Nonconformal Repellers

Abstract

We consider in this chapter a class of nonconformal repellers to which one can apply the almost additive thermodynamic formalism developed in Chapter 10. Namely, we consider the class of repellers satisfying a cone condition, which includes for example repellers with a strongly unstable foliation and repellers with a dominated splitting. In particular, we are interested in the entropy spectrum of the Lyapunov exponents of a repeller. We note that while in the conformal case the multifractal analysis of the Lyapunov exponents is very well understood, quite the contrary happens in the nonconformal case. At least for the class of nonconformal repellers satisfying a cone condition it is still possible to develop substantially a corresponding multifractal analysis. This amounts to verifying that some sequences related to the Lyapunov exponents are almost additive and have bounded variation, which then allows us to apply the theory developed in the former chapter. For simplicity of the exposition, we only consider transformations in the plane.