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Ergodicity and partial hyperbolicity on Seifert manifoldsJul 10 2019We show that conservative partially hyperbolic diffeomorphism isotopic to the identity on Seifert 3-manifolds are ergodic.
A dichotomy for measures of maximal entropy near time-one maps of transitive Anosov flowsApr 16 2019We show that time-one maps of transitive Anosov flows of compact manifolds are accumulated by diffeomorphisms robustly satisfying the following dichotomy: either all of the measures of maximal entropy are non-hyperbolic, or there are exactly two ergodic ... More
SRB measures and Young towers for surface diffeomorphismsMar 29 2019We give geometric conditions that are necessary and sufficient for the existence of Sinai-Ruelle-Bowen (SRB) measures for $C^{1+\alpha}$ surface diffeomorphisms, thus proving a version of the Viana conjecture. As part of our argument we give an original ... More
Transfer operators, atomic decomposition and the BestiaryMar 16 2019Arbieto and S. recently used atomic decomposition to study transfer operators. We give a long list of old and new expanding dynamical systems for which those results can be applied, obtaining the quasi-compactness of transfer operator acting on Besov ... More
Transfer operators and atomic decompositionMar 16 2019Apr 16 2019We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, ... More
Transfer operators and atomic decompositionMar 16 2019We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, ... More
Classic and exotic Besov spaces induced by good gridsMar 16 2019In a previous work we introduced Besov spaces $\mathcal{B}^s_{p,q}$ defined on a measure spaces with a good grid, with $p\in [1,\infty)$, $q\in [1,\infty]$ and $0< s< 1/p$. Here we show that classical Besov spaces on compact homogeneous spaces are examples ... More
Besov-ish spaces through atomic decompositionMar 16 2019We use the method of atomic decomposition to build new families of function spaces, similar to Besov spaces, in measure spaces with grids, a very mild assumption. Besov spaces with low regularity are considered in measure spaces with good grids, and results ... More
Uniform convergence in von Neumann's ergodic theorem in absence of a spectral gapFeb 11 2019Von Neumann's original proof of the ergodic theorem is revisited. A convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit ... More
Uniform convergence in von Neumann's ergodic theorem in absence of a spectral gapFeb 11 2019May 23 2019Von Neumann's original proof of the ergodic theorem is revisited. A convergence rate is established under the assumption that one can control the density of the spectrum of the underlying self-adjoint operator when restricted to suitable subspaces. Explicit ... More
On the rotation sets of generic homeomorphisms on the torus $\mathbb T^d$Jan 02 2019We study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$, $d\ge 2$. In the conservative setting, we prove that there exists a Baire residual subset of the set $\text{Homeo}_{0, \lambda}(\mathbb T^2)$ of conservative ... More
Schwarz reflections and the TricornDec 04 2018We continue our study of the family $\mathcal{S}$ of Schwarz reflection maps with respect to a cardioid and a circle which was started in [LLMM1]. We prove that there is a natural combinatorial bijection between the geometrically finite maps of this family ... More
Nonlinear Decomposition Principle and Fundamental Matrix Solutions for Dynamic Compartmental SystemsNov 29 2018A decomposition principle for nonlinear dynamic compartmental systems is introduced in the present paper. This theory is based on the novel mutually exclusive and exhaustive system and subsystem decomposition methodologies. A deterministic mathematical ... More
Nonlinear Decomposition Principle and Fundamental Matrix Solutions for Dynamic Compartmental SystemsNov 29 2018Mar 13 2019A decomposition principle for nonlinear dynamic compartmental systems is introduced in the present paper. This theory is based on the novel mutually exclusive and exhaustive system and subsystem decomposition methodologies. A deterministic mathematical ... More
