Results 1  10
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18
Decay of Correlations in OneDimensional Dynamics
, 2002
"... We consider multimodal C³ interval maps f satisfying a summability condition on the derivatives Dn along the critical orbits which implies the existence of an absolutely continuous finvariant probability measure µ. If f is nonrenormalizable, µ is mixing and we show that the speed of mixing (decay ..."
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Cited by 48 (17 self)
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We consider multimodal C³ interval maps f satisfying a summability condition on the derivatives Dn along the critical orbits which implies the existence of an absolutely continuous finvariant probability measure µ. If f is nonrenormalizable, µ is mixing and we show that the speed of mixing (decay of correlations) is strongly related to the rate of growth of the sequence (Dn) as n → ∞. We also give sufficient conditions for µ to satisfy the Central Limit Theorem. This applies for example to the quadratic Fibonacci map which is shown to have subexponential decay of correlations.
On the uniform hyperbolicity of some nonuniformly hyperbolic systems
 2003, p1303–1309. YONGLUO CAO, STEFANO LUZZATTO, AND ISABEL RIOS
"... Abstract. We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability are necessarily uniformly expanding. We also present a version of ..."
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Cited by 19 (3 self)
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Abstract. We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability are necessarily uniformly expanding. We also present a version of this result for diffeomorphisms with nonuniformly hyperbolic sets.
Equilibrium states for nonuniformly expanding maps. Ergodic Theory & Dynamical Systems
, 2003
"... We construct equilibrium states, including measures of maximal entropy, for a large (open) class of nonuniformly expanding maps on compact manifolds. Moreover, we study uniqueness of these equilibrium states, as well as some of their ergodic properties. 1 ..."
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Cited by 13 (1 self)
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We construct equilibrium states, including measures of maximal entropy, for a large (open) class of nonuniformly expanding maps on compact manifolds. Moreover, we study uniqueness of these equilibrium states, as well as some of their ergodic properties. 1
Existence, uniqueness and stability of equilibrium states for nonuniformly expanding maps
, 2008
"... Abstract. We prove existence of finitely many ergodic equilibrium states for a large class of nonuniformly expanding local homeomorphisms on compact manifolds and Hölder continuous potentials with not very large oscillation. No Markov structure is assumed. If the transformation is topologically mix ..."
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Cited by 13 (7 self)
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Abstract. We prove existence of finitely many ergodic equilibrium states for a large class of nonuniformly expanding local homeomorphisms on compact manifolds and Hölder continuous potentials with not very large oscillation. No Markov structure is assumed. If the transformation is topologically mixing there is a unique equilibrium state, it is exact and satisfies a nonuniform Gibbs property. Under mild additional assumptions we also prove that the equilibrium states vary continuously with the dynamics and the potentials (statistical stability) and are also stable under stochastic perturbations of the transformation. 1.
Equilibrium states for partially hyperbolic horseshoes, Ergodic Theory Dynam.
 Systems
, 2011
"... Abstract We study ergodic properties of invariant measures for the partially hyperbolic horseshoes, introduced in We prove that every ergodic measure is hyperbolic, but the set of Lyapunov exponents in the central direction has gap: all ergodic invariant measures have negative exponent, with the e ..."
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Cited by 6 (0 self)
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Abstract We study ergodic properties of invariant measures for the partially hyperbolic horseshoes, introduced in We prove that every ergodic measure is hyperbolic, but the set of Lyapunov exponents in the central direction has gap: all ergodic invariant measures have negative exponent, with the exception of one ergodic measure with positive exponent. As a consequence, we obtain the existence of equilibrium states for any continuous potential. We also prove that there exists a phase transition for the smooth family of potentials given by φ t = t log DF  E c .
LARGE DEVIATIONS FOR SEMIFLOWS OVER A NONUNIFORMLY EXPANDING BASE
, 2006
"... We obtain a large deviation bound for continuous observables on suspension semiflows over a nonuniformly expanding base transformation with nonflat singularities or criticalities, where the roof function defining the suspension behaves like the logarithm of the distance to the singular/critical ..."
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Cited by 5 (0 self)
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We obtain a large deviation bound for continuous observables on suspension semiflows over a nonuniformly expanding base transformation with nonflat singularities or criticalities, where the roof function defining the suspension behaves like the logarithm of the distance to the singular/critical set of the base map. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the semiflow, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast as time goes to infinity. Suspension semiflows model the dynamics of flows admitting crosssections, where the dynamics of the base is given by the Poincaré return map and the roof function is the return time to the crosssection. The results are applicable in particular to semiflows modeling the geometric Lorenz attractors and the Lorenz flow, as well as other semiflows with multidimensional nonuniformly expanding base with nonflat singularities and/or criticalities under slow recurrence rate conditions to this singular/critical set. We are also able to obtain exponentially fast escape rates from subsets without full measure.
Birkhoff spectra for onedimensional maps with some hyperbolicity, in preparation
"... Abstract. We study the multifractal analysis for smooth dynamical systems in dimension one. It is characterized the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of hyperbolic measures for a topologically mixing C 2 map mode ..."
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Cited by 4 (2 self)
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Abstract. We study the multifractal analysis for smooth dynamical systems in dimension one. It is characterized the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of hyperbolic measures for a topologically mixing C 2 map modelled by an abstract dynamical system. A characterization which corresponds to above is also given for the ergodic basins of invariant probability measures. And it is shown that the complement of the set of quasiregular points carries full Hausdorff dimension. 1.