Almost Additive Sequences

Abstract

We consider in this chapter the particular class of almost additive sequences and we develop to a larger extent the nonadditive thermodynamic formalism in this setting. We note that any almost additive sequence is asymptotically subadditive, in the sense of Chapter 7, and thus we may expect stronger thermodynamic properties than those in the general nonadditive thermodynamic formalism as well as in the subadditive thermodynamic formalism. This includes a discussion of the existence and uniqueness of equilibrium and Gibbs measures, both for repellers and for hyperbolic sets. On the other hand, the class of almost additive sequences is still sufficiently general to allow nontrivial applications, in particular to the multifractal analysis of the Lyapunov exponents associated to nonconformal repellers (see Chapter 11). Further applications to multifractal analysis are described in Chapter 12. In order to avoid unnecessary technicalities, we first develop the theory for repellers. We then explain how the proofs of the corresponding results for hyperbolic sets and more generally for continuous maps with upper semicontinuous entropy can be obtained from the proofs for repellers. In particular, we describe some regularity properties of the topological pressure for continuous maps with upper semicontinuous entropy.