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Scale recurrence lemma and dimension formula for Cantor sets in the complex plane Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-03-25 CARLOS GUSTAVO T. DE A. MOREIRA, ALEX MAURICIO ZAMUDIO ESPINOSA
We prove a multidimensional conformal version of the scale recurrence lemma of Moreira and Yoccoz [Stable intersections of regular Cantor sets with large Hausdorff dimensions. Ann. of Math. (2)154(1) (2001), 45–96] for Cantor sets in the complex plane. We then use this new recurrence lemma, together with Moreira’s ideas in [Geometric properties of images of Cartesian products of regular Cantor sets
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Rigidity of pressures of Hölder potentials and the fitting of analytic functions through them Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-03-18 LIANGANG MA, MARK POLLICOTT
The first part of this work is devoted to the study of higher derivatives of pressure functions of Hölder potentials on shift spaces with finitely many symbols. By describing the derivatives of pressure functions via the central limit theorem for the associated random processes, we discover some rigid relationships between derivatives of various orders. The rigidity imposes obstructions on fitting
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Approximate homomorphisms and sofic approximations of orbit equivalence relations Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-03-15 BEN HAYES, SRIVATSAV KUNNAWALKAM ELAYAVALLI
We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphisms. We record applications of this result to recover various known stability
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Measure transfer and S-adic developments for subshifts Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-03-11 NICOLAS BÉDARIDE, ARNAUD HILION, MARTIN LUSTIG
Based on previous work of the authors, to any S-adic development of a subshift X a ‘directive sequence’ of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given S-adic development. The issuing rich picture enables one to deduce results about X with unexpected directness
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Invariant measures for -free systems revisited Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-03-08 AURELIA DYMEK, JOANNA KUŁAGA-PRZYMUS, DANIEL SELL
For $\mathscr {B} \subseteq \mathbb {N} $ , the $ \mathscr {B} $ -free subshift $ X_{\eta } $ is the orbit closure of the characteristic function of the set of $ \mathscr {B} $ -free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $ X_{\eta } $ , have their analogues for $ X_{\eta } $ as well. In particular, we settle in
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Bohr chaoticity of principal algebraic actions and Riesz product measures Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-03-06 AI HUA FAN, KLAUS SCHMIDT, EVGENY VERBITSKIY
For a continuous $\mathbb {N}^d$ or $\mathbb {Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $\mathbb {Z}$ actions of positive entropy are Bohr chaotic. The same is proved for principal algebraic actions of $\mathbb {Z}^d$ with
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Non-integrability of the restricted three-body problem Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-03-06 KAZUYUKI YAGASAKI
The problem of non-integrability of the circular restricted three-body problem is very classical and important in the theory of dynamical systems. It was partially solved by Poincaré in the nineteenth century: he showed that there exists no real-analytic first integral which depends analytically on the mass ratio of the second body to the total and is functionally independent of the Hamiltonian. When
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Patterson–Sullivan theory for groups with a strongly contracting element Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-03-05 RÉMI COULON
Using Patterson–Sullivan measures, we investigate growth problems for groups acting on a metric space with a strongly contracting element.
