Abstract

This is a survey article on the results about the Hausdorff dimension or Lebesque measure of the attractors of some non-invertible hyperbolic maps and other fractals of overlapping construction.

Introduction

It is well known that under some regularity conditions we can use the pressure formula [MM, PeWe] to compute the Hausdorff dimension of dynamically defined Cantor-sets of the real line.

In Section 2 we examine whether or not the same is true if the cylinders of the Cantor-set under consideration intersect each other.

In Section 3 we consider attractors of some hyperbolic non-invertible maps of the plane (whose cylinders intersect each other). When we compute their Hausdorff dimension we face a similar problem to that considered in section 2. Furthermore, we see how we can trace back the computation of the Hausdorff dimension of attractors of some axiom-A diffeomorphisms of the space (generalized solenoids) to the problem of computation of the Hausdorff dimension of the attractors of non-invertible hyperbolic endomorphisms of the plane.

We denote the Hausdorff and the box dimension of a set F by dimH(F). dimB(F) respectively. (For the definition of Hausdorff and box dimension see [Falcb1].

The non-overlapping case

Here we give a brief review of the most important results when the cylinders of the dynamically defined Cantor-set are well separated; that is they are disjoint or in the self-similar case the open set condition holds.