Nonlinear Decomposition Principle and Fundamental Matrix Solutions for Dynamic Compartmental SystemsNov 29 2018May 10 2019A decomposition principle for nonlinear dynamic compartmental systems is introduced in the present paper. This theory is based on the novel mutually exclusive and exhaustive system and subsystem decomposition methodologies. A deterministic mathematical ... More
Nonlinear Decomposition Principle and Fundamental Matrix Solutions for Dynamic Compartmental SystemsNov 29 2018Jun 07 2019A decomposition principle for nonlinear dynamic compartmental systems is introduced in the present paper. This theory is based on the mutually exclusive and exhaustive, analytical and dynamic, novel system and subsystem partitioning methodologies. A deterministic ... More
Non vanishing of theta functions and sets of small multiplicative energyOct 12 2018Let $\chi$ range over the $(p-1)/2$ even Dirichlet characters modulo a prime $p$ and denote by $\theta (x,\chi)$ the associated theta series. The asymptotic behaviour of the second and fourth moments proved by Louboutin and the author implies that there ... More
Unique ergodicity of the horocycle flow on Riemannnian foliationsOct 10 2018A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore implies minimality. The horocycle flow ... More
Entropy on normed semigroups (A unifying approach to entropy)Aug 11 2018We present a unifying approach to the study of entropies in Mathematics, such as measure entropy, topological entropy, algebraic entropy, set-theoretic entropy. We take into account discrete dynamical systems, that is, pairs $(X,T)$, where $X$ is the ... More
Bowen factors, their degree, and codings of surface diffeomorphismsJul 11 2018We show that symbolic finite-to-one extensions of the type constructed by O. Sarig for surface diffeomorphisms induce H\"older-continuous conjugacies on large sets, sometimes preserving transitivity. We deduce this from their Bowen property. This notion, ... More
On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard MapsJul 06 2018The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition $h_*$ for the topological entropy of $T$. We prove that $h_*$ is not smaller than the value given by ... More
Boundary rigidity of negatively-curved asymptotically hyperbolic surfacesMay 14 2018In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.
Optimal switching sequence for switched linear systemsMay 12 2018Jul 01 2018We study the following optimization problem over a dynamical system that consists of several linear subsystems: Given a finite set of n-by-n matrices and an n-dimensional vector, find a sequence of K matrices, each chosen from the given set of matrices, ... More
Geometry of limits of zeros of polynomial sequences of type $(1,1)$May 07 2018In this paper, we study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is either an arc, or a circle, ... More
Quadratic Irrationals, Closed Geodesics on the Modular Surface and Dynamical Zeta FunctionsMay 03 2018We show that generating functions associated to the sequence of convergents of a quadratic irrational are related in a natural way to the dynam- ical zeta function of a hyperbolic automorphism of the 2-torus. As a corollary, this shows that the L\'evy ... More
Zeta Function at Zero for Surfaces with BoundaryMar 29 2018Jul 25 2018We study the Ruelle zeta function at zero for negatively curved oriented surfaces with boundary. At zero, the zeta function has a zero and its multiplicity is shown to be determined by the Euler characteristic of the surface. This is shown by considering ... More
Commensurating actions for groups of piecewise continuous transformationsMar 22 2018We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the ... More
Complex Cellular StructuresFeb 21 2018We introduce the notion of a \emph{complex cell}, a complexification of the cells/cylinders used in real tame geometry. Complex cells are equipped with a natural notion of holomorphic extension, and the hyperbolic geometry of a cell within its extension ... More
Complex Cellular StructuresFeb 21 2018Apr 18 2019We introduce the notion of a complex cell, a complexification of the cells/cylinders used in real tame geometry. For $\delta\in(0,1)$ and a complex cell $\mathcal{C}$ we define its holomorphic extension $\mathcal{C}\subset\mathcal{C}^\delta$, which is ... More
