Topological Pressures on Subsets for Nonautonomous Dynamical Systems

We introduce and study the relation between Pesin-Pitskel topological pressure on an arbitrary subset and measure theoretic pressure of Borel probability measure for nonautonomous dynamical systems,which is an extension of the classical definition of Bowen topological entropy.We show that the Pesin-Pitskel topological pressure can be determined by the local pressures of measures in nonautonomous case and establish a variational principle for Pesin-Pitskel topological pressure on compact subsets in the context of nonautonomous dynamical systems.

Variational Principle for Topological Pressure on Subsets of Free Semigroup Actions

We investigate the relations between Pesin–Pitskel topological pressure on an arbitrary subset and measure-theoretic pressure of Borel probability measures for finitely generated semigroup actions.Let(X,G)be a system,where X is a compact metric space and G is a finite family of continuous maps on X.Given a continuous function f on X,we define Pesin–Pitskel topological pressure PG(Z,f)for any subset Z■X and measure-theoretical pressure Pμ,G(X,f)for anyμ∈M(X),where M(X)denotes the set of all Borel probability measures on X.For any non-empty compact subset Z of X,we show that PG(Z,f)=sup{Pμ,G(X,f):μ∈M(X),μ(Z)=1}.

Sub-additive Unstable Topological Pressure of Diffeomorphisms

In this paper,we study unstable topological pressure for C^(1)-smooth partially hyperbolic diffeomorphisms with sub-additive potentials.Moreover,without any additional assumption,we have established the expected variational principle which connects this unstable topological pressure and the unstable measure theoretic entropy,as well as the corresponding Lyapunov exponent.

SUB-ADDITIVE PRESSURE ON A BOREL SET

The goal of this paper is to investigate topological conditional pressure of a continuous transformation as defined for sub-additive potentials. This study presents a vari- ational inequality for sub-additive topological conditional pressure on a closed subset, which is the other form of the variational principle for the sub-additive topological pressure pre- sented by Cao, Feng, and Huang in [9]. Moreover, under some additional assumptions, this result can be generalized to the non-compact case.

THERMODYNAMIC FORMALISM FOR SUBADDITIVE SEQUENCE OF DISCONTINUOUS FUNCTIONS

The topological pressure for subadditive sequence of discontinuous functions is defined on any invariant subset having a nested family of subsets in the compact metric space. Two subadditive variational principles associated with two different relatively weak conditions are developed for the defined topological pressure. As an application, we give an example on systems with nonzero Lyapunov exponents.

A Note on a Result of Pesin

Pesin gives an equivalent definition of the pressures defined respectively by open covering and Bowen covering of the dynamical system which plays an important role in the studies on dimension theory in dynamical systems.In this note,we show a gap of his proof and present a proof for this result.