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An embedding theorem for subshifts over amenable groups with the comparison property Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-03-05 ROBERT BLAND
We obtain the following embedding theorem for symbolic dynamical systems. Let G be a countable amenable group with the comparison property. Let X be a strongly aperiodic subshift over G. Let Y be a strongly irreducible shift of finite type over G that has no global period, meaning that the shift action is faithful on Y. If the topological entropy of X is strictly less than that of Y and Y contains
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Tracial weights on topological graph algebras Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-03-05 JOHANNES CHRISTENSEN
We describe two kinds of regular invariant measures on the boundary path space $\partial E$ of a second countable topological graph E, which allows us to describe all extremal tracial weights on $C^{*}(E)$ which are not gauge-invariant. Using this description, we prove that all tracial weights on the C $^{*}$ -algebra $C^{*}(E)$ of a second countable topological graph E are gauge-invariant when E is
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Invariant measures of Toeplitz subshifts on non-amenable groups Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-03-04 PAULINA CECCHI BERNALES, MARÍA ISABEL CORTEZ, JAIME GÓMEZ
Let G be a countable residually finite group (for instance, ${\mathbb F}_2$ ) and let $\overleftarrow {G}$ be a totally disconnected metric compactification of G equipped with the action of G by left multiplication. For every $r\geq 1$ , we construct a Toeplitz G-subshift $(X,\sigma ,G)$ , which is an almost one-to-one extension of $\overleftarrow {G}$ , having r ergodic measures $\nu _1, \ldots ,\nu
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On the stochastic bifurcations regarding random iterations of polynomials of the form Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-26 TAKAYUKI WATANABE
In this paper, we consider random iterations of polynomial maps $z^{2} + c_{n}$ , where $c_{n}$ are complex-valued independent random variables following the uniform distribution on the closed disk with center c and radius r. The aim of this paper is twofold. First, we study the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness
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Shifts of finite type on locally finite groups Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-26 JADE RAYMOND
In this work we prove that every shift of finite type (SFT), sofic shift, and strongly irreducible shift on locally finite groups has strong dynamical properties. These properties include that every sofic shift is an SFT, every SFT is strongly irreducible, every strongly irreducible shift is an SFT, every SFT is entropy minimal, and every SFT has a unique measure of maximal entropy, among others. In
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Upper, down, two-sided Lorenz attractor, collisions, merging, and switching Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-21 DIEGO BARROS, CHRISTIAN BONATTI, MARIA JOSÉ PACIFICO
We present a modified version of the well-known geometric Lorenz attractor. It consists of a $C^1$ open set ${\mathcal O}$ of vector fields in ${\mathbb R}^3$ having an attracting region ${\mathcal U}$ satisfying three properties. Namely, a unique singularity $\sigma $ ; a unique attractor $\Lambda $ including the singular point and the maximal invariant in ${\mathcal U}$ has at most two chain recurrence
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Measures of maximal entropy of bounded density shifts Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-20 FELIPE GARCÍA-RAMOS, RONNIE PAVLOV, CARLOS REYES
We find sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. We also prove that every measure of maximal entropy of a bounded density shift is fully supported. As a consequence of this, we obtain that bounded density shifts are surjunctive.
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On denseness of horospheres in higher rank homogeneous spaces Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-19 OR LANDESBERG, HEE OH
Let $ G $ be a connected semisimple real algebraic group and $\Gamma
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Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-15 TULLIO CECCHERINI-SILBERSTEIN, MICHEL COORNAERT, XUAN KIEN PHUNG
Let G be a group and let V be an algebraic variety over an algebraically closed field K. Let A denote the set of K-points of V. We introduce algebraic sofic subshifts ${\Sigma \subset A^G}$ and study endomorphisms $\tau \colon \Sigma \to \Sigma $ . We generalize several results for dynamical invariant sets and nilpotency of $\tau $ that are well known for finite alphabet cellular automata. Under mild
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Sufficient conditions for non-zero entropy of closed relations Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-15 IZTOK BANIČ, RENE GRIL ROGINA, JUDY KENNEDY, VAN NALL
We introduce the notions of returns and well-aligned sets for closed relations on compact metric spaces and then use them to obtain non-trivial sufficient conditions for such a relation to have non-zero entropy. In addition, we give a characterization of finite relations with non-zero entropy in terms of Li–Yorke and DC2 chaos.
