Showing papers in "Ergodic Theory and Dynamical Systems in 2004"
TL;DR: In this article, the authors consider topological dynamical systems that arise from locally compact Abelian groups on compact spaces of translation bounded measures and show that such a system has a pure point dynamical spectrum if and only if its diffraction spectrum is pure point.
Abstract: Certain topological dynamical systems that arise from actions of -compact locally compact Abelian groups on compact spaces of translation bounded measures are considered. Such a measure dynamical system is shown to have a pure point dynamical spectrum if and only if its diffraction spectrum is pure point.
215 citations
TL;DR: In this paper, Arnold affirme and partiellement demontre that, for le modele newtonien du Systeme solaire a $n\geq 2$ planetes dans l'espace, si la masse des planetes est suffisamment petite par rapport a celle du Soleil, il existe, dans leespace des phases au voisinage des mouvements kepleriens circulaires coplanaires, un sous-ensemble de mesure de Lebesgue strictement positive de
Abstract: V. I. Arnold (Petits denominateurs et problemes de stabilite du mouvement en mecanique classique et en mecanique celeste. Usp. Mat. Nauk. 18 (1963), 91–192 (en russe)) a affirme et partiellement demontre que, pour le modele newtonien du Systeme solaire a $n\geq 2$ planetes dans l'espace, si la masse des planetes est suffisamment petite par rapport a celle du Soleil, il existe, dans l'espace des phases au voisinage des mouvements kepleriens circulaires coplanaires, un sous-ensemble de mesure de Lebesgue strictement positive de conditions initiales conduisant a des mouvements quasiperiodiques a 3n - 1 frequences. Cet article detaille la demonstration que M. R. Herman a exposee de ce theoreme (Demonstration d'un theoreme de V. I. Arnold, Seminaire de Systemes Dynamiques et manuscrits, 1998).
168 citations
TL;DR: In this article, it was shown that the rate of decay of correlations with respect to the absolutely continuous invariant probability measure $\mu$ is polynomial with the same degree $1/\gamma-1$ for Lipschitz functions.
Abstract: We consider a piecewise smooth expanding map f on the unit interval that has the form $f(x)=x+x^{1+\gamma}+o(x^{1+\gamma})$ near 0, where $0 . We prove by showing both lower and upper bounds that the rate of decay of correlations with respect to the absolutely continuous invariant probability measure $\mu$ is polynomial with the same degree $1/\gamma-1$ for Lipschitz functions. We also show that the density function h of $\mu$ has the order $x^{-\gamma}$ as $x\to 0$ . Perron–Frobenius operators are the main tool used for proofs.
150 citations
TL;DR: An overview and some new applications of the approximation by conjugation method introduced by Anosov and Katok more than 30 years ago (Trans. Moscow Math. Soc. as discussed by the authors ).
Abstract: We present an overview and some new applications of the approximation by conjugation method introduced by Anosov and Katok more than 30 years ago (Trans. Moscow Math. Soc. 23 (1970), 1–35). Michel Herman made important contributions to the development and applications of this method beginning from the construction of minimal and uniquely ergodic diffeomorphisms jointly with Fathi (Asterisque 49 (1977), 37–59) and continuing with exotic invariant sets of rational maps of the Riemann sphere (J. London Math. Soc. (2) 34 (1986), 375–384) and the construction of invariant tori with non-standard and unexpected behavior in the context of KAM theory (Pitman Research Notes Mathematical Series 243 (1992); Proc. Int. Congr. Mathematicians (Berlin, 1998) Vol. 11, 797–808). Recently the method has been experiencing a revival. Some of the new results presented in the paper illustrate variety of uses for tools available for a long time, others exploit new methods, in particular the possibility of mixing in the context of Liouvillean dynamics discovered by the first author (Ergod. Th. & Dynam. Sys. 22 (2002) 437–468; Proc. Amer. Math. Soc. 130 (2002), 103–109).
129 citations
TL;DR: In this paper, it was shown that the vector space of homogeneous quasi-morphisms on a group of diffeomorphisms preserves a given area form, and that for any compact oriented surfacewe, an infinite family of independent homogeneous quasimorphisms can be constructed explicitly for each surface.
Abstract: For any compact oriented surfacewe consider the group of diffeomorphisms ofwhich preserve a given area form. In this paper we show that the vector space of homogeneous quasi-morphisms on this group has infinite dimension. This result is proved by constructing explicitly and for each surface an infinite family of independent homogeneous quasi-morphisms. These constructions use simple arguments related to linking properties of the orbits of the diffeomorphisms.
