $\htop^B(T, K)$ denotes Bowen's topological entropy of $K$, and $\htop^P(T, K)$ the packing topological entropy of $K$. Furthermore, when $\htop(T)<\infty$, the first equality remains valid when $K$ is replaced by an arbitrarily analytic subset of $X$. The second equality always extends to any analytic subset of $X$.
Subfields: Analytical Chemistry, Cell Biology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/e8a507018a24e3b75494d25f51982ac00081c47e
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Citations in the corpus
- Variational principle for weighted topological pressure (2014, background): “3, by adapting and elaborating the approach in [17].”
- Bowen entropy for actions of amenable groups (2014, methodology): “For the proof of this theorem, we use the idea from [4], which is a dynamical corresponding of the classical result in fractal geometry.”