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On a self-embedding problem for self-similar sets Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-14 JIAN-CI XIAO
Let $K\subset {\mathbb {R}}^d$ be a self-similar set generated by an iterated function system $\{\varphi _i\}_{i=1}^m$ satisfying the strong separation condition and let f be a contracting similitude with $f(K)\subseteq K$ . We show that $f(K)$ is relatively open in K if all $\varphi _i$ share a common contraction ratio and orthogonal part. We also provide a counterexample when the orthogonal parts
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Stable laws for random dynamical systems Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-14 ROMAIN AIMINO, MATTHEW NICOL, ANDREW TÖRÖK
In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval $[0,1]$ in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure $\nu $ . We consider a class of non-square-integrable observables $\phi $ , mostly of form $\phi (x)=d(x,x_0)^{-{1}/{\alpha
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Dimension estimates and approximation in non-uniformly hyperbolic systems Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-12 JUAN WANG, YONGLUO CAO, YUN ZHAO
Let $f: M\rightarrow M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional compact smooth Riemannian manifold M and $\mu $ a hyperbolic ergodic f-invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of $\mu $ , which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If $\mu $ is a
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Regularity and linear response formula of the SRB measures for solenoidal attractors Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-06 CARLOS BOCKER, RICARDO BORTOLOTTI, ARMANDO CASTRO
We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose densities are regular and vary differentiably with respect to the dynamical system. The maps we consider are skew-products given by $T(x,y) = (E (x), C(x,y))$ , where E is an expanding map of $\mathbb {T}^u$ and C is a contracting map on each fiber. If $\inf |\!\det DT| \inf
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Actions of discrete amenable groups into the normalizers of full groups of ergodic transformations Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-05 TOSHIHIKO MASUDA
We apply the Evans–Kishimoto intertwining argument to the classification of actions of discrete amenable groups into the normalizer of a full group of an ergodic transformation. Our proof does not depend on the types of ergodic transformations.
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Lifting generic points Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-05 TOMASZ DOWNAROWICZ, BENJAMIN WEISS
Let $(X,T)$ and $(Y,S)$ be two topological dynamical systems, where $(X,T)$ has the weak specification property. Let $\xi $ be an invariant measure on the product system $(X\times Y, T\times S)$ with marginals $\mu $ on X and $\nu $ on Y, with $\mu $ ergodic. Let $y\in Y$ be quasi-generic for $\nu $ . Then there exists a point $x\in X$ generic for $\mu $ such that the pair $(x,y)$ is quasi-generic
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Multiplicity of topological systems Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-02-05 DAVID BURGUET, RUXI SHI
We define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number k of real continuous functions $f_1,\ldots , f_k$ such that the functions $f_i\circ T^n$ , $n\in {\mathbb {Z}}$ , $1\leq i\leq k,$ span a dense linear vector space in the space of real continuous functions on X endowed with the supremum norm. We study some properties of topological systems with
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Eliminating Thurston obstructions and controlling dynamics on curves Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-01-17 MARIO BONK, MIKHAIL HLUSHCHANKA, ANNINA ISELI
Every Thurston map $f\colon S^2\rightarrow S^2$ on a $2$ -sphere $S^2$ induces a pull-back operation on Jordan curves $\alpha \subset S^2\smallsetminus {P_f}$ , where ${P_f}$ is the postcritical set of f. Here the isotopy class $[f^{-1}(\alpha )]$ (relative to ${P_f}$ ) only depends on the isotopy class $[\alpha ]$ . We study this operation for Thurston maps with four postcritical points. In this case
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Joint partial equidistribution of Farey rays in negatively curved manifolds and trees Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2024-01-08 JOUNI PARKKONEN, FRÉDÉRIC PAULIN
We prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat–Tits trees.
