Quadratic maps without asymptotic measure

Hofbauer, Franz; Keller, Gerhard

doi:10.1007/bf02096761

Cited by 113 publications

(73 citation statements)

References 19 publications

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“…Observe that for C 2 maps in our conditions with finitely many smoothness domains, or with derivative of bounded variation, it is well known that there exists a unique ergodic absolutely continuous invariant probability measure µ with bounded density [35,53]. Since the function log dist(x, S) is Leb-integrable we also have that this function is µ-integrable.…”

Section: Suspension Semiflows Over Piecewise Expanding Maps With Singmentioning

confidence: 86%

Large deviations bound for semiflows over a non-uniformly expanding base

Araújo

2007

Bull Braz Math Soc, New Series

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We obtain a exponential large deviation upper bound for continuous observables on suspension semiflows over a non-uniformly expanding base transformation with non-flat 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.The arguments need the base transformation to exhibit exponential slow recurrence to the singular set which, in all known examples, implies exponential decay of correlations.Suspension semiflows model the dynamics of flows admitting cross-sections, where the dynamics of the base is given by the Poincaré return map and the roof function is the return time to the cross-section. The results are applicable in particular * The author was partially supported by CNPq-Brazil and FCT-Portugal through CMUP and POCI/ MAT/61237/2004. 1 to semiflows modeling the geometric Lorenz attractors and the Lorenz flow, as well as other semiflows with multidimensional non-uniformly 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.

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Section: Suspension Semiflows Over Piecewise Expanding Maps With Singmentioning

confidence: 86%

Large deviations bound for semiflows over a non-uniformly expanding base

Araújo

2007

Bull Braz Math Soc, New Series

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“…We emphasize that there are examples of unimodal maps satisfying all the conditions of the theorem except the local diffeomorphism assumption, for which the conclusion fails to hold; see [5,6].…”

Section: Introductionmentioning

confidence: 99%

On the uniform hyperbolicity of some nonuniformly hyperbolic systems

Alves

Araújo

Saussol

2002

Proc. Amer. Math. Soc.

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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 (probability one with respect to every invariant probability measure) are necessarily uniformly expanding. We also present a version of this result for diffeomorphisms with nonuniformly hyperbolic sets.

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“…The third ingredient are real quadratic maps without asymptotic measure constructed by Hofbauer and Keller [3]. Denote byω a (m) the set of all weak accumulation points of the sequence of probability measures (n −1 n−1 k=0 m • T −k a ) n>0 , where m denotes the normalized Lebesgue measure on I. Theorem 1 of [3] provides an uncountable family of parameters a for which the set of ergodic measures inω a (m) is infinite ( 3 ).…”

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confidence: 99%

“…Denote byω a (m) the set of all weak accumulation points of the sequence of probability measures (n −1 n−1 k=0 m • T −k a ) n>0 , where m denotes the normalized Lebesgue measure on I. Theorem 1 of [3] provides an uncountable family of parameters a for which the set of ergodic measures inω a (m) is infinite ( 3 ). Such maps do not, in particular, have a limit measure, because m • T −k a = P k 1 · m so that the existence of a limit measure for T a (in the sense of the above definition) would implyω a (m) = {µ}.…”

mentioning

confidence: 99%

“…The missing link that combines these results to produce examples of completely mixing maps without limit measure is the observation that the maps constructed in [3,4] are topologically mixing. For a unimodal interval map topological mixing is equivalent to the nondecomposability of its kneading ( 1 ) More precisely, Bruin and Hawkins assume that the map T a has no Cantor attractor in the sense of Milnor.…”

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confidence: 99%

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Completely mixing maps without limit measure

Keller

2004

Colloq. Math.

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Abstract. We combine some results from the literature to give examples of completely mixing interval maps without limit measure.Let X be a compact metric space with Borel σ-algebra B and equipped with some Borel measure m. Consider a transformation T : X → X that is non-singular with respect to m, which means that m(T −1 A) = 0 wheneverWe adopt the following definitions:• The system (X, B, m, T ) is completely mixing if lim n→∞ P n f = 0 for each f ∈ L 1 m with ¡ f dm = 0.• A probability measure µ on B is a limit measure for (X, B, m, T ) if for each probability density h ∈ L 1 m the measures P n h · m converge weakly to µ, in other words, if• If a system (X, B, m, T ) is completely mixing and has a nontrivial limit measure µ (i.e. µ is not a one-point mass), then µ is called a stochastic attractor for the system. • A probability measure µ on B is a Sinai-Ruelle-Bowen measure for the system (X, B, m, T ) if for each ϕ ∈ C(X),

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Quadratic maps without asymptotic measure

Cited by 113 publications

References 19 publications

Large deviations bound for semiflows over a non-uniformly expanding base

Large deviations bound for semiflows over a non-uniformly expanding base

On the uniform hyperbolicity of some nonuniformly hyperbolic systems

Completely mixing maps without limit measure

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