The dimension spectrum of graph directed Markov systemsFeb 04 2018In this paper we study the dimension spectrum of general conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We perform a comprehensive study of the dimension spectrum addressing questions regarding its ... More
Regularity of Kleinian limit sets and Patterson-Sullivan measuresDec 12 2017We consider several (related) notions of geometric regularity in the context of limit sets of geometrically finite Kleinian groups and associated Patterson-Sullivan measures. We begin by computing the upper and lower regularity dimensions of the Patterson-Sullivan ... More
Upper large deviations bound for singular-hyperbolic attracting setsNov 27 2017Dec 10 2018We obtain a exponential large deviation upper bound for continuous observables on suspension semiflows over a non-uniformly expanding base transformation with non-flat singularities and/or discontinuities, where the roof function defining the suspension ... More
Nonautonomous Conley Index Theory: The Homology Index and Attractor-Repeller decompositionsNov 13 2017In a previous work, the author established a nonautonomous Conley index based on the interplay between a nonautonomous evolution operator and its skew-product formulation. This index is refined to obtain a Conley index for families of nonautonomous evolution ... More
Intermittent behaviors in weakly coupled map latticesNov 04 2017In this paper, we study intermittent behaviors of coupled piecewise-expanding map lattices with two nodes and a weak coupling. We show that the successive phase transition between ordered and disordered phases occurs for almost every orbit. That is, we ... More
Statistical stability of mostly expanding diffeomorphismsOct 22 2017Dec 11 2017We study how physical measures vary with the underlying dynamics in the open class of $C^r$, $r>1$, strong partially hyperbolic diffeomorphisms for which the central Lyapunov exponents of every Gibbs $u$-state is positive. If transitive, such a diffeomorphism ... More
Quantum-classical correspondence on associated vector bundles over locally symmetric spacesOct 12 2017Jul 10 2018For a compact Riemannian locally symmetric space $\mathcal M$ of rank one and an associated vector bundle $\mathbf V_\tau$ over the unit cosphere bundle $S^\ast\mathcal M$, we give a precise description of those classical (Pollicott-Ruelle) resonant states ... More
Quantum-classical correspondence on associated vector bundles over locally symmetric spacesOct 12 2017Mar 12 2019For a compact Riemannian locally symmetric space $\mathcal M$ of rank one and an associated vector bundle $\mathbf V_\tau$ over the unit cosphere bundle $S^\ast\mathcal M$, we give a precise description of those classical (Pollicott-Ruelle) resonant states ... More
X-ray Transform and Boundary Rigidity for Asymptotically Hyperbolic ManifoldsSep 15 2017We consider the boundary rigidity problem for asymptotically hyperbolic manifolds. We show injectivity of the X-ray transform in several cases and consider the non-linear inverse problem which consists of recovering a metric from boundary measurements ... More
Principal Floquet subspaces and exponential separations of type II with applications to random delay differential equationsMay 03 2017This paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type ... More
Critical exponents of normal subgroups, the spectrum of group extended transfer operators, and Kazhdan distanceFeb 20 2017For a pinched Hadamard manifold $X$ and a discrete group of isometries $\Gamma$ of $X$, the critical exponent $\delta_\Gamma$ is the exponential growth rate of the orbit of a point in $X$ under the action of $\Gamma$. We show that the critical exponent ... More
Local perturbations of conservative $C^1$-diffeomorphismsDec 20 2016A number of techniques have been developed to perturb the dynamics of $C^1$-diffeomorphisms and to modify the properties of their periodic orbits. For instance, one can locally linearize the dynamics, change the tangent dynamics, or create local homoclinic ... More
Asymptotic Schur orthogonality in hyperbolic groups with application to monotonyOct 19 2016Nov 19 2016We prove a generalization of Schur orthogonality relations for certain classes of representations of Gromov hyperbolic groups. We apply the obtained results to show that representations of non-abelian free groups associated to the Patterson-Sullivan measures ... More