118 citations
TL;DR: In this article, it was shown that for an open dense subset of dominated linear cocycles over a hyperbolic transformation and for any invariant probability with continuous local product structure (including all equilibrium states of Holder continuous potentials), all Oseledets subspaces are one-dimensional.
Abstract: We exhibit an explicit criterion for the simplicity of the Lyapunov spectrum of linear cocycles, either locally constant or dominated, over hyperbolic (Axiom A) transformations. This criterion is expressed by a geometric condition on the cocycle's behaviour over periodic points and associated homoclinic orbits. It allows us to prove that for an open dense subset of dominated linear cocycles over a hyperbolic transformation and for any invariant probability with continuous local product structure (including all equilibrium states of Holder continuous potentials), all Oseledets subspaces are one-dimensional. Moreover, the complement of this subset has infinite codimension and, thus, is avoided by any generic family of cocycles described by finitely many parameters.This improves previous results of Bonatti, Gomez–Mont and Viana where it was shown that some Lyapunov exponent is non-zero, in a similar setting and also for an open dense subset.
116 citations
84 citations
TL;DR: In this paper, the authors prove several new results about AF-equivalence relations and relate these to minimal Z-actions, which is crucial for the study of the topological orbit structure of more general countable group actions on Cantor sets.
Abstract: We prove several new results about AF-equivalence relations and relate these to Cantor minimal systems (i.e. to minimal Z-actions). The results we obtain turn out to be crucial for the study of the topological orbit structure of more general countable group actions (as homeomorphisms) on Cantor sets, which will be the topic of a forthcoming paper. In all this, Bratteli diagrams and their dynamical interpretation are indispensable tools.
82 citations
TL;DR: For planar Lorentz processes with a finite horizon, Schmidt and Conze as discussed by the authors proved a local central limit theorem (CLT) and recurrence for the case d = 2, finite horizon.
Abstract: For Young systems, i.e. for hyperbolic systems without/with singularities satisfying Young's axioms (Lai-Sang Young, Ann. Math.147 (1998), 585–650), which imply exponential decay of correlations and the central limit theorem (CLT), a local CLT is proven. In fact, a unified version of the local CLT is found, covering, among others, the absolutely continuous and arithmetic cases. For planar Lorentz process with a finite horizon, this result implies (a) a local CLT and (b) recurrence. For the latter case (d = 2, finite horizon), combining the global CLT with abstract ergodic theoretic ideas, K. Schmidt (C. R. Acad. Sci. Paris Ser. 1 Math. 372(9) (1998), 837–842) and J.-P. Conze (Ergod. Th. & Dynam. Sys.19(5) (1999), 1233–1245) could already establish recurrence.
76 citations
TL;DR: In this paper, the effect of various graphical constructions on the associated graph C*-algebras was explored in the study of flow equivalence for topological Markov chains.
Abstract: This paper explores the effect of various graphical constructions upon the associated graph C*-algebras. The graphical constructions in question arise naturally in the study of flow equivalence for topological Markov chains. We prove that out-splittings give rise to isomorphic graph algebras, and in-splittings give rise to strongly Morita equivalent C*-algebras. We generalize the notion of a delay as defined in (D. Drinen, Preprint, Dartmouth College, 2001) to form in-delays and out-delays. We prove that these constructions give rise to Morita equivalent graph C*-algebras. We provide examples which suggest that our results are the most general possible in the setting of the C*-algebras of arbitrary directed graphs.
73 citations
TL;DR: In this article, the complexity function of an open cover along some sequences of natural numbers is used to characterize mild mixing, strong scattering and scattering, and it is shown that strongly mixing systems are disjoint from minimal uniformly rigid (respectively minimal rigid) systems.
Abstract: In Blanchard et al (Topological complexity. Ergod. Th. & Dynam. Sys.20 (2000), 641–662), the authors introduced the notion of scattering and a weaker notion of 2-scattering. It is an open question whether the two notions are equivalent. The question is answered affirmatively in this paper. Using the complexity function of an open cover along some sequences of natural numbers, we characterize mild mixing, strong scattering and scattering. We show that mildly mixing (respectively strongly mixing) systems are disjoint from minimal uniformly rigid (respectively minimal rigid) systems.Moreover, assuming minimality we show that a dynamical system is full-scattering (respectively mildly mixing or weakly mixing) if and only if it is strongly mixing (respectively IP*-transitive or with the lower Banach density 1.
TL;DR: In this article, a thermodynamic formalism of potentials of the form $-t\log|F_\lambda'| was developed, where F is the natural map associated with the corresponding map.
