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

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" $P_{\pi }(\varphi )$ for $\varphi \in C({\mathbb{R}}^d)$ 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 $K$.