Asymptotic Schur orthogonality in hyperbolic groups with application to monotonyOct 19 2016We prove a generalization of Schur orthogonality relations for certain classes of representations of Gromov hyperbolic groups. We apply the obtained results to show that representations of non-abelian free groups associated to the Patterson-Sullivan measures ... More
On some generalizations of skew-shifts on $\mathbb{T}^2$Oct 11 2016In this paper we investigate maps of the two-torus $\mathbb{T}^2$ of the form $T(x,y)=(x+\omega,g(x)+f(y))$ for Diophantine $\omega\in\mathbb{T}$ and for a class of maps $f,g:\mathbb{T}\to\mathbb{T}$, where each $g$ is strictly monotone and of degree ... More
On the Global Dynamics of an Electroencephalographic Mean Field Model of the NeocortexOct 11 2016May 05 2017This paper investigates the global dynamics of a mean field model of the electroencephalogram developed by Liley et al., 2002. The model is presented as a system of coupled ordinary and partial differential equations with periodic boundary conditions. ... More
On the Global Dynamics of an Electroencephalographic Mean Field Model of the NeocortexOct 11 2016This paper investigates the global dynamics of a mean field model of the electroencephalogram developed by Liley \emph{et al.}, 2002. The model is presented as a system of coupled ordinary and partial differential equations with periodic boundary conditions. ... More
Statistical properties of the maximal entropy measure for partially hyperbolic attractorsOct 05 2016We show the existence and uniqueness of the maximal entropy probability measure for partially hyperbolic diffeomorphisms which are semi-conjugate to nonuniformly expanding maps. Using the theory of projective metric on cones we then prove exponential ... More
Arcwise connectedness of the set of ergodic measures of hereditary shiftsOct 03 2016Oct 07 2016We show that the set of ergodic invariant measures of a shift space with a safe symbol (this includes all hereditary shifts) is arcwise connected when endowed with the $d$-bar metric. As a consequence the set of ergodic measures of such a shift is also ... More
Characterization of Exact Lumpability for Vector Fields on Smooth ManifoldsJul 05 2016Jul 06 2016We characterize the exact lumpability of smooth vector fields on smooth manifolds. We derive necessary and sufficient conditions for lumpability and express them from four different perspectives, thus simplifying and generalizing various results from ... More
Perspectives on Kuperberg flowsJul 04 2016The "Seifert Conjecture" asks, "Does every non-singular vector field on the 3-sphere ${\mathbb S}^3$ have a periodic orbit?" In a celebrated work, Krystyna Kuperberg gave a construction of a smooth aperiodic vector field on a plug, which is then used ... More
A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functionsJun 29 2016Jun 06 2018Over the last few years Pohl (partly jointly with coauthors) developed dual `slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $\Gamma\backslash\mathbb{H}$ ... More
Isomorphisms between eigenspaces of slow and fast transfer operatorsJun 29 2016For any Hecke triangle surface $\Gamma\backslash\mathbb{H}$ of finite or infinite area and any finite-dimensional unitary representation $\chi$ of the Hecke triangle group $\Gamma$ there had been constructed two families of Ruelle-like transfer operators ... More
Entropy of $C^1$ diffeomorphisms without a dominated splittingJun 06 2016Nov 06 2017A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local $C^1$-perturbations. Our main theorem shows that, when one works in the $C^1$ topology, the entropy of such ... More
Entropy of $C^1$ diffeomorphisms without a dominated splittingJun 06 2016A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local $C^1$-perturbations. Our main theorem shows that, when one works in the $C^1$ topology, the entropy of such ... More
Degree, mixing, and absolutely continuous spectrum of cocycles with values in compact Lie groupsMay 13 2016We consider skew products $$T_\phi:X\times G\to X\times G,~~(x,g)\mapsto(F_1(x),g\;\!\phi(x)),$$ where $X$ is a compact manifold with probability measure, $G$ a compact Lie group with Lie algebra $\frak g$, $F_1:X\to X$ the time-one map of a measure-preserving ... More
