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Functions for relative maximization
 Dynamical Systems
"... Abstract We introduce functions for relative maximization in a general context: the beta and alpha applications. After a systematic study of different kinds of regularities, we investigate how to approximate certain values of these functions using periodic orbits. We also show that the differential ..."
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Abstract We introduce functions for relative maximization in a general context: the beta and alpha applications. After a systematic study of different kinds of regularities, we investigate how to approximate certain values of these functions using periodic orbits. We also show that the differential of an alpha application determines the asymptotic behavior of the optimal trajectories.
MULTIFRACTAL ANALYSIS OF GIBBS MEASURES FOR NONUNIFORMLY EXPANDING MAPS
"... Abstract. We will consider the local dimension spectrum of a Gibbs measure on a nonuniformly hyperbolic system of MannevillePomeau type. We will present the spectrum in three ways: using invariant measures, uniformly hyperbolic ergodic measures and equilibrium states. All three presentations are ..."
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Abstract. We will consider the local dimension spectrum of a Gibbs measure on a nonuniformly hyperbolic system of MannevillePomeau type. We will present the spectrum in three ways: using invariant measures, uniformly hyperbolic ergodic measures and equilibrium states. All three presentations are well known for uniformly hyperbolic systems. The theory of multifractal analysis for hyperbolic conformal dynamical systems is now extremely well developed. There are complete results for local dimension of Gibbs measures, Lyapunov spectra and Birkhoff spectra. For a general description see [14] and for more specific and very general results see [1] and [13]. In the situation of nonuniformly hyperbolic systems there have also been several papers however there is no such complete picture. In the case of local dimensions there are results by Nakaishi on the measure of maximal entropy, [12], some results by Pollicott and Weiss [15], by Kesseböhmer and Stratmann ([9] and [10]), by Yuri in [19] and in the case of complex dynamics by Bryne, [2]. The aim of this paper is to obtain a complete spectra for the local dimension of Gibbs measures for nonuniformly hyperbolic systems. We will be looking at cases where there exist parabolic periodic points but no critical points. Well known examples of such maps include the MannevillePomeau map and the Farey map. The methods used will be adapted from the papers [5] and [8] where the Lyapunov and Birkhoff spectra of such maps are considered. 1. Notation and results We consider nonuniformly expanding onedimensional Markov maps. More precisely, let I = [0, 1]. Let {Ii}, i = 1,..., p be closed subintervals of I with disjoint interiors. Let A be a p × p matrix consisting of