[Paper Review] Variational principles for topological entropies of subsets

De‐Jun Feng, Wen Huang|arXiv (Cornell University)|Dec 6, 2010

Mathematical Dynamics and Fractals26 references5 citations

TL;DR

This paper establishes variational principles linking Bowen’s and packing topological entropies of subsets in topological dynamical systems to measure-theoretic entropies of invariant and non-invariant measures. It proves that for any non-empty compact subset $K$, the Bowen entropy $h_{ m top}^{B}(T,K)$ equals the supremum of lower measure-theoretic entropies $\underline{h}_{\mu}(T)$ over all Borel probability measures $\mu$ with $\mu(K)=1$, and similarly for packing entropy using upper entropies. The results extend to analytic sets under mild conditions.

ABSTRACT

Let $(X,T)$ be a topological dynamical system. We define the measure-theoretical lower and upper entropies $\underline{h}_μ(T)$, $\bar{h}_μ(T)$ for any $μ\in M(X)$, 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 $$\htop^B(T, K)= \sup \{\underline{h}_μ(T): μ\in M(X),\; μ(K)=1\}, $$ $$\htop^P(T, K)= \sup \{\bar{h}_μ(T): μ\in M(X),\; μ(K)=1\}. $$ 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)

Motivation & Objective

• To extend the classical variational principle for topological entropy to non-invariant compact and analytic subsets of a topological dynamical system.
• To define and analyze measure-theoretic lower and upper entropies $\underline{h}_{\mu}(T)$ and $\overline{h}_{\mu}(T)$ for arbitrary Borel probability measures $\mu \in M(X)$, not necessarily $T$-invariant.
• To establish variational principles for Bowen’s topological entropy $h_{\rm top}^{B}(T,K)$ and packing topological entropy $h_{\rm top}^{P}(T,K)$ of compact and analytic subsets $K \subset X$.
• To investigate the behavior of topological entropy on analytic sets, showing that the entropy of an analytic set is the supremum of the entropies over its compact subsets.
• To resolve the challenge that for non-invariant sets, $h_{\rm top}^{B}(T,K) > 0$ may hold even when $\mu(K) = 0$ for all $T$-invariant measures $\mu$, by shifting focus to general measures in $M(X)$.

Proposed method

• Define measure-theoretic lower and upper entropies $\underline{h}_{\mu}(T)$ and $\overline{h}_{\mu}(T)$ via pointwise entropies $\underline{h}_{\mu}(T,x)$ and $\overline{h}_{\mu}(T,x)$, based on the asymptotic decay rate of measure of $n$-ball neighborhoods under the Bowen metric $d_n$.
• Use the Bowen metric $d_n(x,y) = \max_{0 \leq k < n} d(T^k x, T^k y)$ to define the entropy of a set $K$ via covering and packing of $\epsilon$-balls in the $d_n$-metric.
• For compact $K$, prove $h_{\rm top}^{B}(T,K) = \sup \{ \underline{h}_{\mu}(T) : \mu \in M(X), \mu(K) = 1 \}$ using a recursive construction of measures $\mu_i$ supported on nested compact sets $K_i$, with controlled entropy decay.
• Establish the existence of a weak-star limit measure $\tilde{\mu}$ supported on a compact subset $K \subset Z$, where $Z$ is analytic, and show $\overline{h}_{\mu}(T) \geq s$ for the normalized $\mu = \tilde{\mu}/\tilde{\mu}(K)$.
• Prove that for analytic $Z$, the topological entropy $h_{\rm top}^{B}(T,Z)$ is the supremum of $h_{\rm top}^{B}(T,K)$ over all compact $K \subset Z$, using a Cantor diagonal argument and nested compact sets.
• Use the fact that $\prod_{n=1}^\infty (1 + 2^{-n}) < \infty$ to control measure distortion across scales, ensuring the existence of a uniform constant $C$ in the entropy estimate $\mu_i(F_i) \leq C \mu_j(F_i)$ for $j > i$.

Experimental results

Research questions

• RQ1 Can a variational principle be established for Bowen’s topological entropy $h_{\rm top}^{B}(T,K)$ of a non-invariant compact set $K$ using measure-theoretic entropies of general Borel probability measures $\mu$ with $\mu(K)=1$?
• RQ2 Does the variational principle for packing topological entropy $h_{\rm top}^{P}(T,K)$ hold for compact sets $K$ when using upper measure-theoretic entropies $\overline{h}_{\mu}(T)$?
• RQ3 Can the variational principle for $h_{\rm top}^{B}(T,Z)$ be extended from compact sets to analytic subsets $Z$ of $X$, especially when $h_{\rm top}(T) < \infty$?
• RQ4 Is the topological entropy of an analytic set $Z$ equal to the supremum of the entropies of its compact subsets?
• RQ5 What is the role of analyticity in extending variational principles beyond compact $T$-invariant sets, particularly when $T$-invariant measures assign zero measure to $Z$?

Key findings

• For any non-empty compact subset $K \subset X$, the Bowen topological entropy satisfies $h_{\rm top}^{B}(T,K) = \sup \{ \underline{h}_{\mu}(T) : \mu \in M(X), \mu(K) = 1 \}$, establishing a variational principle using lower measure-theoretic entropy.
• For any analytic subset $Z \subset X$, the packing topological entropy satisfies $h_{\rm top}^{P}(T,Z) = \sup \{ h_{\rm top}^{P}(T,K) : K \subset Z \text{ compact} \}$, showing that the entropy of an analytic set is determined by its compact subsets.
• When $h_{\rm top}(T) < \infty$, the variational principle for Bowen entropy extends to analytic sets: $h_{\rm top}^{B}(T,Z) = \sup \{ \underline{h}_{\mu}(T) : \mu \in M(X), \mu(Z) = 1 \}$, even if $Z$ is not $T$-invariant.
• The upper measure-theoretic entropy $\overline{h}_{\mu}(T)$ satisfies $\overline{h}_{\mu}(T) \geq s$ for a constructed measure $\mu$ supported on a compact subset $K \subset Z$, where $s$ is the entropy of the set $Z$, proving the existence of a measure achieving the entropy supremum.
• The construction of a sequence of measures $\mu_i$ with controlled decay $\mu_i(\overline{B}(x,\gamma_i)) \sim e^{-m_i(x)s}$ and uniform distortion bound $C = \prod_{n=1}^\infty (1 + 2^{-n}) < \infty$ ensures the existence of a weak-star limit measure $\tilde{\mu}$ with $\tilde{\mu}(K) \in [1, 2C]$.
• For any $z \in K$, there exists a sequence $k_i \to \infty$ such that $\mu(B_{k_i}(z,\epsilon)) \leq C e^{-k_i s} / \tilde{\mu}(K)$, which implies $\overline{h}_{\mu}(T) \geq s$, proving the variational principle for packing entropy.