Abstract: We deal with all the mappings $f_\lambda(z)=\lambda e^z$ that have an attracting periodic orbit. We consider the set $J_r(f_\lambda)$ consisting of those points of the Julia set of $f_\lambda$ that do not escape to infinity under positive iterates of $f_\lambda$ . Our ultimate result is that the function $\lambda\mapsto{\rm HD}(J_r(f_\lambda))$ is real-analytic. In order to prove it we develop the thermodynamic formalism of potentials of the form $-t\log|F_\lambda'|$ , where $F_\lambda$ is the natural map associated with $f_\lambda$ closely related to the corresponding map introduced in Urbanski and Zdunik (2001). The formalism includes appropriately defined topological pressure, Perron–Frobenius operators, and geometric and invariant generalized conformal measures (Gibbs states). We show that our Perron–Frobenius operators are quasicompact and that they embed into a family of operators depending holomorphically on an appropriate parameter, and we obtain several other properties of these operators. We prove an appropriate version of Bowen's formula that the Hausdorff dimension of the set $J_r(f_\lambda)$ is equal to the unique zero of the pressure function. Since the formula for the topological pressure is independent of the set J r ( f ), Bowen's formula also indicates that J r ( f ) is the right set to deal with. We also study in detail the properties of quasiconformal conjugacies between the maps $f_\lambda$ . As a byproduct of our main course of reasoning we prove stochastic properties of the dynamical system generated by $F_\lambda$ and the invariant Gibbs states $\mu_t$ such as the Central Limit Theorem and exponential decay of correlations.
TL;DR: In this article, the authors introduced the notion of a space with measured walls, generalizing the concept of a spaces with walls due to Haglund and Paulin (Simplicite de groupes d'automorphisms d'espaces a courbure negative).
Abstract: We introduce the notion of a space with measured walls, generalizing the concept of a space with walls due to Haglund and Paulin (Simplicite de groupes d'automorphismes d'espaces a courbure negative. Geom. Topol. Monograph1 (1998), 181–248). We observe that if a locally compact group G acts properly on a space with measured walls, then G has the Haagerup property. We conjecture that the converse holds and we prove this conjecture for the following classes of groups: discrete groups with the Haagerup property, closed subgroups of SO(n, 1), groups acting properly on real trees, SL2(K) where K is a global field and amenable groups.
TL;DR: In this paper, a survey of various developments of the classical KAM theory in dynamical systems to infinite-dimensional lattice models and Hamiltonian PDEs is presented, with particular emphasis on Nekhoroshev stability and growth of higher Sobolev norms.
Abstract: This is a partial survey of various developments of the classical KAM theory in dynamical systems to infinite-dimensional lattice models and Hamiltonian PDEs. Particular emphasis is given on Nekhoroshev stability and growth of higher Sobolev norms. Some new results are also presented.
TL;DR: In this paper, Gevrey perturbations H of a completely integrable non-degenerate Gevreys Hamiltonian H0 were considered given a Cantor set.
Abstract: We consider Gevrey perturbations H of a completely integrable non-degenerate Gevrey Hamiltonian H0. Given a Cantor set .
TL;DR: In this article, it was shown that for all rational functions f on the Riemann sphere and potential −t ln |f � |,t ≥ 0 all the notions of pressure introduced in Przytycki (Proc. Amer. Math. Soc. 351(5) (1999), 2081-2099) coincide.
Abstract: We prove that for all rational functions f on the Riemann sphere and potential −t ln |f � | ,t ≥ 0 all the notions of pressure introduced in Przytycki (Proc. Amer. Math. Soc. 351(5) (1999), 2081-2099) coincide. In particular, we get a new simple proof of the equality between the hyperbolic Hausdorff dimension and the minimal exponent of conformal measure on a Julia set. We prove that these pressures are equal to the pressure definedwith theuse of periodicorbitsunderan assumptionthat therearenotmanyperiodic orbits with Lyapunov exponent close to 1 moving close together, in particular under the Topological Collet-Eckmann condition. In Appendix A, we discuss the case t< 0.
TL;DR: In this paper, a general theory of transformations of and [0, 1] with independent s-adic digits is studied, and it is proved that any continuous function from the previously mentioned class is a DP function, despite the fact that it may have a very complicated local structure.
Abstract: This article is devoted to the development of a general theory of transformations of and [0, 1]. A class of distribution functions of random variables with independent s-adic digits is studied in detail. It is proved that any absolutely continuous function from the previously mentioned class is a DP function, despite the fact that it may have a very complicated local structure. Necessary, respectively, sufficient conditions for dimension preservation are also given for singular functions. Relations between the entropy of transformations and their DP properties are investigated.Examples and counterexamples are provided,and some applications are discussed.
