507 results
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2. Addendum to our paper ‘Decreasing sequences of sigma fields: product type, standard, and substandard’
- Author
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M. Smorodinsky and J. Feldman
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,Calculus ,Addendum ,Sigma ,Product type ,Mathematics - Published
- 2002
3. Corrigendum to the paper ‘Decidability of the isomorphism problem for stationary AF-algebras and the associated ordered simple dimension groups’
- Author
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Ola Bratteli, Fred W. Roush, Ki Hang Kim, and Palle E. T. Jorgensen
- Subjects
Combinatorics ,Group isomorphism ,Order isomorphism ,Isomorphism extension theorem ,Applied Mathematics ,General Mathematics ,Subgraph isomorphism problem ,Induced subgraph isomorphism problem ,Isomorphism ,Graph isomorphism ,Decidability ,Mathematics - Published
- 2002
4. Correction to the paper ‘Entire functions of slow growth whose Julia set coincides with the plane’
- Author
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Alexandre Eremenko and Walter Bergweiler
- Subjects
TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Plane (geometry) ,Applied Mathematics ,General Mathematics ,Entire function ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Geometry ,Julia set ,Slow growth ,Mathematics - Abstract
In our paper [1], in the proof of Proposition 1, we implicitly assume that the polynomial P is monic, although later we apply this proposition to polynomials which are not monic. The following corrections should be made in the proof of Proposition 1.
- Published
- 2001
5. A remark on R. Moeckel's paper ‘Geodesies on modular surfaces and continued fractions’
- Author
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Toshihiro Nakanishi
- Subjects
Algebra ,Thesaurus (information retrieval) ,business.industry ,Applied Mathematics ,General Mathematics ,Modular design ,business ,Mathematics - Abstract
It is shown that a result by Moeckel holds not only for admissible subgroups of SL (2, ℤ), but also for arbitrary subgroups of finite index.
- Published
- 1989
6. Corrections to the paper ‘On orbits of unipotent flows on homogeneous spaces’
- Author
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S. G. Dani
- Subjects
Homogeneous ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Unipotent ,Mathematics - Abstract
The author regrets that there are certain errors in [1] and would like to give the following corrections.
- Published
- 1986
7. Chaotic behavior of the p-adic Potts–Bethe mapping II
- Author
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Otabek Khakimov and Farrukh Mukhamedov
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Chaotic ,Mathematics - Abstract
The renormalization group method has been developed to investigate p-adic q-state Potts models on the Cayley tree of order k. This method is closely related to the examination of dynamical behavior of the p-adic Potts–Bethe mapping which depends on the parameters q, k. In Mukhamedov and Khakimov [Chaotic behavior of the p-adic Potts–Behte mapping. Discrete Contin. Dyn. Syst.38 (2018), 231–245], we have considered the case when q is not divisible by p and, under some conditions, it was established that the mapping is conjugate to the full shift on $\kappa _p$ symbols (here $\kappa _p$ is the greatest common factor of k and $p-1$ ). The present paper is a continuation of the forementioned paper, but here we investigate the case when q is divisible by p and k is arbitrary. We are able to fully describe the dynamical behavior of the p-adic Potts–Bethe mapping by means of a Markov partition. Moreover, the existence of a Julia set is established, over which the mapping exhibits a chaotic behavior. We point out that a similar result is not known in the case of real numbers (with rigorous proofs).
- Published
- 2021
8. Topologically mixing tiling of generated by a generalized substitution
- Author
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Tyler M. White
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Substitution (logic) ,Mixing (physics) ,Mathematics - Abstract
This paper presents sufficient conditions for a substitution tiling dynamical system of $\mathbb {R}^2$ , generated by a generalized substitution on three letters, to be topologically mixing. These conditions are shown to hold on a large class of tiling substitutions originally presented by Kenyon in 1996. This problem was suggested by Boris Solomyak, and many of the techniques that are used in this paper are based on the work by Kenyon, Sadun, and Solomyak [Topological mixing for substitutions on two letters. Ergod. Th. & Dynam. Sys.25(6) (2005), 1919–1934]. They studied one-dimensional tiling dynamical systems generated by substitutions on two letters and provided similar conditions sufficient to ensure that one-dimensional substitution tiling dynamical systems are topologically mixing. If a tiling dynamical system of $\mathbb {R}^2$ satisfies our conditions (and thus is topologically mixing), we can construct additional topologically mixing tiling dynamical systems of $\mathbb {R}^2$ . By considering the stepped surface constructed from a tiling $T_\sigma $ , we can get a new tiling of $\mathbb {R}^2$ by projecting the surface orthogonally onto an irrational plane through the origin.
