Relative local variational principles for subadditive potentials

TL;DR

This paper establishes two relative local variational principles for topological pressure functions associated with subadditive potentials, linking these pressures to local measure-theoretic conditional entropies in dynamical systems.

Contribution

It introduces new variational principles for subadditive potentials and proves properties like upper semi-continuity and affinity of related entropy maps.

Findings

Proves two relative local variational principles for topological pressure.

Shows the upper semi-continuity of entropy maps.

Demonstrates that the relative local pressure determines local measure-theoretic conditional entropies.

Abstract

We prove two relative local variational principles of topological pressure functions $P(T,\mathcal{F},\mathcal{U},y)$ and$P(T,\mathcal{F},\mathcal{U}|Y)$ for a given factor map $\pi$, an open cover $\mathcal{U} $ and a subadditive sequence of real-valued continuous functions $\mathcal{F}$. By proving the upper semi-continuity and affinity of the entropy maps $h_{\{\cdot\}}(T,\mathcal{U}\mid Y)$ and $h^+_{\{\cdot\}}(T,\mathcal{U}\mid Y)$ on the space of all invariant Borel probability measures, we show that the relative local pressure $P(T,\mathcal{\{\cdot\}},\mathcal{U}|Y)$ for subadditive potentials determines the local measure-theoretic conditional entropies.