Projection Pressure and Bowen's Equation for a Class of Self-similar Fractals with Overlap Structure

TL;DR

This paper introduces the concept of projection pressure for affine iterated function systems with overlaps, establishes a variational principle, and demonstrates that Bowen's equation can determine the Hausdorff dimension of the attractor.

Contribution

It defines projection pressure for self-similar fractals with overlaps and proves its relation to Bowen's equation for dimension calculation.

Findings

Projection pressure satisfies a variational principle.

The zero of projection pressure solves Bowen's equation.

Hausdorff dimension of the attractor can be obtained from Bowen's equation.

Abstract

Let $\{S_i\}_{i=1}^{l}$ be an iterated function system(IFS) on $\mathbb{R}^d$ with attractor K. Let $\pi$ be the canonical projection. In this paper we define a new concept called "projection pressure" $P_\pi(\phi)$ for $\phi\in C(\mathbb{R}^d)$ under certain affine IFS, and show the variational principle about the projection pressure. Furthermore we check that the unique zero root of "projection pressure" still satisfies Bowen's equation when each $S_i$ is the similar map with the same compression ratio. Using the root of Bowen's equation, we can get the Hausdorff dimension of the attractor $K$.