- Published
- 2021
9. Multiplicative constants and maximal measurable cocycles in bounded cohomology
- Author
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Marco Moraschini, Alessio Savini, Moraschini M., and Savini A.
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Applied Mathematics ,General Mathematics ,Multiplicative function ,Lattice ,Geometric Topology (math.GT) ,Cohomology ,Mathematics - Geometric Topology ,Maximal cocycle ,Mathematics::Quantum Algebra ,Bounded function ,FOS: Mathematics ,Bounded cohomology ,Boundary map ,Invariant (mathematics) ,Zimmer cocycle ,Mathematics - Abstract
Multiplicative constants are a fundamental tool in the study of maximal representations. In this paper we show how to extend such notion, and the associated framework, to measurable cocycles theory. As an application of this approach, we define and study the Cartan invariant for measurable $\textup{PU}(m,1)$-cocycles of complex hyperbolic lattices., Comment: 35 pages; Major corrections along the paper following the referee's suggestions. To appear in Ergod. Theory Dyn. Syst
- Published
- 2021
10. Local limit theorems in relatively hyperbolic groups I: rough estimates
- Author
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Matthieu Dussaule
- Subjects
Pure mathematics ,Series (mathematics) ,010201 computation theory & mathematics ,Spectral radius ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Limit (mathematics) ,0101 mathematics ,Random walk ,01 natural sciences ,Mathematics - Abstract
This is the first of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this first paper, we prove rough estimates for the Green function. Along the way, we introduce the notion of relative automaticity which will be useful in both papers and we show that relatively hyperbolic groups are relatively automatic. We also define the notion of spectral positive recurrence for random walks on relatively hyperbolic groups. We then use our estimates for the Green function to prove that $p_n\asymp R^{-n}n^{-3/2}$ for spectrally positive-recurrent random walks, where $p_n$ is the probability of going back to the origin at time n and where R is the inverse of the spectral radius of the random walk.
- Published
- 2021
11. Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems
- Author
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Lior Fishman, Tushar Das, Mariusz Urbański, and David Simmons
- Subjects
Class (set theory) ,Pure mathematics ,Conjecture ,Mathematics - Number Theory ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,Diophantine equation ,010102 general mathematics ,11J13, 11J83, 28A75, 37F35 ,Open set ,Dynamical Systems (math.DS) ,Rational function ,01 natural sciences ,Measure (mathematics) ,010101 applied mathematics ,Hausdorff dimension ,FOS: Mathematics ,Number Theory (math.NT) ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Abstract
We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis [{\it Invent. Math.} {\bf 138}(3) (1999), 451--494] resolving Sprind\v zuk's conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. {\it Selecta Math.} {\bf 10} (2004), 479--523], hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson--Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW's sufficient conditions for extremality. In the first of this series of papers [{\it Selecta Math.} {\bf 24}(3) (2018), 2165--2206], we introduce and develop a systematic account of two classes of measures, which we call {\it quasi-decaying} and {\it weakly quasi-decaying}. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, as well as proving the ``inherited exponent of irrationality'' version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson--Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying., Comment: Link to Part I: arXiv:1504.04778
- Published
- 2020
12. Bernoulliness of when is an irrational rotation: towards an explicit isomorphism
- Author
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Christophe Leuridan
- Subjects
Rational number ,Lebesgue measure ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Diophantine approximation ,01 natural sciences ,Irrational rotation ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,Bernoulli scheme ,Isomorphism ,0101 mathematics ,Real number ,Unit interval ,Mathematics - Abstract
Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\unicode[STIX]{x1D703}$: the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ This condition can be relaxed again into $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}| where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$. Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.
