Abstract. This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials −t log |Df |, for the largest possible interval of parameters t. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are also obtained.

We introduce a definition of pressure for almost-additive sequences of continuous functions defined over (non-compact) countable Markov shifts. The variational principle is proved. Under certain assumptions we prove the existence of Gibbs and equilibrium measures. Applications are given to the study of maximal Lypaunov exponents of product of matrices.

Abstract. In this paper we study two types of means of the entries of a nonnegative matrix: the permanental mean, which is defined using permanents, and the scaling mean, which is defined in terms of an optimization problem. We explore relations between these two means, making use of important results by Ergorychev and Falikman (the van der Waerden conjecture), Friedland, Sinkhorn, and others. We also define a scaling mean for functions in a much more general context. Our main result is a Law of Large Permanents, a pointwise ergodic theorem for permanental means of dynamically defined matrices that expresses the limit as a functional scaling mean. The concepts introduced in this paper are general enough so to include as particular cases certain classical types of means, as for example symmetric means and Muirhead means. As a corollary, we reobtain a formula of Halász and Székely for the limit of the symmetric means of a stationary random process.

This paper is devoted to the study of dimension theory, in particular multifractal analysis, for multimodal maps. We describe the Lyapunov spectrum and study the multifractal spectrum of pointwise dimension. The lack of regularity of the thermodynamic formalism for this class of maps is reflected in the phase transitions of the spectra.

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