Emergence for diffeomorphisms with nonzero Lyapunov exponents

TL;DR

This paper investigates the size and structure of points with high pointwise emergence in certain smooth dynamical systems, establishing bounds on their Hausdorff dimension related to hyperbolic measures, especially for SRB measures.

Contribution

It provides new lower bounds on the Hausdorff dimension of high emergence sets for $C^{1+eta}$ diffeomorphisms with hyperbolic measures, including full dimension results for SRB measures.

Findings

Lower bound on Hausdorff dimension in terms of unstable dimension

High emergence set has full Hausdorff dimension for SRB measures

Results apply to $C^{1+eta}$ diffeomorphisms with hyperbolic measures

Abstract

We consider the set of points with high pointwise emergence for $C^{1+\alpha}$ diffeomorphisms preserving a hyperbolic measure. We find a lower bound on the Hausdorff dimension of this set in terms of unstable Hausdorff dimension of the hyperbolic measure. If the measure is an SRB, we prove that the set of points with high emergence has full Hausdorff dimension.