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- W2319936723 abstract "The dynamics of endomorphisms (i.e non-invertible smooth maps)presents many significant differences from that of diffeomorphisms, aswell as from the dynamics of expanding maps. There are numerousconcrete examples of hyperbolic endomorphisms. Many methods cannotbe used here due to overlappings in the fractal set and to theexistence of (possibly infinitely) many local unstable manifoldsgoing through the same point. First we will present the general problems andexplain how to constructcertain useful limit measures for atomic measures supported on variousprehistories. These limit measures are in many cases shown to beequal to certain equilibrium measures for Holder potentials. Weobtain thus an analogue of the SRB measure, namely an inverse SRBmeasure in the case of a hyperbolic repeller, or of an Anosovendomorphism. We study then the 1-sided Bernoullicity (or lack ofit) for certain measures invariant to endomorphisms, and give aClassification Theorem for the ergodic and metric types ofbehaviour of perturbations of a class of maps on their respectivebasic sets, in terms of the values of the stable dimension. Wegive also relations between thermodynamic formalism and fractaldimensions (Hausdorff dimension of stable/unstable intersectionswith basic sets, stable/unstable box dimensions, dimension of theglobal unstable set for endomorphisms). Applications to certain nonlinear evolution models are also given in the end." @default.
- W2319936723 created "2016-06-24" @default.
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- W2319936723 date "2012-01-01" @default.
- W2319936723 modified "2023-10-18" @default.
- W2319936723 title "Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms" @default.
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- W2319936723 doi "https://doi.org/10.3934/dcds.2012.32.2485" @default.
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