304 research outputs found

    Multiple mixing from weak hyperbolicity by the Hopf argument

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    International audienceWe show that using only weak hyperbolicity (no smoothness, compactness or exponential rates) the Hopf argument produces multiple mixing in an elementary way. While this recovers classical results with far simpler proofs, the point is the broader applicability implied by the weak hypotheses. Some of the results can also be viewed as establishing ''mixing implies multiple mixing'' outside the classical hyperbolic context

    The linear request problem

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    We propose a simple approach to a problem introduced by Galatolo and Pollicott, consisting in perturbing a dynamical system in order for its absolutely continuous invariant measure to change in a prescribed way. Instead of using transfer operators, we observe that restricting to an infinitesimal conjugacy already yields a solution. This allows us to work in any dimension and dispense from any dynamical hypothesis. In particular, we don't need to assume hyperbolicity to obtain a solution, although expansion moreover ensures the existence of an infinite-dimensional space of solutions.Comment: v2: the approach has been further simplified, only basic differential calculus is in fact needed instead of basic PD

    ORBIT GROWTH OF CONTACT STRUCTURES AFTER SURGERY

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    Investigation of the effects of a contact surgery construction and of invariance of contact homology reveals a rich new field of inquiry at the intersection of dynamical systems and contact geometry. We produce contact 3-flows not topologically orbit-equivalent to any algebraic flow, including examples on many hyperbolic 3-manifolds, and we show how the surgery produces dynamical complexity for any Reeb flow compatible with the resulting contact structure. This includes exponential complexity when neither the surg-ered flow nor the surgered manifold are hyperbolic. We also demonstrate the use in dynamics of contact homology, a powerful tool in contact geometry

    Degeneration of Dynamical Degrees in Families of Maps

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    The dynamical degree of a dominant rational map f:PNPNf:\mathbb{P}^N\rightarrow\mathbb{P}^N is the quantity δ(f):=lim(degfn)1/n\delta(f):=\lim(\text{deg} f^n)^{1/n}. We study the variation of dynamical degrees in 1-parameter families of maps fTf_T. We make a conjecture and ask two questions concerning, respectively, the set of tt such that: (1) δ(ft)δ(fT)ϵ\delta(f_t)\le\delta(f_T)-\epsilon; (2) δ(ft)<δ(fT)\delta(f_t)<\delta(f_T); (3) δ(ft)<δ(fT)\delta(f_t)<\delta(f_T) and δ(gt)<δ(gT)\delta(g_t)<\delta(g_T) for "independent" families of maps. We give a sufficient condition for our conjecture to hold and prove that it is true for monomial maps. We describe non-trivial families of maps for which our questions have affirmative and negative answers.Comment: 18 pages. This is an expanded version of the article publishd in Acta Arithmetica. It contains a corrected statement and full proof of Propostion 11(c

    Prevalence of non-Lipschitz Anosov foliations

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