Dimension Theory of Some Non-Markovian Repellers Part I: A Gentle Introduction

Abstract

Michael Barnsley introduced a family of fractals sets which are repellers of piecewise affine systems. The study of these fractals was motivated by certain problems that arose in fractal image compression, but the results we obtained can be applied for the computation of the Hausdorff dimension of the graph of some functions, like generalized Takagi functions and fractal interpolation functions. In this paper, we introduce this class of fractals and present the tools in the one-dimensional dynamics and nonconformal fractal theory that are needed to investigate them. This is the first part in a series of two papers. In the continuation, there will be more proofs and we apply the tools introduced here to study some fractal function graphs.