Effective SPR property for surface diffeomorphisms and three-dimensional vector fields

TL;DR

This paper establishes an effective SPR property for smooth surface diffeomorphisms and proves finiteness of equilibrium measures for certain three-dimensional flows, linking entropy, stable/unstable manifolds, and measure properties.

Contribution

It introduces an effective SPR property for smooth surface diffeomorphisms and proves finiteness of equilibrium measures for specific three-dimensional flows, advancing understanding of dynamical systems.

Findings

Large entropy ergodic measures have large measure on points with long stable and unstable manifolds.

Effective SPR property established for smooth surface diffeomorphisms.

Finiteness of equilibrium measures for 3D flows with low variation potentials.

Abstract

In this paper, we prove that ergodic measures with large entropy give uniformly large measure to the set of points with simultaneously long unstable and long stable manifolds. As a consequence, for $C^{\infty}$ surface diffeomorphisms, we establish an effective version of the SPR property. For $C^{\infty}$ three-dimensional flows without singularities, we prove the finiteness of equilibrium measures for admissible potentials whose variation is strictly less than half of the topological entropy.