Dimension bounds for invariant measures of bi-Lipschitz iterated function systems

TL;DR

This paper establishes bounds on the Hausdorff and packing dimensions of invariant measures for bi-Lipschitz iterated function systems, linking these bounds to entropy and Lyapunov exponents under certain conditions.

Contribution

It provides the first explicit bounds for dimensions of invariant measures in bi-Lipschitz IFS with the strong open set condition, extending dimension theory in fractal geometry.

Findings

Bounds for Hausdorff and packing dimensions are expressed as ratios of entropy to Lyapunov exponents.

Both upper and lower bounds are derived under the strong open set condition.

Results apply to systems with finite or infinite average-contracting bi-Lipschitz maps.

Abstract

We study probabilistic iterated function systems (IFS), consisting of a finite or infinite number of average-contracting bi-Lipschitz maps on R^d. If our strong open set condition is also satisfied, we show that both upper and lower bounds for the Hausdorff and packing dimensions of the invariant measure can be found. Both bounds take on the familiar form of ratio of entropy to the Lyapunov exponent.