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Non-existence of sublinear diffusion for a class of torus homeomorphisms Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2021-01-18 GUILHERME SILVA SALOMÃO; FABIO ARMANDO TAL
We prove that, if f is a homeomorphism of the 2-torus isotopic to the identity whose rotation set is a non-degenerate segment and f has a periodic point, then it has uniformly bounded deviations in the direction perpendicular to the segment.
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Realization of analytic moduli for parabolic Dulac germs Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2021-01-18 PAVAO MARDEŠIĆ; MAJA RESMAN
In a previous paper [P. Mardešić and M. Resman. Analytic moduli for parabolic Dulac germs. Russian Math. Surveys, to appear, 2021, arXiv:1910.06129v2.] we determined analytic invariants, that is, moduli of analytic classification, for parabolic generalized Dulac germs. This class contains parabolic Dulac (almost regular) germs, which appear as first-return maps of hyperbolic polycycles. Here we solve
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On subshifts with slow forbidden word growth Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2021-01-12 RONNIE PAVLOV
In this work, we treat subshifts, defined in terms of an alphabet $\mathcal {A}$ and (usually infinite) forbidden list $\mathcal {F}$, where the number of n-letter words in $\mathcal {F}$ has ‘slow growth rate’ in n. We show that such subshifts are well behaved in several ways; for instance, they are boundedly supermultiplicative in the sense of Baker and Ghenciu [Dynamical properties of S-gap shifts
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Value distribution of derivatives in polynomial dynamics Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2021-01-05 YÛSUKE OKUYAMA; GABRIEL VIGNY
For every $m\in \mathbb {N}$, we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in $\mathbb {C}\setminus \{0\}$ under the $m$th order derivatives of the iterates of a polynomials $f\in \mathbb {C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of f with pole at $\infty $. We also establish non-archimedean and
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Generic rotation sets Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-29 SEBASTIÁN PAVEZ-MOLINA
Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$. In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological
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Invariant densities for random systems of the interval Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-29 CHARLENE KALLE; MARTA MAGGIONI
For random piecewise linear systems T of the interval that are expanding on average we construct explicitly the density functions of absolutely continuous T-invariant measures. If the random system uses only expanding maps our procedure produces all invariant densities of the system. Examples include random tent maps, random W-shaped maps, random $\beta $-transformations and random Lüroth maps with
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Minimal flows with arbitrary centralizer Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-29 ANDY ZUCKER
Given a G-flow X, let $\mathrm{Aut}(G, X)$, or simply $\mathrm{Aut}(X)$, denote the group of homeomorphisms of X which commute with the G action. We show that for any pair of countable groups G and H with G infinite, there is a minimal, free, Cantor G-flow X so that H embeds into $\mathrm{Aut}(X)$. This generalizes results of [2, 7].
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The number of ergodic measures for transitive subshifts under the regular bispecial condition Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-29 MICHAEL DAMRON; JON FICKENSCHER
If $\mathcal {A}$ is a finite set (alphabet), the shift dynamical system consists of the space $\mathcal {A}^{\mathbb {N}}$ of sequences with entries in $\mathcal {A}$, along with the left shift operator S. Closed S-invariant subsets are called subshifts and arise naturally as encodings of other systems. In this paper, we study the number of ergodic measures for transitive subshifts under a condition
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On Polish groups admitting non-essentially countable actions Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-29 ALEXANDER S. KECHRIS; MACIEJ MALICKI; ARISTOTELIS PANAGIOTOPOULOS; JOSEPH ZIELINSKI
It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups
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Chaos and frequent hypercyclicity for weighted shifts Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-28 STÉPHANE CHARPENTIER; KARL GROSSE-ERDMANN; QUENTIN MENET
Bayart and Ruzsa [Difference sets and frequently hypercyclic weighted shifts. Ergod. Th. & Dynam. Sys. 35 (2015), 691–709] have recently shown that every frequently hypercyclic weighted shift on $\ell ^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Frequently hypercyclic operators. Trans. Amer. Math. Soc. 358 (2006), 5083–5117], who constructed a non-chaotic frequently
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Diffeomorphism cocycles over partially hyperbolic systems Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-28 VICTORIA SADOVSKAYA
We consider Hölder continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold $\mathcal {M}$. We obtain several results for this setting. If a cocycle is bounded in $C^{1+\gamma }$, we show that it has a continuous invariant family of $\gamma $-Hölder Riemannian metrics on $\mathcal {M}$. We establish continuity of a measurable
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Generic homeomorphisms have full metric mean dimension Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-28 MARIA CARVALHO; FAGNER B. RODRIGUES; PAULO VARANDAS
We prove that for $C^0$-generic homeomorphisms, acting on a compact smooth boundaryless manifold with dimension greater than one, the upper metric mean dimension with respect to the smooth metric coincides with the dimension of the manifold. As an application, we show that the upper box dimension of the set of periodic points of a $C^0$-generic homeomorphism is equal to the dimension of the manifold
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Mixing operators with prescribed unimodular eigenvalues Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-28 H.-P. BEISE; L. FRERICK; J. MÜLLER
For arbitrary closed countable subsets Z of the unit circle examples of topologically mixing operators on Hilbert spaces are given which have a densely spanning set of eigenvectors with unimodular eigenvalues restricted to Z. In particular, these operators cannot be ergodic in the Gaussian sense.