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Slices of the Takagi function Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-12-20 ROOPE ANTTILA, BALÁZS BÁRÁNY, ANTTI KÄENMÄKI
We show that the Hausdorff dimension of any slice of the graph of the Takagi function is bounded above by the Assouad dimension of the graph minus one, and that the bound is sharp. The result is deduced from a statement on more general self-affine sets, which is of independent interest. We also prove that Marstrand’s slicing theorem on the graph of the Takagi function extends to all slices if and only
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Levy and Thurston obstructions of finite subdivision rules Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-12-15 INSUNG PARK
For a post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm. Especially when a finite subdivision rule has polynomial growth of edge subdivisions, the algorithm terminates very quickly, and the existence of a Levy cycle is
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Bracket words along Hardy field sequences Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-12-14 JAKUB KONIECZNY, CLEMENS MÜLLNER
We study bracket words, which are a far-reaching generalization of Sturmian words, along Hardy field sequences, which are a far-reaching generalization of Piatetski-Shapiro sequences $\lfloor n^c \rfloor $ . We show that sequences thus obtained are deterministic (that is, they have subexponential subword complexity) and satisfy Sarnak’s conjecture.
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A dynamical classification for crossed products of fiberwise essentially minimal zero-dimensional dynamical systems Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-12-11 PAUL HERSTEDT
We prove that crossed products of fiberwise essentially minimal zero-dimensional dynamical systems, a class that includes systems in which all orbit closures are minimal, have isomorphic K-theory if and only if the dynamical systems are strong orbit equivalent. Under the additional assumption that the dynamical systems have no periodic points, this gives a classification theorem including isomorphism
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Equilibrium states for non-uniformly expanding skew products Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-12-11 GREGORY HEMENWAY
We study equilibrium states for a class of non-uniformly expanding skew products, and show how a family of fiberwise transfer operators can be used to define the conditional measures along fibers of the product. We prove that the pushforward of the equilibrium state onto the base of the product is itself an equilibrium state for a Hölder potential defined via these fiberwise transfer operators.
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Invariant measures for substitutions on countable alphabets Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-12-11 WEBERTY DOMINGOS, SÉBASTIEN FERENCZI, ALI MESSAOUDI, GLAUCO VALLE
In this work, we study ergodic and dynamical properties of symbolic dynamical system associated to substitutions on an infinite countable alphabet. Specifically, we consider shift dynamical systems associated to irreducible substitutions which have well-established properties in the case of finite alphabets. Based on dynamical properties of a countable integer matrix related to the substitution, we
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Random and mean Lyapunov exponents for Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-12-11 DIEGO ARMENTANO, GAUTAM CHINTA, SIDDHARTHA SAHI, MICHAEL SHUB
We consider orthogonally invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices with the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu and Shub [On random
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Conditional intermediate entropy and Birkhoff average properties of hyperbolic flows Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-11-14 XIAOBO HOU, XUETING TIAN
Katok [Lyapunov exponents, entropy and periodic points of diffeomorphisms. Publ. Math. Inst. Hautes Études Sci.51 (1980), 137–173] conjectured that every $C^{2}$ diffeomorphism f on a Riemannian manifold has the intermediate entropy property, that is, for any constant $c \in [0, h_{\mathrm {top}}(f))$ , there exists an ergodic measure $\mu $ of f satisfying $h_{\mu }(f)=c$ . In this paper, we obtain
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Constructing Birkhoff sections for pseudo-Anosov flows with controlled complexity Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-11-14 CHI CHEUK TSANG
