Variational principles for topological entropies of subsets

Abstract

Let (X, T ) be a topological dynamical system. We define the measure-theoretical lower and upper entropies h μ (T ), h μ (T ) for any μ ∈ 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 thatwhere h B top (T , K) denotes the Bowen topological entropy of K, and h P top (T , K) the packing topological entropy of K. Furthermore, when h top (T ) < ∞, the first equality remains valid when K is replaced by any analytic subset of X. The second equality always extends to any analytic subset of X.

Slow Entropy for Noncompact Sets and Variational Principle

“…In 1983 Brin and Katok [8] gave a topological version of the Shannon-McMillan-Breiman theorem with a local decomposition of the measure-theoretical entropy. Recently, Feng and Huang [9] gave a certain variational relation between Bowen's topological entropy and measure-theoretic entropy for arbitrary non-invariant compact set. i.e.…”

“…where K is any non-empty compact subset of (X, T ), h μ (T ) is the measure-theoretical lower entropy of Borel probability measure μ (see [8,9]). The name slow entropy was introduced into dynamical systems by Katok and Thouvenot [10], Hochman [11] for Z k -actions.…”

Pattern generation problems arising in multiplicative integer systems

“…Recently, Feng and Huang [16] defined several topological entropies of subsets Z in a topological dynamical system (X, T ). In particular, they defined the upper capacity topological entropy h [16]. If Z is T -invariant and compact, then they are coincident.…”

A Formula of Packing Pressure of a Factor Map

“…Inspired by the approach of defining of the topological entropy of non-compact subset, Feng and Huang [10] introduced the notion of packing entropy in dynamical systems, which resembles packing dimension. An understanding of both the topological entropy and the packing entropy of a set provides the basis for substantially better understanding of the underlying dynamical behavior of the set.…”

“…Barreira and Schmeling [13] have showed that BS dimension is the unique root of topological pressure function. Inspired by Pesin [12] and Feng and Huang [10], Wang and Chen [14] generalized it to packing topological pressure. In [14], Wang and Chen also introduced packing version of BS dimension and called it BSP dimension.…”