Anosov representations and dominated splittingsMay 05 2016We provide a link between Anosov representations introduced by Labourie and dominated splitting of linear cocycles. This allows us to obtain equivalent characterizations for Anosov representations and to recover recent results due to Gu\'eritaud-Guichard-Kassel-Wienhard ... More
Anosov representations and dominated splittingsMay 05 2016Jun 13 2017We provide a link between Anosov representations introduced by Labourie and dominated splitting of linear cocycles. This allows us to obtain equivalent characterizations for Anosov representations and to recover recent results due to Gu\'eritaud-Guichard-Kassel-Wienhard ... More
Conformal graph directed Markov systems on Carnot groupsMay 04 2016We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, we develop the thermodynamic formalism and show that, under natural hypotheses, ... More
A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theoryApr 27 2016We prove a complex Ruelle-Perron-Frobenius theorem for Markov shifts over an infinite alphabet, whence extending results by M. Pollicott from the finite to the infinite alphabet setting. As an application we obtain an extension of renewal theory in symbolic ... More
Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observablesMar 31 2016Jun 20 2017We consider a smooth one-parameter family $t \to f_t$ of diffeomorphisms with compact transitive Axiom A attractors. Our first result (corrected) is that for any function $G$ in the Sobolev space $H^r_p$, with $p>1$ and $0<r<1/p$, the map $R(t)$ sending ... More
Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observablesMar 31 2016Nov 27 2016We consider a smooth one-parameter family t -> f_t of diffeomorphisms with compact transitive Axiom A attractors. Our first result is that for any function G in the Sobolev space H^r_p, with p>1 and 0<r<1/p, the map R(t) sending t to the average of G ... More
Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observablesMar 31 2016We consider a smooth one-parameter family t -> f_t of diffeomorphisms with compact transitive Axiom A attractors. Our first result is that for any function G in the Sobolev space H^r_p, with p>1 and 0<r<1/p, the map R(t) sending t to the average of G ... More
Measure-theoretical properties of center foliationsMar 11 2016Center foliations of partially hyperbolic diffeomorphisms may exhibit pathological behavior from a measure-theoretical viewpoint: quite often, the disintegration of the ambient volume measure along the center leaves consists of atomic measures. We add ... More
Generalized beta-transformations and the entropy of unimodal mapsFeb 10 2016Feb 17 2016Generalized beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformation $x \mapsto \beta x ~\pmod 1$, where $\beta>1$, and replacing some of the branches with branches of constant negative slope. If ... More
Generalized beta-transformations and the entropy of unimodal mapsFeb 10 2016Jan 11 2017Generalized beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformation $x \mapsto \beta x ~\pmod 1$, where $\beta>1$, and replacing some of the branches with branches of constant negative slope. If ... More
Semigroup actions of expanding mapsJan 17 2016Jun 22 2016We consider semigroups of Ruelle-expanding maps, parameterized by random walks on the free semigroup, with the aim of examining their complexity and exploring the relation between intrinsic properties of the semigroup action and the thermodynamic formalism ... More
Topological Entropy on Points without Physical-like BehaviourDec 07 2015Apr 20 2016We study a class of asymptotically entropy-expansive $C^1$ diffeomorphisms with dominated splitting on a compact manifold $M$, that satisfy the specification property. This class includes, in particular, transitive Anosov diffeomorphisms and time-one ... More
Dynamical properties of the absolute period foliationNov 25 2015We show that the absolute period foliation of the principal stratum of abelian differentials on a surface of genus at least 3 is ergodic. We also investigate the absolute period foliation of affine invariant manifolds.
Geometric Lorenz flows with historic behaviorNov 17 2015Aug 18 2016We will show that, in the the geometric Lorenz flow, the set of initial states which give rise to orbits with historic behavior is residual in a trapping region.
