Marcelo R.R. Alves

Published and accepted articles

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● Braid stability and the Hofer metric.

    (with Matthias Meiwes)        

     To appear in Annales Henri Lebesgue

     Available here arXiv:2112.11351.

    In this article, we show that the braid type of a set of 1-periodic orbits of a non-degenerate Hamiltonian diffeomorphism on a surface is stable under perturbations that are sufficiently small with respect to the Hofer metric. We call this new phenomenon braid stability for the Hofer metric.  We also establish a purely dynamical result: given any surface diffeomorphism 𝜙 the supremum of the topological entropy of the braid types realized as periodic orbits of 𝜙 equals the topological entropy of 𝜙.

       The combination of these two results implies that the topological entropy is lower-semi-continuous with respect to the Hofer metric on non-degenerate Hamiltonian diffeomorphisms.

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● Entropy collapse versus entropy rigidity for Reeb and Finsler flows.

   (with Alberto Abbondandolo, Murat Saglam and Felix Schlenk)  

    Published in Selecta Mathematica (open access)

   On every closed contact manifold there exist contact forms with volume one whose Reeb flows have arbitrarily small topological entropy. In contrast, for many closed manifolds there is a uniform positive lower bound for the topological entropy of (not necessarily reversible) normalized Finsler geodesic flows.

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● C0-Robustness of topological entropy for geodesic flows.
    (with Lucas Dahinden, Matthias Meiwes and Louis Merlin)        Available here arXiv:2109.03917.
    To appear in Journal of Fixed Point Theory and Applications in the special volume celebrating Viterbo's 60th birthday.

    In this paper, we study the regularity of topological entropy, as a function on the space of Riemannian metrics endowed with the C0 topology. We establish several instances of entropy robustness (persistence of entropy non-vanishing after C0 small perturbations). A large part of this paper is dedicated to metrics on the 2-dimensional torus, for which our main results are that metrics with a contractible closed geodesic have robust entropy and that metrics with robust positive entropy on the torus are C∞ generic.

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● Reeb orbits that force topological entropy.       (with Abror Pirnapasov)

     Published in Ergodic Theory and Dynamical Systems.
     Also available here arXiv:2004.08106.

     We develop a forcing theory of topological entropy for Reeb flows in dimension 3. A transverse link L in a co-orientable closed contact 3-manifold (Y,ξ) is said to force topological entropy if (Y,ξ) admits a Reeb flow with vanishing topological entropy, and every Reeb flow on (Y,ξ) realizing L as a set of periodic Reeb orbits has positive topological entropy. Our main results establish topological conditions on a transverse link which imply that L forces topological entropy. These conditions are formulated in terms of two Floer theoretical invariants: the cylindrical contact homology on the complement of a transverse link introduced by Momin, and the Legendrian contact homology on the complement of a transverse link, introduced in my thesis and developed here. We then use these results to show that on every contact 3-manifold that admits a Reeb flow with vanishing topological entropy, there exists transverse links that force topological entropy.

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●  Cylindrical contact homology and topological entropy.    

      Geometry & Topology 20 (2016) 3519–3569

      We establish a relation between the growth of the cylindrical contact homology of a contact manifold and the topological entropy of Reeb flows on this manifold. We show that if a contact manifold (M,ξ) admits a hypertight contact form λ for which the cylindrical contact homology has exponential homotopical growth rate, then the Reeb flow of every contact form on (M,ξ) has positive topological entropy. Using this result, we provide numerous new examples of contact 3–manifolds on which every Reeb flow has positive topological entropy.

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●  Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds.    

       Journal of Modern Dynamics 10, 497-509

      Let (M,ξ) be a compact contact 3-manifold and assume that there exists a contact form α on (M,ξ) whose Reeb flow is Anosov. We show this implies that every Reeb flow on (M,ξ) has positive topological entropy. Our argument builds on previous work of the author and the paper "Counting orbits of Anosov flows in free homotopy classes" of Barthelmé and Fenley. This result combined with the work "Contact Anosov flows on hyperbolic 3–manifolds" of Foulon and Hasselblatt is then used to obtain the first examples of hyperbolic contact 3-manifolds on which every Reeb flow has positive topological entropy.

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●  Legendrian contact homology and topological entropy.    

       Journal of Topology and Analysis 11, n° 1, 53–108

      In this paper we study the growth rate of a version of Legendrian contact homology, which we call strip Legendrian contact homology, in 3-dimensional contact manifolds and its relation to the topological entropy of Reeb flows. We show that: if for a pair of Legendrian knots in a contact 3-manifold (M,ξ) the strip Legendrian contact homology is defined and has exponential homotopical growth with respect to the action, then every Reeb flow on (M,ξ) has positive topological entropy. This has the following dynamical consequence: for all Reeb flows (even degenerate ones) on (M,ξ) the number of hyperbolic periodic orbits grows exponentially with respect to the period. We show that for an infinite family of 3-manifolds, infinitely many different contact structures exist that possess a pair of Legendrian knots for which the strip Legendrian contact homology has exponential growth rate.

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●  Dynamically exotic contact spheres in dimensions≥ 7.         (joint with Matthias Meiwes)

        Commentarii Mathematici Helvetici 94, n° 3, 569–622                       

     We exhibit the first examples of contact structures on (odd-dimensional) spheres of dimension ≥ 7, all equipped with their standard smooth structures, for which every Reeb flow has positive topological entropy. As a new technical tool for the study of the volume growth of Reeb flows we introduce the notion of algebraic growth of wrapped Floer homology. Its power stems from its stability under several geometric operations on Liouville domains.

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●  Topological entropy for Reeb vector fields in dimension three via open book decompositions.                                (joint with Vincent Colin and Ko Honda)

       Journal de l’École polytechnique — Mathématiques 6, 119-148                         

      Given an open book decomposition of a contact three manifold (M, ξ) with pseudo-Anosov monodromy and fractional Dehn twist coefficient c = k/n, we construct a Legendrian knot Λ close to the stable foliation of a page, together with a small Legendrian pushoff Λ. When k ≥ 5, we apply the techniques developed in the article "Reeb vector fields and open book decompositions" by Colin and Honda to show that the strip Legendrian contact homology of Λ → Λ is well-defined and has an exponential growth property. Combining these with my previous work we obtain that all Reeb vector fields for ξ have positive topological entropy.

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