Packing entropy for fixed-point free flows

Abstract

Let (X, φ) be a compact flow without fixed points. We define the packing topological entropy h P top (φ, K) on subsets of X through considering all the possible reparametrizations of time. For fixed-point free flows, we prove the following result: for any non-empty compact subset K of X,, µ is a Borel probability measure onX}, where h µ (φ) denotes the upper local entropy for a Borel probability measure µ on X.

Variational principles for topological pressures on subsets

“…where h P top (Z, T), h µ (T), and h µ (T) denote respectively the packing topological entropy of Z, measure-theoretical lower and upper entropies of µ. Since then, Feng-Huang's variational principles have been extended to different systems and topological pressures; we refer the reader to [10][11][12][13][14][15][16][17][18] for more details. Tang et al [14] generalized Feng-Huang's variational principle of Bowen topological entropy to Pesin-Pitskel topological pressure: if Z ⊂ X is nonempty and compact then…”

Variational principles for topological pressures on subsets

“…Inspired by the variational relation between topological entropy and measure-theoretical entropy, Feng and Huang [7] introduced measure-theoretical lower and upper entropies and packing topological entropy, and they obtained two variational principles for Bowen entropy and packing entropy: if Z ⊂ X is nonempty and compact then h B top (Z, T ) = sup{h µ (T ) : µ ∈ M(X ), µ(Z) = 1}, h P top (Z, T ) = sup{h µ (T ) : µ ∈ M(X ), µ(Z) = 1}, where h P top (Z, T ), h µ (T ), and h µ (T ) denote respectively the packing topological entropy of Z, measure-theoretical lower and upper entropies of µ. Since then, Feng-Huang's variational principles have been extended to different systems and topological pressures; we refer the reader to [21,24,5,27,19,25,11,12,8] for more details. Tang, Cheng, and Zhao [19] generalized Feng-Huang's variational principle of Bowen topological entropy to Pesin-Pitskel topological pressure: if Z ⊂ X is nonempty and compact then…”