Linear response for intermittent mapsAug 11 2015Feb 16 2016We consider the one parameter family $\alpha \mapsto T_\alpha$ ($\alpha \in [0,1)$) of Pomeau-Manneville type interval maps $T_\alpha(x)=x(1+2^\alpha x^\alpha)$ for $x \in [0,1/2)$ and $T_\alpha(x)=2x-1$ for $x \in [1/2, 1]$, with the associated absolutely ... More
Central limit theorem for generalized Weierstrass functionsAug 09 2015Jan 05 2018Let $f$ be a $C^{2+\epsilon}$ expanding map of the circle and $v$ be a $C^{1+\epsilon}$ real function of the circle. Consider the twisted cohomological equation $v(x) = \alpha (f(x)) - Df(x) \alpha (x)$ which has a unique bounded solution $\alpha$. We ... More
Central limit theorem for generalized Weierstrass functionsAug 09 2015Jul 13 2016Let $f$ be a $C^{2+\epsilon}$ expanding map of the circle and $v$ be a $C^{1+\epsilon}$ real function of the circle. Consider the twisted cohomological equation $v(x) = \alpha (f(x)) - Df(x) \alpha (x)$ which has a unique bounded solution $\alpha$. We ... More
The limit set for discrete complex hyperbolic groupsJun 26 2015Jun 14 2016Given a discrete subgroup $\Gamma$ of $PU(1,n)$ it acts by isometries on the unit complex ball $\Bbb{H}^n_{\Bbb{C}}$, in this setting a lot of work has been done in order to understand the action of the group. However when we look at the action of $\Gamma$ ... More
Möbius disjointness for homogeneous dynamicsJun 25 2015Oct 16 2016We prove Sarnak's M\"obius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected Lie groups. Namely, we show that if $G$ is any such group, $\Gamma\subset G$ a lattice, and $u\in G$ an Ad-unipotent element, then ... More
Spectral properties of horocycle flows for surfaces of constant negative curvatureJun 21 2015Mar 13 2016We consider flows, called $W^{\rm u}$ flows, whose orbits are the unstable manifolds of a codimension one Anosov flow. Under some regularity assumptions, we give a short proof of the strong mixing property of $W^{\rm u}$ flows and we show that $W^{\rm ... More
Teichmueller flow and Weil-Petersson flowMay 05 2015For a non-exceptional oriented surface S let Q(S) be the moduli space of area one quadratic differentials. We show that there is a Borel subset E of Q(S) which is invariant under the Teichmueller flow F^t and of full measure for every invariant Borel ... More
Ring homeomorphisms and prime endsMar 30 2015We show that every homeomorphic $W^{1,1}_{\rm loc}$ solution $f$ of a Beltrami equation $\overline{\partial}f=\mu\,\partial f$ in a domain $D\subseteq\Bbb C$ is the so--called ring $Q-$homeomorphism with $Q(z)=K^T_{\mu}(z, z_0)$ where $K^T_{\mu}(z, z_0)$ ... More
Transitivity of conservative diffeomorphisms isotopic to Anosov on $\mathbb{T}^3$Mar 23 2015We prove transitivity for volume preserving $C^{1+}$ diffeomorphisms on $\mathbb{T}^3$ which are isotopic to a linear Anosov automorphism along a path of weakly partially hyperbolic diffeomorphisms.
Dominated Pesin theory: convex sum of hyperbolic measuresMar 19 2015Jul 19 2017In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: ... More
Dominated Pesin theory: convex sum of hyperbolic measuresMar 19 2015Feb 22 2016In the uniformly hyperbolic setting it is well known that the measure supported on periodic orbits is dense in the convex space of all invariant measure. In this paper we consider the reverse question, in the non-uniformly hyperbolic setting: assuming ... More
The Dirichlet problem and prime endsMar 14 2015Mar 29 2015It is developed the theory of the boundary behavior of homeomorphic solutions of the Beltrami equations ${\bar{\partial}}f=\mu\,{\partial}f$ of the Sobolev class $W^{1,1}_{\rm loc}$ with respect to prime ends of domains. On this basis, under certain conditions ... More
Central limit theorem for the modulus of continuity of averages of observables on transversal families of piecewise expanding unimodal mapsMar 04 2015Apr 22 2016We prove that the Newton quotient of the average R(t) of a lipschitzian function (with non vanishing variation) with respect to the SRB measure on a transversal family f_t of piecewise expanding unimodal maps, after an appropriated normalization, converges ... More
Symbolic dynamics, automorphic functions, and Selberg zeta functions with unitary representationsMar 02 2015Jun 08 2016Using Hecke triangle surfaces of finite and infinite area as examples, we present techniques for thermodynamic formalism approaches to Selberg zeta functions with unitary finite-dimensional representations $(V,\chi)$ for hyperbolic surfaces (orbifolds) ... More
Robust criterion for the existence of nonhyperbolic ergodic measuresFeb 23 2015We give explicit $C^1$-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with one-dimensional center ... More