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Uniformly positive entropy of induced transformations Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-28 NILSON C. BERNARDES; UDAYAN B. DARJI; RÔMULO M. VERMERSCH
Let $(X,T)$ be a topological dynamical system consisting of a compact metric space X and a continuous surjective map $T : X \to X$ . By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does the induced system $({\mathcal {M}}(X),\widetilde {T})$ on the space of Borel probability measures endowed with the weak* topology. This result can be seen as a
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Symbolic dynamics for non-uniformly hyperbolic systems Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-17 YURI LIMA
This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical
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Strong submeasures and applications to non-compact dynamical systems Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-14 TUYEN TRUNG TRUONG
A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the
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Steady Euler flows and Beltrami fields in high dimensions Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-07 ROBERT CARDONA
Using open books, we prove the existence of a non-vanishing steady solution to the Euler equations for some metric in every homotopy class of non-vanishing vector fields of any odd-dimensional manifold. As a corollary, any such field can be realized in an invariant submanifold of a contact Reeb field on a sphere of high dimension. The solutions constructed are geodesible and hence of Beltrami type
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Balanced strong shift equivalence, balanced in-splits, and eventual conjugacy Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-04 KEVIN AGUYAR BRIX
We introduce the notion of balanced strong shift equivalence between square non-negative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a
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Decompositions and measures on countable Borel equivalence relations Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-04 RUIYUAN CHEN
We show that the uniform measure-theoretic ergodic decomposition of a countable Borel equivalence relation $(X, E)$ may be realized as the topological ergodic decomposition of a continuous action of a countable group $\Gamma \curvearrowright X$ generating E. We then apply this to the study of the cardinal algebra $\mathcal {K}(E)$ of equidecomposition types of Borel sets with respect to a compressible
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Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-04 SHOTA OSADA
We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J. 120 (2003), 515–575] and Shirai
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Sarnak’s conjecture for sequences of almost quadratic word growth Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-04 REDMOND MCNAMARA
We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many words which occur with positive logarithmic density. This allows us to
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Continuity of Lyapunov exponents for non-uniformly fiber-bunched cocycles Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-02 CATALINA FREIJO; KARINA MARIN
We provide conditions that imply the continuity of the Lyapunov exponents for non-uniformly fiber-bunched cocycles in $SL(2,\mathbb {R})$ . The main theorem is an extension of the result of Backes, Brown and Butler and gives a partial answer to a conjecture of Marcelo Viana.
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Non-singular -actions: an ergodic theorem over rectangles with application to the critical dimensions Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-12-02 ANTHONY H. DOOLEY; KIERAN JARRETT
We adapt techniques developed by Hochman to prove a non-singular ergodic theorem for $\mathbb {Z}^d$ -actions where the sums are over rectangles with side lengths increasing at arbitrary rates, and in particular are not necessarily balls of a norm. This result is applied to show that the critical dimensions with respect to sequences of such rectangles are invariants of metric isomorphism. These invariants
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Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measure Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-20 MAREIKE WOLFF
We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial f both have finite Lebesgue measure. Essentially, these conditions are designed such that $|f(z)|\ge \exp (|z|^\alpha )$ for some $\alpha>0$ and all z outside a set of finite Lebesgue measure.
