Sub-additive unstable topological pressure on saturated subsets

TL;DR

This paper investigates sub-additive pressures in partially hyperbolic dynamical systems, establishing equalities for unstable Bowen topological pressure on saturated sets and linking capacity pressure with system-wide pressure.

Contribution

It extends the understanding of unstable topological pressures by relating pressures on saturated sets to invariant measures and system-wide pressures under certain properties.

Findings

Unstable Bowen topological pressure equals the infimum over invariant measures.

Unstable topological capacity pressure coincides with the system's topological pressure.

Results apply to $C^1$-smooth partially hyperbolic diffeomorphisms.

Abstract

In this paper, we continue our investigation on sub-additive pressures for $C^1$-smooth partially hyperbolic diffeomorphisms. Under the assumption of unstable almost product property, we show that the unstable Bowen topological pressure on the saturated set of a given non-empty compact connected set of invariant measures equals the infimum of the summation of unstable metric entropy and the corresponding \emph{Lyapunov exponent}, where the infimum is taken over all invariant measures inside the compact connected set above. Moreover, we also show that the unstable topological capacity pressure on the saturated sets above coincide with the unstable topological pressure of the whole system.