Ergodic lifts and overlap numbers

TL;DR

This paper investigates the properties of skew product lifts and overlap numbers for equilibrium measures, providing formulas, estimates, and structural insights, with applications to systems with overlaps and dimension theory.

Contribution

It introduces computable formulas for overlap numbers, analyzes the structure of Rokhlin measures, and estimates the box dimension of invariant measures in systems with overlaps.

Findings

Derived formulas for overlap numbers in systems with overlaps

Established relations between Rokhlin measures and fiber partitions

Provided an estimate for the box dimension of invariant measures

Abstract

We study skew product lifts and overlap numbers for equilibrium measures \mu_\psi of H\"older continuous potentials \psi on such lifts. We find computable formulas and estimates for the overlap numbers in several concrete significant cases of systems with overlaps. In particular we obtain iterated systems which are asymptotically irrational-to-1 and absolutely continuous on their limit sets. Then we look into the general structure of the Rokhlin conditional measures of \mu_\psi with respect to different fiber partitions associated to the lift \Phi, and find relations between them. Moreover we prove an estimate on the box dimension of a certain associated invariant measure \nu_\psi on the limit set \Lambda by using the overlap number of \mu_\psi.