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Expansive dynamics on locally compact groups Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-18 BRUCE P. KITCHENS
Let $\mathcal {G}$ be a second countable, Hausdorff topological group. If $\mathcal {G}$ is locally compact, totally disconnected and T is an expansive automorphism then it is shown that the dynamical system $(\mathcal {G}, T)$ is topologically conjugate to the product of a symbolic full-shift on a finite number of symbols, a totally wandering, countable-state Markov shift and a permutation of a countable
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Complete regularity of Ellis semigroups of $\mathbb{Z} $ -actions Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-13 MARCY BARGE; JOHANNES KELLENDONK
It is shown that the Ellis semigroup of a $\mathbb Z$ -action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$ - or $\mathbb R$ -action for which forward proximality and backward proximality are transitive
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Computable Følner monotilings and a theorem of Brudno Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-13 NIKITA MORIAKOV
A theorem of Brudno says that the Kolmogorov–Sinai entropy of an ergodic subshift over $\mathbb {N}$ equals the asymptotic Kolmogorov complexity of almost every word in the subshift. The purpose of this paper is to extend this result to subshifts over computable groups that admit computable regular symmetric Følner monotilings, which we introduce in this work. For every $d \in \mathbb {N}$ , the groups
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Kneading sequences for toy models of Hénon maps Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-13 ERMERSON ARAUJO
The purpose of this article is to study the relation between combinatorial equivalence and topological conjugacy, specifically how a certain type of combinatorial equivalence implies topological conjugacy. We introduce the concept of kneading sequences for a setting that is more general than one-dimensional dynamics: for the two-dimensional toy model family of Hénon maps introduced by Benedicks and
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On equicontinuous factors of flows on locally path-connected compact spaces Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-13 NIKOLAI EDEKO
We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm
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Closures of locally divergent orbits of maximal tori and values of homogeneous forms Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-05 GEORGE TOMANOV
Let ${\mathbf {G}}$ be a semisimple algebraic group over a number field K, $\mathcal {S}$ a finite set of places of K, $K_{\mathcal {S}}$ the direct product of the completions $K_{v}, v \in \mathcal {S}$ , and ${\mathcal O}$ the ring of $\mathcal {S}$ -integers of K. Let $G = {\mathbf {G}}(K_{\mathcal {S}})$ , $\Gamma = {\mathbf {G}}({\mathcal O})$ and $\pi :G \rightarrow G/\Gamma $ the quotient map
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Divergent trajectories in arithmetic homogeneous spaces of rational rank two Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-05 NATTALIE TAMAM
Let G be a semisimple real algebraic group defined over ${\mathbb {Q}}$ , $\Gamma $ be an arithmetic subgroup of G, and T be a maximal ${\mathbb {R}}$ -split torus. A trajectory in $G/\Gamma $ is divergent if eventually it leaves every compact subset. In some cases there is a finite collection of explicit algebraic data which accounts for the divergence. If this is the case, the divergent trajectory
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Number-theoretic positive entropy shifts with small centralizer and large normalizer Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-04 M. BAAKE; Á. BUSTOS; C. HUCK; M. LEMAŃCZYK; A. NICKEL
Higher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centralizer and normalizer of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralizers,
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The set of points with Markovian symbolic dynamics for non-uniformly hyperbolic diffeomorphisms Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-04 SNIR BEN OVADIA
The papers [O. M. Sarig. Symbolic dynamics for surface diffeomorphisms with positive entropy. J. Amer. Math. Soc. 26(2) (2013), 341–426] and [S. Ben Ovadia. Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds. J. Mod. Dyn. 13 (2018), 43–113] constructed symbolic dynamics for the restriction of $C^r$ diffeomorphisms to a set $M'$ with full measure for all sufficiently
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Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-03 THOMAS BARTHELMÉ; SERGIO R. FENLEY; STEVEN FRANKEL; RAFAEL POTRIE
We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques
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Mean equicontinuity and mean sensitivity on cellular automata Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-03 LUGUIS DE LOS SANTOS BAÑOS; FELIPE GARCíA-RAMOS
We show that a cellular automaton (or shift-endomorphism) on a transitive subshift is either almost equicontinuous or sensitive. On the other hand, we construct a cellular automaton on a full shift (hence a transitive subshift) that is neither almost mean equicontinuous nor mean sensitive.