TL;DR: In this paper, a thermodynamic and multifractal formalism for general classes of potential functions and their average growth rates is presented. But these formalisms are restricted to certain geometrically finite Kleinian groups which may have parabolic elements of different ranks.
Abstract: We elaborate thermodynamic and multifractal formalisms for general classes of potential functions and their average growth rates. We then apply these formalisms to certain geometrically finite Kleinian groups which may have parabolic elements of different ranks. We show that for these groups our revised formalisms give access to a description of the spectrum of ‘homological growth rates’ in terms of Hausdorff dimension. Furthermore, we derive necessary and sufficient conditions for the existence of ‘thermodynamic phase transitions’.
TL;DR: In this article, the authors studied the closed Gamma-invariant subsets of V-0 under the condition that the Zariski closure of $Gamma$ is semi-simple.
Abstract: Let V be a finite-dimensional vector space over $\\mathbb{R}$and let $\\Gamma \\subset \\mathrm{GL}(V)$ be a semigroup. We study the closed $\\Gamma$-invariant subsets of V-\\{0\\} under the condition that the Zariski closure of $\\Gamma$ is semi-simple. We use the results to show that, if $\\Gamma\\subset\\mathrm{SL}(\\mathbb{R}^d)$ acts on $\\mathbb{T}^d$ by automorphisms, then the orbits of $\\Gamma$ are finite or dense.
TL;DR: In this paper, it was shown that one-dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an invariant measure, and that the rate of decay of correlations is determined, in some situations, by the average rate at which typical points start to exhibit exponential growth of the derivative.
Abstract: We show that one-dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive, some power of f is mixing and, in particular, the correlation of Holder continuous observables decays to zero. The main objective of this paper is to show that the rate of decay of correlations is determined, in some situations, by the average rate at which typical points start to exhibit exponential growth of the derivative.
TL;DR: In this paper, it was shown that in the case when the fast motions are hyperbolic flows for each freezed slow variable, the Lebesgue measure of initial conditions with bad averaging approximation tends to zero exponentially fast.
Abstract: In the study of systems which combine slow and fast motions, the averaging principle suggests that a good approximation of the slow motion on long time intervals can be obtained by averaging its parameters over the fast variables. When the slow and fast motions depend on each other (fully coupled), as is usually the case, for instance, in perturbations of Hamiltonian systems, the averaging prescription cannot always be applied, and when it does work, this is usually only in some averaged with respect to initial conditions sense. In this paper we first give necessary and sufficient conditions for the averaging principle to hold (in the above sense) and then, relying on some large deviations arguments, verify them in the case when the fast motions are hyperbolic (Axiom A) flows for each freezed slow variable. It turns out that in this situation the Lebesgue measure of initial conditions with bad averaging approximation tends to zero exponentially fast as the parameter tends to zero.
TL;DR: The main result of the paper states that entropy pairs for the measure $\mu$ can be defined using either $h_\mu^+$ or $h-\mu-$ and it is proved that both $h_.\mu+$ and $h.\mu-$ have an ergodic decomposition and is used to prove a local Abramov formula for $h_{mu^-$.
Abstract: Let ( X , T ) be a topological dynamical system and let $\mu$ be a T -invariant probability measure on X . In this paper, we study two properties of the notions of measure theoretical entropy for a measurable cover $\mathcal{U},\ h_\mu^+(\mathcal{U},T)$ and $h_\mu^-(\mathcal{U},T)$ introduced by P. P. Romagnoli ( Ergod. Th. & Dynam. Sys . 23 (2003), 1601–1610). The main result of the paper states that entropy pairs for the measure $\mu$ can be defined using either $h_\mu^+$ or $h_\mu^-$ . We also prove that both $h_\mu^+$ and $h_\mu^-$ have an ergodic decomposition and we use it to prove a local Abramov formula for $h_\mu^-$ .
TL;DR: In this paper, Marco and Sauzin generalized the results of Lochak-Neishtadt and Poschel to the quasiconvex case, where the steepness indices are all equal to one and the same exponents 1/2n were obtained.