- Published
- 2020
13. Type classification of extreme quantized characters
- Author
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Ryosuke Sato
- Subjects
Pure mathematics ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Context (language use) ,01 natural sciences ,Representation theory ,Quantization (physics) ,symbols.namesake ,Character (mathematics) ,Operator algebra ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Quantum ,Mathematics ,Von Neumann architecture - Abstract
The notion of quantized characters was introduced in our previous paper as a natural quantization of characters in the context of asymptotic representation theory forquantum groups. As in the case of ordinary groups, the representation associated with any extreme quantized character generates a von Neumann factor. From the viewpoint of operator algebras (and measurable dynamical systems), it is natural to ask what is the Murray–von Neumann–Connes type of the resulting factor. In this paper, we give a complete solution to this question when the inductive system is of quantum unitary groups $U_{q}(N)$.
- Published
- 2019
14. Weak containment of measure-preserving group actions
- Author
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Alexander S. Kechris and Peter Burton
- Subjects
Containment (computer programming) ,Group action ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Calculus ,Measure (physics) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Weak equivalence ,Mathematics - Abstract
This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.
- Published
- 2019
15. Local rigidity of higher rank non-abelian action on torus
- Author
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Zhenqi Jenny Wang
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Rigidity (electromagnetism) ,Applied Mathematics ,General Mathematics ,Torus ,Abelian group ,Mathematics - Abstract
In this paper, we show local smooth rigidity for higher rank ergodic nilpotent action by toral automorphisms. In former papers all examples for actions enjoying the local smooth rigidity phenomenon are higher rank and have no rank-one factors. In this paper we give examples of smooth rigidity of actions having rank-one factors. The method is a generalization of the KAM (Kolmogorov–Arnold–Moser) iterative scheme.
- Published
- 2017
16. Purely exponential growth of cusp-uniform actions
- Author
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Wenyuan Yang
- Subjects
Cusp (singularity) ,Pure mathematics ,Lemma (mathematics) ,Mathematics::Dynamical Systems ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Metric Geometry (math.MG) ,Group Theory (math.GR) ,Dynamical Systems (math.DS) ,01 natural sciences ,Mathematics - Metric Geometry ,Exponential growth ,0103 physical sciences ,Shadow ,FOS: Mathematics ,Primary 20F65, 20F67 ,Countable set ,010307 mathematical physics ,Preprint ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Abstract
Suppose that a countable group $G$ admits a cusp-uniform action on a hyperbolic space $(X,d)$ such that $G$ is of divergent type. The main result of the paper is characterizing the purely exponential growth type of the orbit growth function by a condition introduced by Dal'bo-Otal-Peign\'e. For geometrically finite Cartan-Hadamard manifolds with pinched negative curvature this condition ensures the finiteness of Bowen-Margulis-Sullivan measures. In this case, our result recovers a theorem of Roblin (in a weaker form). Our main tool is the Patterson-Sullivan measures on the Gromov boundary of $X$, and a variant of the Sullivan shadow lemma called partial shadow lemma. This allows us to prove that the purely exponential growth of either cones, or partial cones or horoballs is also equivalent to the condition of Dal'bo-Otal-Peign\'e. These results are further used in the paper \cite{YANG7}., Comment: Version 2: 34 pages, 2 figures. Sections 4 and 5 was rewritten following suggestions of the referee. Paper accepted by Ergodic Theory and Dynamical Systems
- Published
- 2017
17. Quadratic stochastic operators and zero-sum game dynamics
- Author
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Rasul N. Ganikhodjaev, U. U. Jamilov, and Nasir Ganikhodjaev
- Subjects
Discrete mathematics ,Volterra operator ,Simplex ,Applied Mathematics ,General Mathematics ,Volterra integral equation ,Quasinormal operator ,Semi-elliptic operator ,symbols.namesake ,Operator (computer programming) ,Zero-sum game ,symbols ,Invariant (mathematics) ,Mathematics - Abstract
In this paper we consider the set of all extremal Volterra quadratic stochastic operators defined on a unit simplex $S^{4}$ and show that such operators can be reinterpreted in terms of zero-sum games. We show that an extremal Volterra operator is non-ergodic and an appropriate zero-sum game is a rock-paper-scissors game if either the Volterra operator is a uniform operator or for a non-uniform Volterra operator $V$ there exists a subset $I\subset \{1,2,3,4,5\}$ with $|I|\leq 2$ such that $\sum _{i\in I}(V^{n}\mathbf{x})_{i}\rightarrow 0,$ and the restriction of $V$ on an invariant face ${\rm\Gamma}_{I}=\{\mathbf{x}\in S^{m-1}:x_{i}=0,i\in I\}$ is a uniform Volterra operator.
