Search

This paper studies the relations between Pesin–Pitskel topological pressure on an arbitrary subset and measure theoretic pressure of Borel probability measures, which extends Feng and Huang's recent result on entropies [13] for pressures. More precisely, this paper defines the measure theoretic pressure P μ ( T , f ) for any Borel probability measure, and shows that P B ( T , f , K ) = sup ⁡ { P μ ( T , f ) : μ ∈ M ( X ) , μ ( K ) = 1 } , where M ( X ) is the space of all Borel probability measures, K ⊆ X is a non-empty compact subset and P B ( T , f , K ) is the Pesin–Pitskel topological pressure on K . Furthermore, if Z ⊆ X is an analytic subset, then P B ( T , f , Z ) = sup ⁡ { P B ( T , f , K ) : K ⊆ Z is compact } . This paper also shows that Pesin–Pitskel topological pressure can be determined by the measure theoretic pressure. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the lowers-topological entropy to distinguish zero entropy systems. That this quantity is an invariant factor under topological conjugacy and a power rule is shown. Some examples are given to show that the lower entropy dimension can attain any value in (0, 1), and are different with the upper one and the entropy dimension in the sense of Bowen. A counterexample is used to indicate that the product rule does not hold, and the lowers-topological entropy of the subsystem for the non-wandering set can be strictly less than that of the system when 0

The purpose of this paper is to define and study conditional pressure for subadditive potentials which is an extension of conditional entropy. This study reveals a variational principle of topological conditional pressure for subadditive potentials. Besides, we discuss the topological conditional pressure for asymptotically subadditive potentials. The main idea of the proof is quite similar to that of Cao, Feng and Huang's approximations. [ABSTRACT FROM AUTHOR]