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Periodic point free homeomorphisms and irrational rotation factors Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-03 ALEJANDRO KOCSARD
We provide a complete characterization of periodic point free homeomorphisms of the $2$ -torus admitting irrational circle rotations as topological factors. Given a homeomorphism of the $2$ -torus without periodic points and exhibiting uniformly bounded rotational deviations with respect to a rational direction, we show that annularity and the geometry of its non-wandering set are the only possible
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Unstable pressure and u-equilibrium states for partially hyperbolic diffeomorphisms Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-03 HUYI HU; WEISHENG WU; YUJUN ZHU
Unstable pressure and u-equilibrium states are introduced and investigated for a partially hyperbolic diffeomorphism f. We define the unstable pressure $P^{u}(f, \varphi )$ of f at a continuous function $\varphi $ via the dynamics of f on local unstable leaves. A variational principle for unstable pressure $P^{u}(f, \varphi )$ , which states that $P^{u}(f, \varphi )$ is the supremum of the sum of the
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Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-11-03 HIROKAZU MARUHASHI
Let $M\stackrel {\rho _0}{\curvearrowleft }S$ be a $C^\infty $ locally free action of a connected simply connected solvable Lie group S on a closed manifold M. Roughly speaking, $\rho _0$ is parameter rigid if any $C^\infty $ locally free action of S on M having the same orbits as $\rho _0$ is $C^\infty $ conjugate to $\rho _0$ . In this paper we prove two types of result on parameter rigidity. First
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Finitely many physical measures for sectional-hyperbolic attracting sets and statistical stability Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-30 VITOR ARAUJO
We show that a sectional-hyperbolic attracting set for a Hölder- $C^{1}$ vector field admits finitely many physical/SRB measures whose ergodic basins cover Lebesgue almost all points of the basin of topological attraction. In addition, these physical measures depend continuously on the flow in the $C^{1}$ topology, that is, sectional-hyperbolic attracting sets are statistically stable. To prove these
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Foliations and conjugacy, II: the Mendes conjecture for time-one maps of flows Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-30 JORGE GROISMAN; ZBIGNIEW NITECKI
A diffeomorphism of theplane is Anosov if it has a hyperbolic splitting at every point of the plane. In addition to linear hyperbolic automorphisms, translations of the plane also carry an Anosov structure (the existence of Anosov structures for plane translations was originally shown by White). Mendes conjectured that these are the only topological conjugacy classes for Anosov diffeomorphisms in the
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Krieger’s finite generator theorem for actions of countable groups III Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-30 ANDREI ALPEEV; BRANDON SEWARD
We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving (p.m.p.) actions of countablegroups introduced in Part I [B. Seward. Krieger’s finite generator theorem for actions of countable groups I. Invent. Math. 215(1) (2019), 265–310]. In this paper we prove a non-ergodic finite generator theorem and use it to establish sub-additivity and semicontinuity properties
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Lagrangian foliations and Anosov symplectomorphisms on Kähler manifolds Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-30 M. J. D. HAMILTON; D. KOTSCHICK
We investigate parallel Lagrangian foliations on Kähler manifolds. On the one hand, we show that a Kähler metric admitting a parallel Lagrangian foliation must be flat. On the other hand, we give many examples of parallel Lagrangian foliations on closed flat Kähler manifolds which are not tori. These examples arise from Anosov automorphisms preserving a Kähler form.
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A strongly irreducible affine iterated function system with two invariant measures of maximal dimension Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-30 IAN D. MORRIS; CAGRI SERT
A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb {R}^{d}$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor, which arise as the projections of shift-invariant measures on the coding space, this self-similar
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Topological flows for hyperbolic groups Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-30 RYOKICHI TANAKA
Weshow that for every non-elementary hyperbolic group the Bowen–Margulis current associated with a strongly hyperbolic metric forms a unique group-invariant Radon measure class of maximal Hausdorff dimension on the boundary square. Applications include a characterization of roughly similar hyperbolic metrics via mean distortion.
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$L^{q}$ -spectra of measures on planar non-conformal attractors Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-26 KENNETH J. FALCONER; JONATHAN M. FRASER; LAWRENCE D. LEE
We study the $L^{q}$ -spectrum of measures in the plane generated by certain nonlinear maps. In particular, we consider attractors of iterated function systems consisting of maps whose components are $C^{1+\alpha }$ and for which the Jacobian is a lower triangular matrix at every point subject to a natural domination condition on the entries. We calculate the $L^{q}$ -spectrum of Bernoulli measures
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Examples of exponentially many collisions in a hard ball system Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-23 DMITRI BURAGO; SERGEI IVANOV
Consider the system of n identical hard balls in ${\mathbb {R}}^3$ moving freely and colliding elastically. We show that there exist initial conditions such that the number of collisions is exponential in n.
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Limits of geodesic push-forwards of horocycle invariant measures Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-23 GIOVANNI FORNI
We prove several general conditional convergence results on ergodic averages for horocycle and geodesic subgroups of any continuous $\operatorname {SL}(2, \mathbb {R})$ -action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $\operatorname {SL}(2, \mathbb {R})$ -action on the moduli space of Abelian differentials. By our argument we can
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Classification of partially hyperbolic diffeomorphisms under some rigid conditions Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-22 PABLO D. CARRASCO; ENRIQUE PUJALS; FEDERICO RODRIGUEZ-HERTZ
Consider a three-dimensional partially hyperbolic diffeomorphism. It is proved that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) an Anosov diffeomorphism, a (generalized) skew-product or the time-one map of an Anosov flow, thus recovering a well-known classification conjecture of the second author to this restricted setting.