- Published
- 2014
18. Ruelle operator with weakly contractive iterated function systems
- Author
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Yuan-Ling Ye
- Subjects
Sequence ,Pure mathematics ,Operator (computer programming) ,Iterated function system ,Dynamical systems theory ,Triple system ,Applied Mathematics ,General Mathematics ,Lipschitz continuity ,Mathematics - Abstract
The Ruelle operator has been studied extensively both in dynamical systems and iterated function systems (IFSs). Given a weakly contractive IFS $(X, \{w_j\}_{j=1}^m)$ and an associated family of positive continuous potential functions $\{p_j\}_{j=1}^m$, a triple system $(X, \{w_j\}_{j=1}^m, \{p_j\}_{j=1}^m)$is set up. In this paper we study Ruelle operators associated with the triple systems. The paper presents an easily verified condition. Under this condition, the Ruelle operator theorem holds provided that the potential functions are Dini continuous. Under the same condition, the Ruelle operator is quasi-compact, and the iterations sequence of the Ruelle operator converges with a specific geometric rate, if the potential functions are Lipschitz continuous.
- Published
- 2012
19. Dynamical profile of a class of rank-one attractors
- Author
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Qiudong Wang and Lai Sang Young
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Rank (linear algebra) ,Dynamical systems theory ,Differential equation ,Applied Mathematics ,General Mathematics ,Lyapunov exponent ,Nonlinear Sciences::Chaotic Dynamics ,symbols.namesake ,Attractor ,symbols ,Ergodic theory ,Large deviations theory ,Central limit theorem ,Mathematics - Abstract
This paper contains results on the geometric and ergodic properties of a class of strange attractors introduced by Wang and Young [Towards a theory of rank one attractors. Ann. of Math. (2) 167 (2008), 349–480]. These attractors can live in phase spaces of any dimension, and have been shown to arise naturally in differential equations that model several commonly occurring phenomena. Dynamically, such systems are chaotic; they have controlled non-uniform hyperbolicity with exactly one unstable direction, hence the name rank-one. In this paper we prove theorems on their Lyapunov exponents, Sinai–Ruelle–Bowen (SRB) measures, basins of attraction, and statistics of time series, including central limit theorems, exponential correlation decay and large deviations. We also present results on their global geometric and combinatorial structures, symbolic coding and periodic points. In short, we build a dynamical profile for this class of dynamical systems, proving that these systems exhibit many of the characteristics normally associated with ‘strange attractors’.
- Published
- 2012
20. Strong renewal theorems and Lyapunov spectra forα-Farey andα-Lüroth systems
- Author
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Marc Kesseböhmer, Sara Munday, and Bernd O. Stratmann
- Subjects
Lyapunov function ,Pure mathematics ,Gauss map ,Computer Science::Information Retrieval ,Applied Mathematics ,General Mathematics ,symbols.namesake ,Number theory ,symbols ,Countable set ,Farey sequence ,Ergodic theory ,Partition (number theory) ,Mathematics ,Unit interval - Abstract
In this paper, we introduce and study theα-Farey map and its associated jump transformation, theα-Lüroth map, for an arbitrary countable partitionαof the unit interval with atoms which accumulate only at the origin. These maps represent linearized generalizations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic theoretical properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have calledα-sum-level sets for theα-Lüroth map. Similar results have previously been obtained for the Farey map and the Gauss map by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of theα-Farey map and theα-Lüroth map in terms of the thermodynamical formalism. We show how to derive these spectra and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen partitionα.
