Ledrappier-Young formulae for a family of nonlinear attractors

TL;DR

This paper establishes that a class of invariant measures on nonlinear attractors are exact dimensional and their dimensions follow a Ledrappier-Young formula, extending understanding of measure dimensions in complex dynamical systems.

Contribution

It proves that pushforward quasi-Bernoulli measures on nonlinear attractors are exact dimensional and satisfy a Ledrappier-Young formula, broadening the scope of dimension theory in dynamical systems.

Findings

Measures are exact dimensional.

Dimensions satisfy Ledrappier-Young formula.

Applicable to nonlinear, non-conformal attractors.

Abstract

We study a natural class of invariant measures supported on the attractors of a family of nonlinear, non-conformal iterated function systems introduced by Falconer, Fraser and Lee. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for H\"older continuous potentials. We show that these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier-Young formula.