We introduce a new method of constructing Birkhoff sections for pseudo-Anosov flows, which uses the connection between pseudo-Anosov flows and veering triangulations. This method allows for explicit constructions, as well as control over the Birkhoff section in terms of its Euler characteristic and the complexity of the boundary orbits. In particular, we show that any transitive pseudo-Anosov flow
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Smooth models for certain fibered partially hyperbolic systems Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-11-13 MEG DOUCETTE
We prove that under restrictions on the fiber, any fibered partially hyperbolic system over a nilmanifold is leaf conjugate to a smooth model that is isometric on the fibers and descends to a hyperbolic nilmanifold automorphism on the base. One ingredient is a result of independent interest generalizing a result of Hiraide: an Anosov homeomorphism of a nilmanifold is topologically conjugate to a hyperbolic
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Stabilized automorphism group of odometers and of Toeplitz subshifts Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-11-13 JENNIFER N. JONES-BARO
We characterize the stabilized automorphism group for odometers and Toeplitz subshifts, and then prove an invariance property of the stabilized automorphism group of these dynamical systems. Namely, we prove the isomorphism invariance of the primes for which the p-adic valuation of the period structure tends to infinity. A particular case of interest is that for torsion-free odometers, the stabilized
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Equidistribution of rational subspaces and their shapes Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-11-10 MENNY AKA, ANDREA MUSSO, ANDREAS WIESER
To any k-dimensional subspace of $\mathbb {Q}^n$ one can naturally associate a point in the Grassmannian $\mathrm {Gr}_{n,k}(\mathbb {R})$ and two shapes of lattices of rank k and $n-k$ , respectively. These lattices originate by intersecting the k-dimensional subspace and its orthogonal with the lattice $\mathbb {Z}^n$ . Using unipotent dynamics, we prove simultaneous equidistribution of all of these
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Markov capacity for factor codes with an unambiguous symbol Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-11-07 GUANGYUE HAN, BRIAN MARCUS, CHENGYU WU
In this paper, we first give a necessary and sufficient condition for a factor code with an unambiguous symbol to admit a subshift of finite type restricted to which it is one-to-one and onto. We then give a necessary and sufficient condition for the standard factor code on a spoke graph to admit a subshift of finite type restricted to which it is finite-to-one and onto. We also conjecture that for
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Geometrical representation of subshifts for primitive substitutions Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-11-03 PAUL MERCAT
For any primitive substitution whose Perron eigenvalue is a Pisot unit, we construct a domain exchange that is measurably conjugate to the subshift. Additionally, we give a condition for the subshift to be a finite extension of a torus translation. For the particular case of weakly irreducible Pisot substitutions, we show that the subshift is either a finite extension of a torus translation or its
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A mechanism for ejecting a horseshoe from a partially hyperbolic chain recurrence class Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-11-03 CHRISTIAN BONATTI, KATSUTOSHI SHINOHARA
We give a $C^1$ -perturbation technique for ejecting an a priori given finite set of periodic points preserving a given finite set of homo/heteroclinic intersections from a chain recurrence class of a periodic point. The technique is first stated under a simpler setting called a Markov iterated function system, a two-dimensional iterated function system in which the compositions are chosen in a Markovian
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Distortion element in the automorphism group of a full shift Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-10-23 ANTONIN CALLARD, VILLE SALO
We show that there is a distortion element in a finitely generated subgroup G of the automorphism group of the full shift, namely an element of infinite order whose word norm grows polylogarithmically. As a corollary, we obtain a lower bound on the entropy dimension of any subshift containing a copy of G, and that a sofic shift’s automorphism group contains a distortion element if and only if the sofic