- Published
- 2011
21. Dimension of the generalized 4-corner set and its projections
- Author
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Balázs Bárány
- Subjects
Discrete mathematics ,Set (abstract data type) ,Iterated function system ,Dimension (vector space) ,Applied Mathematics ,General Mathematics ,Computation ,Hausdorff dimension ,Hausdorff space ,Dimension theory ,Calculus ,Fixed point ,Mathematics - Abstract
In the last two decades, considerable attention has been paid to the dimension theory of self-affine sets. In the case of generalized 4-corner sets (see Figure 1), the iterated function systems obtained as the projections of self-affine systems have maps of common fixed points. In this paper, we extend our result [B. Bárány. On the Hausdorff dimension of a family of self-similar sets with complicated overlaps. Fund. Math. 206 (2009), 49–59], which introduced a new method of computation of the box and Hausdorff dimensions of self-similar families where some of the maps have common fixed points. The extended version of our method presented in this paper makes it possible to determine the box dimension of the generalized 4-corner set for Lebesgue-typical contracting parameters.
- Published
- 2011
22. Differentiating potential functions of SRB measures on hyperbolic attractors
- Author
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Miaohua Jiang
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Derivative ,Chain rule ,Measure (mathematics) ,Manifold ,Volume form ,symbols.namesake ,Attractor ,Jacobian matrix and determinant ,symbols ,Differentiable function ,Mathematics - Abstract
The derivation of Ruelle’s derivative formula of the SRB measure depends largely on the calculation of the derivative of the unstable Jacobian. Although Ruelle’s derivative formula is correct, the proofs in the original paper and its corrigendum are not complete. In this paper, we re-visit the differentiation process of the unstable Jacobian and provide a complete derivation of its derivative formula. Our approach is to extend the volume form provided by the SRB measure on local unstable manifolds to a system of Hölder continuous local Riemannian metrics on the manifold so that under this system of local metrics, the unstable Jacobian becomes differentiable with respect to the base point and its derivative with respect to the map can be obtained by the chain rule.
- Published
- 2011
23. Invariant rigid geometric structures and expanding maps
- Author
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Yong Fang
- Subjects
Chaotic dynamical systems ,Pure mathematics ,Closed manifold ,Rigidity (electromagnetism) ,Homogeneous ,Applied Mathematics ,General Mathematics ,Invariant (mathematics) ,Algorithm ,Mathematics - Abstract
In the first part of this paper, we consider several natural problems about locally homogeneous rigid geometric structures. In particular, we formulate a notion of topological completeness which is adapted to the study of global rigidity of chaotic dynamical systems. In the second part of the paper, we prove the following result: let φ be a C∞ expanding map of a closed manifold. If φ preserves a topologically complete C∞ rigid geometric structure, then φ is C∞ conjugate to an expanding infra-nilendomorphism.
- Published
- 2011
24. An uncountable Furstenberg–Zimmer structure theory
- Author
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Asgar Jamneshan, Jamneshan, Asgar (ORCID 0000-0002-1450-6569 & YÖK ID 332404), College of Sciences, and Department of Mathematics
- Subjects
Applied Mathematics ,General Mathematics ,Structure theory ,Measure preserving systems ,Ergodic theory ,Mathematics - Abstract
Furstenberg-Zimmer structure theory refers to the extension of the dichotomy between the compact and weakly mixing parts of a measure-preserving dynamical system and the algebraic and geometric descriptions of such parts to a conditional setting, where such dichotomy is established relative to a factor and conditional analogs of those algebraic and geometric descriptions are sought. Although the unconditional dichotomy and the characterizations are known for arbitrary systems, the relative situation is understood under certain countability and separability hypotheses on the underlying groups and spaces. The aim of this article is to remove these restrictions in the relative situation and establish a Furstenberg-Zimmer structure theory in full generality. As an independent byproduct, we establish a connection between the relative analysis of systems in ergodic theory and the internal logic in certain Boolean topoi., A.J. was supported by DFG-research fellowship JA 2512/3-1. A.J. offers his thanks to Terence Tao for suggesting this project, many helpful discussions, and his encouragement and support. He is grateful to Pieter Spaas for several helpful discussions. A.J. thanks Markus Haase for organizing an online workshop on structural ergodic theory where the results of this paper and the parallel work could be discussed, and Nikolai Edeko, Markus Haase, and Henrik Kreidler for helpful comments on an early version of the manuscript. A.J. is indebted to the anonymous referee for several useful suggestions and corrections.