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The Assouad spectrum of random self-affine carpets Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-15 JONATHAN M. FRASER; SASCHA TROSCHEIT
We derive the almost sure Assouad spectrum and quasi-Assouad dimension of one-variable random self-affine Bedford–McMullen carpets. Previous work has revealed that the (related) Assouad dimension is not sufficiently sensitive to distinguish between subtle changes in the random model, since it tends to be almost surely ‘as large as possible’ (a deterministic quantity). This has been verified in conformal
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Large deviations and central limit theorems for sequential and random systems of intermittent maps Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-07 MATTHEW NICOL; FELIPE PEREZ PEREIRA; ANDREW TÖRÖK
We obtain large and moderate deviation estimates for both sequential and random compositions of intermittent maps. We also address the question of whether or not centering is necessary for the quenched central limit theorems obtained by Nicol, Török and Vaienti [Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Th. & Dynam. Sys.38(3) (2018), 1127–1153] for random
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On the rotation sets of generic homeomorphisms on the torus ${\mathbb T^d}$ Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-07 HEIDES LIMA; PAULO VARANDAS
We 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 homeomorphisms homotopic to the identity so that the set of points with wild pointwise rotation set is a Baire residual subset in $\mathbb
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Spread out random walks on homogeneous spaces Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-06 ROLAND PROHASKA
A measure on a locally compact group is said to be spread out if one of its convolution powers is not singular with respect to Haar measure. Using Markov chain theory, we conduct a detailed analysis of random walks on homogeneous spaces with spread out increment distribution. For finite volume spaces, we arrive at a complete picture of the asymptotics of the n-step distributions: they equidistribute
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Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-06 SILVIUS KLEIN; XIAO-CHUAN LIU; ALINE MELO
We obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we show that for the one-dimensional torus translation
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Strongly ergodic equivalence relations: spectral gap and type III invariants – CORRIGENDUM Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-10-02 CYRIL HOUDAYER; AMINE MARRAKCHI; PETER VERRAEDT
We obtain a spectral gap characterization of strongly ergodic equivalence relations on standard measure spaces. We use our spectral gap criterion to prove that a large class of skew-product equivalence relations arising from measurable $1$ -cocycles with values in locally compact abelian groups are strongly ergodic. By analogy with the work of Connes on full factors, we introduce the Sd and $\unicode[STIX]{x1D70F}$
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Periodic points for amenable group actions on uniquely arcwise connected continua Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-09-30 ENHUI SHI; XIANGDONG YE
We show that any action of a countable amenable group on a uniquely arcwise connected continuum has a periodic point of order $\leq 2$ .
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A context in which finite or unique ergodicity is generic Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-09-29 ANDY Q. YINGST
We show that for good measures, the set of homeomorphisms of Cantor space which preserve that measure and which have no invariant clopen sets contains a residual set of homeomorphisms which are uniquely ergodic. Additionally, we show that for refinable Bernoulli trial measures, the same set of homeomorphisms contains a residual set of homeomorphisms which admit only finitely many ergodic measures.
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Markov chains with exponential return times are finitary Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-09-29 OMER ANGEL; YINON SPINKA
Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an independent and identically distributed (i.i.d.) process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of ℤ
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Arithmeticity of discrete subgroups Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-09-28 YVES BENOIST
The topic of this course is the discrete subgroups of semisimple Lie groups. We discuss a criterion that ensures that such a subgroup is arithmetic. This criterion is a joint work with Sébastien Miquel, which extends previous work of Selberg and Hee Oh and solves an old conjecture of Margulis. We focus on concrete examples like the group $\mathrm {SL}(d,{\mathbb {R}})$ and we explain how classical
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The conformal measures of a normal subgroup of a cocompact Fuchsian group Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-09-28 OFER SHWARTZ
In this paper we study the conformal measures of a normal subgroup of a cocompact Fuchsian group. In particular, we relate the extremal conformal measures to the eigenmeasures of a suitable Ruelle operator. Using Ancona’s theorem, adapted to the Ruelle operator setting, we show that if the group of deck transformations G is hyperbolic then the extremal conformal measures and the hyperbolic boundary
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Limit drift for complex Feigenbaum mappings Ergod. Theory Dyn. Syst. (IF 1.211) Pub Date : 2020-09-28 GENADI LEVIN; GRZEGORZ ŚWIA̧TEK
We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order $\ell $ of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to $0$ under the dynamics of the tower for corresponding $\ell $ . That in turn depends on the sign of
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