Micromeasure distributions and applications for conformally generated fractalsFeb 19 2015We study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconer's distance set problem and dimensions of projections. Our analysis includes hyperbolic Julia sets, limit sets of Schottky ... More
On boundary behavior of mappings with finite distortion in the planeFeb 05 2015Feb 08 2015In the present paper, it was studied the boundary behavior of the so-called lower Q-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions on the function ... More
Furstenberg transformations on Cartesian products of infinite-dimensional toriJan 26 2015We consider in this note Furstenberg transformations on Cartesian products of infinite-dimensional tori. Under some appropriate assumptions, we show that these transformations are uniquely ergodic with respect to the Haar measure and have countable Lebesgue ... More
A survey of some arithmetic applications of ergodic theory in negative curvatureJan 09 2015This paper is a survey of some arithmetic applications of techniques in the geometry and ergodic theory of negatively curved Riemannian manifolds, focusing on the joint works of the authors. We describe Diophantine approximation results of real numbers ... More
Topological entropy of Lorenz-like flowsDec 03 2014We use entropy theory as a new tool for studying Lorenz-like classes of flows in any dimension. More precisely, we show that every Lorenz-like class is entropy expansive, and has positive entropy which varies continuously with vector fields. We deduce ... More
Amenability, Critical Exponents of Subgroups and Growth of Closed GeodesicsNov 25 2014Jul 21 2015Let $\Gamma$ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold $X$. We show that a normal subgroup $\Gamma_0$ has critical exponent equal to the critical exponent of $\Gamma$ if and only if $\Gamma / \Gamma_0$ ... More
Remarks on the dynamics of the horocycle flow for homogeneous foliations by hyperbolic surfacesOct 27 2014Jul 25 2015This article is a first step towards the understanding of the dynamics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Martinez and A. Verjovsky on the minimality of this flow assuming ... More
Typical properties of periodic Teichmueller geodesicsSep 21 2014Nov 05 2015Call a property for periodic orbits of the Teichmueller flow acting on the moduli space Q of area one abelian differentials on a surface of genus g typical if the growth rate of orbits with this property is maximal. We show that the following property ... More
Typical and atypical properties of periodic Teichmueller geodesicsSep 21 2014Feb 21 2017Consider a component Q of a stratum in the moduli space of area one abelian differentials on a surface of genus g. Call a property P for periodic orbits of the Teichmueller flow typical if the growth rate of orbits with this property is maximal. Typical ... More
Diffeomorphisms with positive metric entropyAug 19 2014Oct 04 2015We obtain a dichotomy for $C^1$-generic, volume preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. hyperbolic and the splitting into stable and unstable spaces ... More
Diffeomorphisms with positive metric entropyAug 19 2014Sep 19 2017We obtain a dichotomy for $C^1$-generic, volume-preserving diffeomorphisms: either all the Lyapunov exponents of almost every point vanish or the volume is ergodic and non-uniformly Anosov (i.e. nonuniformly hyperbolic and the splitting into stable and ... More
Linear response, or elseAug 13 2014Consider a smooth one-parameter family t -> f_t of dynamical systems f_t, with |t|<epsilon. Assume that for all t (or for many t close to t=0) the map f_t admits a unique SRB invariant probability measure m_t. We say that linear response} holds if t -> ... More
On boundary behavior of spatial mappingsAug 03 2014We show that homeomorphisms $f$ in ${\Bbb R}^n$, $n\geqslant3$, of finite distortion in the Orlicz--Sobolev classes $W^{1,\varphi}_{\rm loc}$ with a condition on $\varphi$ of the Calderon type and, in particular, in the Sobolev classes $W^{1,p}_{\rm loc}$ ... More
Generalized Mass-Action Systems and Positive Solutions of Polynomial Equations with Real and Symbolic ExponentsJun 25 2014Dynamical systems arising from chemical reaction networks with mass action kinetics are the subject of chemical reaction network theory (CRNT). In particular, this theory provides statements about uniqueness, existence, and stability of positive steady ... More
A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative ResultsMay 05 2014Dec 04 2014We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on the scattering map (outer) dynamics and on the recurrence property of the (inner) dynamics ... More
A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative ResultsMay 05 2014Apr 25 2017We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the `outer dynamics' along homoclinic orbits to a normally hyperbolic ... More