- Published
- 2022
25. On maximal pattern complexity of some automatic words
- Author
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Pavel V. Salimov and Teturo Kamae
- Subjects
Combinatorics ,Discrete mathematics ,Linear function (calculus) ,Applied Mathematics ,General Mathematics ,Bounded function ,Substitution (logic) ,Value (computer science) ,Function (mathematics) ,Fixed point ,Constant (mathematics) ,Word (group theory) ,Mathematics - Abstract
The pattern complexity of a word for a given pattern S, where S is a finite subset of {0,1,2,…}, is the number of distinct restrictions of the word to S+n (with n=0,1,2,…). The maximal pattern complexity of the word, introduced in the paper of T. Kamae and L. Zamboni [Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Th. & Dynam. Sys.22(4) (2002), 1191–1199], is the maximum value of the pattern complexity of S with #S=k as a function of k=1,2,…. A substitution of constant length on an alphabet is a mapping from the alphabet to finite words on it of constant length not less than two. An infinite word is called a fixed point of the substitution if it stays the same after the substitution is applied. In this paper, we prove that the maximal pattern complexity of a fixed point of a substitution of constant length on {0,1} (as a function of k=1,2,…) is either bounded, a linear function of k, or 2k.
- Published
- 2010
26. Lambda-topology versus pointwise topology
- Author
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Mariusz Urbański, Hiroki Sumi, and Mario Roy
- Subjects
Pointwise ,Pointwise convergence ,Dense set ,Applied Mathematics ,General Mathematics ,Hausdorff dimension ,Metrization theorem ,Natural topology ,Invariant (mathematics) ,Topology ,Axiom of countability ,Mathematics - Abstract
This paper deals with families of conformal iterated function systems (CIFSs). The space CIFS(X,I) of all CIFSs, with common seed space X and alphabet I, is successively endowed with the topology of pointwise convergence and the so-calledλ-topology. We show just how bad the topology of pointwise convergence is: although the Hausdorff dimension function is continuous on a dense Gδ-set, it is also discontinuous on a dense subset of CIFS(X,I). Moreover, all of the different types of systems (irregular, critically regular, etc.), have empty interior, have the whole space as boundary, and thus are dense in CIFS(X,I), which goes against intuition and conception of a natural topology on CIFS(X,I). We then prove how good the λ-topology is: Roy and Urbański [Regularity properties of Hausdorff dimension in infinite conformal IFSs. Ergod. Th. & Dynam. Sys.25(6) (2005), 1961–1983] have previously pointed out that the Hausdorff dimension function is then continuous everywhere on CIFS(X,I). We go further in this paper. We show that (almost) all of the different types of systems have natural topological properties. We also show that, despite not being metrizable (as it does not satisfy the first axiom of countability), the λ-topology makes the space CIFS(X,I) normal. Moreover, this space has no isolated points. We further prove that the conformal Gibbs measures and invariant Gibbs measures depend continuously on Φ∈CIFS(X,I) and on the parameter t of the potential and pressure functions. However, we demonstrate that the coding map and the closure of the limit set are discontinuous on an important subset of CIFS(X,I).