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Katok’s special representation theorem for multidimensional Borel flows Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-10-23 KONSTANTIN SLUTSKY
Katok’s special representation theorem states that any free ergodic measure- preserving $\mathbb {R}^{d}$ -flow can be realized as a special flow over a $\mathbb {Z}^{d}$ -action. It provides a multidimensional generalization of the ‘flow under a function’ construction. We prove the analog of Katok’s theorem in the framework of Borel dynamics and show that, likewise, all free Borel $\mathbb {R}^{d}$
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Some measure rigidity and equidistribution results for β-maps Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-10-23 NEVO FISHBEIN
We prove $\times a \times b$ measure rigidity for multiplicatively independent pairs when $a\in \mathbb {N}$ and $b>1$ is a ‘specified’ real number (the b-expansion of $1$ has a tail or bounded runs of $0$ s) under a positive entropy condition. This is done by proving a mean decay of the Fourier series of the point masses average along $\times b$ orbits. We also prove a quantitative version of this
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On the ergodicity of unitary frame flows on Kähler manifolds Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-10-16 MIHAJLO CEKIĆ, THIBAULT LEFEUVRE, ANDREI MOROIANU, UWE SEMMELMANN
Let $(M,g,J)$ be a closed Kähler manifold with negative sectional curvature and complex dimension $m := \dim _{\mathbb {C}} M \geq 2$ . In this article, we study the unitary frame flow, that is, the restriction of the frame flow to the principal $\mathrm {U}(m)$ -bundle $F_{\mathbb {C}}M$ of unitary frames. We show that if $m \geq 6$ is even and $m \neq 28$ , there exists $\unicode{x3bb} (m) \in (0
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-chain closing lemma for certain partially hyperbolic diffeomorphisms Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-10-11 YI SHI, XIAODONG WANG
For every $r\in \mathbb {N}_{\geq 2}\cup \{\infty \}$ , we prove a $C^r$ -orbit connecting lemma for dynamically coherent and plaque expansive partially hyperbolic diffeomorphisms with one-dimensional orientation preserving center bundle. To be precise, for such a diffeomorphism f, if a point y is chain attainable from x through pseudo-orbits, then for any neighborhood U of x and any neighborhood V
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Exponential multiple mixing for commuting automorphisms of a nilmanifold Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-10-11 TIMOTHÉE BÉNARD, PÉTER P. VARJÚ
Let $l\in \mathbb {N}_{\ge 1}$ and $\alpha : \mathbb {Z}^l\rightarrow \text {Aut}(\mathscr {N})$ be an action of $\mathbb {Z}^l$ by automorphisms on a compact nilmanifold $\mathscr{N}$ . We assume the action of every $\alpha (z)$ is ergodic for $z\in \mathbb {Z}^l\smallsetminus \{0\}$ and show that $\alpha $ satisfies exponential n-mixing for any integer $n\geq 2$ . This extends the results of Gorodnik
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Statistical determinism in non-Lipschitz dynamical systems Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-10-11 THEODORE D. DRIVAS, ALEXEI A. MAILYBAEV, ARTEM RAIBEKAS
We study a class of ordinary differential equations with a non-Lipschitz point singularity that admits non-unique solutions through this point. As a selection criterion, we introduce stochastic regularizations depending on a parameter $\nu $ : the regularized dynamics is globally defined for each $\nu> 0$ , and the original singular system is recovered in the limit of vanishing $\nu $ . We prove that
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Effective rigidity away from the boundary for centrally symmetric billiards Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-09-28 MISHA BIALY
In this paper, we study centrally symmetric Birkhoff billiard tables. We introduce a closed invariant set $\mathcal {M}_{\mathcal {B}}$ consisting of locally maximizing orbits of the billiard map lying inside the region $\mathcal {B}$ bounded by two invariant curves of $4$ -periodic orbits. We give an effective bound from above on the measure of this invariant set in terms of the isoperimetric defect
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An elementary proof that the Rauzy gasket is fractal Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-09-25 MARK POLLICOTT, BENEDICT SEWELL
We present an elementary proof that the Rauzy gasket has Hausdorff dimension strictly smaller than two.
