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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 28 (13 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 14 (2 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 8 (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 4 (3 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.
Dynamical Zeta Functions For SUnimodal Maps
, 1999
"... . Let f be a nonrenormalizable Sunimodal map. We prove that f is a ColletEckmann map if its dynamical zeta function looks like that of a uniformly hyperbolic map. 1. Introduction A unimodal map f : [0; 1] ! [0; 1] is called Sunimodal if f(0) = f(1) = 0 and if it has nonpositive Schwarzian derivat ..."
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Cited by 1 (0 self)
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. Let f be a nonrenormalizable Sunimodal map. We prove that f is a ColletEckmann map if its dynamical zeta function looks like that of a uniformly hyperbolic map. 1. Introduction A unimodal map f : [0; 1] ! [0; 1] is called Sunimodal if f(0) = f(1) = 0 and if it has nonpositive Schwarzian derivative Sf = f 000 f 0 \Gamma 3 2 ( f 00 f 0 ) 2 . For such a map set '(x) := log jf 0 (x)j and 'n (x) := '(x) + '(fx) + \Delta \Delta \Delta + '(f n\Gamma1 x). Let \Pi n = fx 2 [0; 1] : f n (x) = xg and define for t 2 R the zeta function i t (z) = exp 1 X n=1 z n n i n;t where i n;t = X x2\Pi n e (t\Gamma1)' n (x) : Observe that i 0 (z) is just the usual dynamical zeta function. Denote per := inffj(f n ) 0 (x)j 1=n : n ? 0; x 2 \Pi n g : Nowicki and Sands [6] proved that per ? 1 (i.e. f is uniformly hyperbolic on periodic orbits) if and only if f satisfies the ColletEckmann condition (i.e. there are C ? 0 and CE ? 1 such that j(f n ) 0 (fc)j C n ...
Absolutely Continuous Invariant Measures As Equilibrium States For Piecewise Invertible Maps
, 1999
"... . We characterize, for piecewise invertible dynamical systems in arbitrary dimension, the invariant probability measures which are absolutely continuous w.r.t. a given measure m with jacobian J as the equilibrium states w.r.t. \Gamma log J . We assume mainly bounded distortion of log J and negativit ..."
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Cited by 1 (0 self)
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. We characterize, for piecewise invertible dynamical systems in arbitrary dimension, the invariant probability measures which are absolutely continuous w.r.t. a given measure m with jacobian J as the equilibrium states w.r.t. \Gamma log J . We assume mainly bounded distortion of log J and negativity of the pressure of the boundary of the pieces of the map. The proof uses shadowing (defined in [5,7] for Markov extensions) to apply an abstract proposition of F. Ledrappier [27] (introduced to characterize a.c.i.m.'s on the interval) using topological pressure estimates. Contrarily to [15,24], we do not use Markov extensions or the more usual transfer operator techniques. We apply this first to Lebesgue measure and multidimensional piecewise expanding maps with Jacobian not necessarily Holder and satisfying a generic condition which even holds for all piecewise expanding and affine mappings in the plane. As a corollary, we get existence of finitely many ergodic a.c.i.m.'s. Second, we con...
Equilibrium States for Partially Hyperbolic Horseshoes
, 801
"... In this paper, we study ergodic features of invariant measures for the partially hyperbolic horseshoe at the boundary of uniformly hyperbolic diffeomorphisms constructed in [12]. Despite the fact that the nonwandering set is a horseshoe, it contains intervals. We prove that every recurrent point ha ..."
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In this paper, we study ergodic features of invariant measures for the partially hyperbolic horseshoe at the boundary of uniformly hyperbolic diffeomorphisms constructed in [12]. Despite the fact that the nonwandering set is a horseshoe, it contains intervals. We prove that every recurrent point has nonzero Lyapunov exponents and all ergodic invariant measures are hyperbolic. As a consequence, we obtain the existence of equilibrium measures for any continuous potential. We also obtain an example of a family of C ∞ potentials with phase transition. 1