Abstract: In the 1970s, Nekhorochev proved that for an analytic nearly integrable Hamiltonian system with a perturbation of size $\\varepsilon$, the actions linked to the unperturbed Hamiltonian vary only by the order of $\\varepsilon^b$ over a time of the order of $\\exp (C\\varepsilon^{-a})$ for some positive constants a, b and C, provided that the unperturbed Hamiltonian meets some generic transversality condition known as steepness. Among steep systems, convex or quasiconvex systems are easier to describe since the use of energy conservation allows the proof of exponential estimates of stability to be shortened. In this case, Lochak–Neishtadt and Poschel have independently obtained the stability exponents a = b = 1/2n for systems of n degrees of freedom—especially the time exponent (a) is expected to be optimal (see P. Lochak, J.-P. Marco and D. Sauzin. Preprint. Institut de Máthematique de Jussieu, 1999; J.-P. Marco and D. Sauzin. Preprint. Publ. Math. Inst. Hautes Etudes Science, 2001). Moreover, Lochak's study relies on simultaneous Diophantine approximation which gives a very transparent proof. However, the proof in the steep case has rarely been studied since Nekhorochev's original work despite various physical examples where the model Hamiltonian is only steep. Here, we combine the original scheme with a simultaneous Diophantine approximation as in Lochak's proof. This yields significant simplifications with respect to Nekhorochev's reasoning: it also allows the exponents $a=b=(2n p_1\\dotsb p_{n-1})^{-1}$ where $(p_1\\dotsb p_{n-1})$ are the steepness indices of the considered Hamiltonian to be obtained. In the quasiconvex case, the steepness indices are all equal to one and we find the same exponents 1/2n as Lochak–Neishtadt and Poschel, whose results are thus generalized to the steep case.
TL;DR: In this paper, the growth of fibers of coverings of pinched negatively curved Riemannian manifolds was studied, and a Khintchine-Sullivan-type theorem was proved for the Hausdorff measure of the geodesic lines starting from a cusp.
Abstract: We study the growth of fibers of coverings of pinched negatively curved Riemannian manifolds. The applications include counting estimates for horoballs in the universal cover of geometrically finite manifolds with cusps. Continuing our work on diophantine approximation in negatively curved manifolds started in an earlier paper (Math. Zeit.241 (2002), 181–226), we prove a Khintchine–Sullivan-type theorem giving the Hausdorff measure of the geodesic lines starting from a cusp that are well approximated by the cusp returning ones.
TL;DR: In this paper, the authors give explicit construction of certain components of the space of holomorphic foliations of codimension one, in complex projective spaces, associated to some algebraic representations of the affine Lie algebra.
Abstract: In this paper, we give the explicit construction of certain components of the space of holomorphic foliations of codimension one, in complex projective spaces. These components are associated to some algebraic representations of the affine Lie algebra $\mathfrak{aff}(\mathbb{C})$. Some of them, the so-called exceptional or Klein–Lie components, are rigid in the sense that all generic foliations in the component are equivalent (Example 1). In particular, we obtain rigid foliations of all degrees. Some generalizations and open problems are given at the end of §1.
TL;DR: In this paper, the authors constructed a quotient Aubry set isometric to the unit interval, and constructed several related examples of Aubry sets, including a set of quotient sets that is related to the one we present here.
Abstract: We construct a quotient Aubry set isometric to the unit interval. We also construct several related examples of Aubry sets.
TL;DR: In this paper, it was shown that two systems that are topologically conjugate are smooth conjugates, which is somewhat more general than a conjecture of the author in 2002, and related results have been obtained by B. Kalinin and V. Sadovskaia.
Abstract: We consider systems that have some hyperbolicity behavior and which preserve conformal structures on the stable and unstable bundles. We show that two such systems that are topologically conjugate are smoothly conjugate. This is somewhat more general than a conjecture of the author in 2002. Related results have also been obtained by B. Kalinin and V. Sadovskaia.
TL;DR: In this article, it was shown that a rank-one transformation satisfying restricted growth is a mixing transformation if and only if the spacer sequence for the transformation is uniformly ergodic.
Abstract: We prove that a rank-one transformation satisfying a condition called restricted growth is a mixing transformation if and only if the spacer sequence for the transformation is uniformly ergodic. Uniform ergodicity is a generalization of the notion of ergodicity for sequences, in the sense that the mean ergodic theorem holds for a family of what we call dynamical sequences. The application of our theorem shows that the class of polynomial rank-one transformations, rank-one transformations where the spacers are chosen to be the values of a polynomial with some mild conditions on the polynomials, that have restricted growth are mixing transformations, implying, in particular, Adams' result on staircase transformations. Another application yields a new proof that Ornstein's class of rank-one transformations constructed using ‘random spacers’ are almost surely mixing transformations.
TL;DR: In this article, it was shown that special flows over irrational rotations and under functions whose Fourier coefficients are of order O(1/| n |) are disjoint in the sense of Furstenberg from all mixing flows.
Abstract: It is proved that special flows over irrational rotations and under functions whose Fourier coefficients are of order O(1/| n |) are disjoint in the sense of Furstenberg from all mixing flows. This is an essential strengthening of a classical result by Kocergin on the absence of mixing of special flows built over irrational rotations and under bounded variation roof functions.