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On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-09-25 MATHEUS M. CASTRO, VINCENT P. H. GOVERSE, JEROEN S. W. LAMB, MARTIN RASMUSSEN
In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $\mu $ on M where the transition kernel ${\mathcal P}$ admits an eigenfunction $0\leq \eta \in L^1(M,\mu )$ . We find conditions on the transition densities of ${\mathcal P}$ with respect to $\mu $ which ensure that $\eta (x) \mu (\mathrm {d} x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit
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Pointwise convergence in nilmanifolds along smooth functions of polynomial growth Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-09-25 KONSTANTINOS TSINAS
We study the equidistribution of orbits of the form $b_1^{a_1(n)}\cdots b_k^{a_k(n)}\Gamma $ in a nilmanifold X, where the sequences $a_i(n)$ arise from smooth functions of polynomial growth belonging to a Hardy field. We show that under certain assumptions on the growth rates of the functions $a_1,\ldots ,a_k$ , these orbits are equidistributed on some subnilmanifold of the space X. As an application
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Measure-theoretic sequence entropy pairs and mean sensitivity Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-09-19 FELIPE GARCÍA-RAMOS, VÍCTOR MUÑOZ-LÓPEZ
We characterize measure-theoretic sequence entropy pairs of continuous actions of abelian groups using mean sensitivity. This addresses an open question of Li and Yu [On mean sensitive tuples. J. Differential Equations297 (2021), 175–200]. As a consequence of our results, we provide a simpler characterization of Kerr and Li’s independence sequence entropy pairs ( $\mu $ -IN-pairs) when the measure
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Spectral cocycle for substitution tilings Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-09-18 BORIS SOLOMYAK, RODRIGO TREVIÑO
The construction of a spectral cocycle from the case of one-dimensional substitution flows [A. I. Bufetov and B. Solomyak. A spectral cocycle for substitution systems and translation flows. J. Anal. Math.141(1) (2020), 165–205] is extended to the setting of pseudo-self-similar tilings in ${\mathbb R}^d$ , allowing expanding similarities with rotations. The pointwise upper Lyapunov exponent of this
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Strongly mixing systems are almost strongly mixing of all orders Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-09-13 V. BERGELSON, R. ZELADA
We prove that any strongly mixing action of a countable abelian group on a probability space has higher-order mixing properties. This is achieved via the utilization of $\mathcal R$ -limits, a notion of convergence which is based on the classical Ramsey theorem. $\mathcal R$ -limits are intrinsically connected with a new combinatorial notion of largeness which is similar to but has stronger properties
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A set of 2-recurrence whose perfect squares do not form a set of measurable recurrence Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-09-04 JOHN T. GRIESMER
We say that $S\subseteq \mathbb Z$ is a set of k-recurrence if for every measure-preserving transformation T of a probability measure space $(X,\mu )$ and every $A\subseteq X$ with $\mu (A)>0$ , there is an $n\in S$ such that $\mu (A\cap T^{-n} A\cap T^{-2n}\cap \cdots \cap T^{-kn}A)>0$ . A set of $1$ -recurrence is called a set of measurable recurrence. Answering a question of Frantzikinakis, Lesigne
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A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems – CORRIGENDUM Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-09-01 JOEL MOREIRA, FLORIAN K. RICHTER
This is a corrigendum to the paper ‘A spectral refinement of the Bergelson–Host–Kra decomposition and new multiple ergodic theorems’ [3]. Theorem 7.1 in that paper is incorrect as stated, and the error originates with Proposition 7.5, part (iii), which was incorrectly quoted from a paper by Bergelson, Host, and Kra [1]. Consequently, this invalidates the proof of Theorem 4.2, which was used in the
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Disjoint hypercyclicity, Sidon sets and weakly mixing operators Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-08-22 RODRIGO CARDECCIA
We prove that a finite set of natural numbers J satisfies that $J\cup \{0\}$ is not Sidon if and only if for any operator T, the disjoint hypercyclicity of $\{T^j:j\in J\}$ implies that T is weakly mixing. As an application we show the existence of a non-weakly mixing operator T such that $T\oplus T^2\oplus\cdots \oplus T^n$ is hypercyclic for every n.
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Conditional mixing in deterministic chaos Ergod. Theory Dyn. Syst. (IF 0.9) Pub Date : 2023-08-18 CAROLINE L. WORMELL
While on the one hand, chaotic dynamical systems can be predicted for all time given exact knowledge of an initial state, they are also in many cases rapidly mixing, meaning that smooth probabilistic information (quantified by measures) on the system’s state has negligible value for predicting the long-term future. However, an understanding of the long-term predictive value of intermediate kinds of