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SRB measures for partially hyperbolic systems whose central direction is mostly expanding (2000)
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@MISC{Alves00srbmeasures,
author = {José F. Alves and Christian Bonatti and Marcelo Viana},
title = {SRB measures for partially hyperbolic systems whose central direction is mostly expanding},
year = {2000}
}
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Abstract
We construct Sinai-Ruelle-Bowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms -- the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting -- under the assumption that the complementary subbundle is non-uniformly expanding. If the rate of expansion (Lyapunov exponents) is bounded away from zero, then there are only finitely many SRB measures. Our techniques extend to other situations, including certain maps with singularities or critical points, as well as diffeomorphisms having only a dominated splitting (and no uniformly hyperbolic subbundle). 1 Introduction The following approach has been most effective in studying the dynamics of complicated systems: one tries to describe the average time spent by typical orbits in different regions of the phase space. According to the ergodic theorem of Birkhoff, such times are well defined for almost all point, with respect to any invariant probability measure. However, the...
Keyphrases
srb measure hyperbolic system central direction complicated system lyapunov exponent critical point different region many srb measure complementary subbundle typical orbit invariant subbundles average time certain map following approach ergodic theorem tangent bundle invariant probability measure phase space uniformly hyperbolic subbundle hyperbolic set
