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13
A characterization of hyperbolic potentials of rational maps
 Bull. Braz. Math. Soc. (N.S
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EXPANDING MEASURES
, 2008
"... We prove that any C 1+α transformation, possibly with a (nonflat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a ..."
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Cited by 8 (1 self)
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We prove that any C 1+α transformation, possibly with a (nonflat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a famous theorem of Keller [37] for maps of the interval with negative Schwarzian derivative. We also show how to construct an induced Markov map F such that every expanding probability of the initial transformation lifts to an invariant probability of F. The induced time is bounded at each point by the corresponding first hyperbolic time (the first time the dynamics exhibits hyperbolic behavior). In particular, F may be used to study decay of correlations and others statistical properties of the initial map, relative to any expanding
0 NONUNIFORM SPECIFICATION AND LARGE DEVIATIONS FOR WEAK GIBBS MEASURES
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Differentiability of thermodynamical quantities in nonuniformly expanding dynamics
, 2013
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ENTROPY AND POINCARÉ RECURRENCE FROM A GEOMETRICAL VIEWPOINT
, 809
"... Abstract. We study Poincaré recurrence from a purely geometrical viewpoint. In [8] it was proven that the metric entropy is given by the exponential growth rate of return times to dynamical balls. Here we use combinatorial arguments to provide an alternative and more direct proof of this result and ..."
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Abstract. We study Poincaré recurrence from a purely geometrical viewpoint. In [8] it was proven that the metric entropy is given by the exponential growth rate of return times to dynamical balls. Here we use combinatorial arguments to provide an alternative and more direct proof of this result and to prove that minimal return times to dynamical balls grow linearly with respect to its length. Some relations using weighted versions of recurrence times are also obtained for equilibrium states. Then we establish some interesting relations between recurrence, dimension, entropy and Lyapunov exponents of ergodic measures. 1.
Correlation decay and recurrence asymptotics for some robust nonuniformly hyperbolic maps
"... Abstract. We study decay of correlations, the asymptotic distribution of hitting times and fluctuations of the return times for a robust class of multidimensional nonuniformly hyperbolic transformations. Oliveira and Viana [15] proved that there is a unique equilibrium state µ for a large class of ..."
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Abstract. We study decay of correlations, the asymptotic distribution of hitting times and fluctuations of the return times for a robust class of multidimensional nonuniformly hyperbolic transformations. Oliveira and Viana [15] proved that there is a unique equilibrium state µ for a large class of nonuniformly expanding transformations and Hölder continuous potentials with small variation. For an open class of potentials with small variation, we prove quasicompactness of the RuellePerronFrobenius operator in a space Vθ of functions with essential bounded variation that strictly contain Hölder continuous observables. We deduce that the equilibrium states have exponential decay of correlations. Furthermore, we prove exponential asymptotic distribution of hitting times and lognormal fluctuations of the return times around the average hµ(f). 1.
The thermodynamic approach to multifractal analysis
 Ergod.Th. Dynam. Sys
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Multifractal formalism derived from thermodynamics for general dynamical systems
 Electron. Res. Announc. Math. Sci
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