- Published
- 2009
27. Parameter rays in the space of exponential maps
- Author
-
Dierk Schleicher and Markus Förster
- Subjects
Set (abstract data type) ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Orbit (dynamics) ,Structure (category theory) ,Geometry ,Parameter space ,Space (mathematics) ,Mathematics ,Exponential function - Abstract
We investigate the setIof parametersκ∈ℂ for which the singular orbit (0,eκ,…) ofEκ(z):=exp (z+κ) converges to$\infty $. These parameters are organized in curves in parameter space calledparameter rays, together with endpoints of certain rays. Parameter rays are an important tool to understand the detailed structure of exponential parameter space. In this paper, we construct and investigate these parameter rays. Based on these results, a complete classification of the setIis given in the following paper [M. Förster, L. Rempe and D. Schleicher. Classification of escaping exponential maps.Proc. Amer. Math. Soc.136(2008), 651–663].
- Published
- 2009
28. Measure-theoretic and topological entropy of operators on function spaces
- Author
-
TOMASZ DOWNAROWICZ and BARTOSZ FREJ
- Subjects
Pure mathematics ,Operator (computer programming) ,Markov chain ,Function space ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Axiomatic system ,Entropy (information theory) ,Topological entropy ,Uniqueness ,Mathematics ,Probability measure - Abstract
We study entropy of actions on function spaces with the focus on doubly stochastic operators on probability spaces and Markov operators on compact spaces. Using an axiomatic approach to entropy we prove that there is basically only one reasonable measure-theoretic entropy notion on doubly stochastic operators. By “reasonable” we mean extending the KolmogorovSinai entropy on measure-preserving transformations and satisfying some obvious continuity conditions for Hμ. In particular this establishes equality on such operators between the entropy notion introduced by R. Alicki, J. Andries, M. Fannes and P. Tuyls (a version of which was studied also by I.I. Makarov), another one introduced by E. Ghys, R. Langevin and P. Walczak, and our new definition introduced later in this paper. The key tool in proving this uniqueness is the discovery of a very general property of all doubly stochastic operators, which we call asymptotic lattice stability. Unlike the other explicit definitions of entropy mentioned above, ours satisfies many natural requirements already on the level of the function Hμ, and we prove that the limit defining hμ exists. The proof uses an integral representation of a stochastic operator obtained many years ago by A. Iwanik. In the topological part of the paper we introduce three natural definitions of topological entropy for Markov operators on C(X). Then we prove that all three are equal. Finally, we establish the partial variational principle: the topological entropy of a Markov operator majorizes the measure-theoretic entropy of this operator with respect to any of its invariant probability measures.
- Published
- 2005
29. Lyapunov 1-forms for flows
- Author
-
Eduard Zehnder, Janko Latschev, Thomas Kappeler, Michael Farber, University of Zurich, Farber, M, and Forschungsinstitut für Mathematik Zürich
- Subjects
Cech cohomology ,Lyapunov function ,Class (set theory) ,Pure mathematics ,LIAPUNOW-GLEICHUNGEN (MATRIZENGLEICHUNGEN) ,GEODÄTISCHE FLÜSSE (DIFFERENTIALGEOMETRIE) ,LYAPUNOV EQUATIONS (MATRIX EQUATIONS) ,GEODESIC FLOWS (DIFFERENTIAL GEOMETRY) ,Generalization ,General Mathematics ,chain recurrent set ,Dynamical Systems (math.DS) ,Set (abstract data type) ,symbols.namesake ,510 Mathematics ,2604 Applied Mathematics ,Chain (algebraic topology) ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,ddc:510 ,Mathematics - Dynamical Systems ,Čech cohomology ,2600 General Mathematics ,Lyapunov functions ,Mathematics ,Applied Mathematics ,510 Mathematik ,10123 Institute of Mathematics ,Compact space ,Flow (mathematics) ,theorem by Conley ,symbols - Abstract
In this paper we find conditions which guarantee that a given flow $\Phi$ on a compact metric space $X$ admits a Lyapunov one-form $\omega$ lying in a prescribed \v{C}ech cohomology class $\xi\in \check H^1(X;\R)$. These conditions are formulated in terms of the restriction of $\xi$ to the chain recurrent set of $\Phi$. The result of the paper may be viewed as a generalization of a well-known theorem of C. Conley about the existence of Lyapunov functions., Comment: 27 pages, 3 figures. This revised version incorporates a few minor improvements
- Published
- 2004
30. Non-standard real-analytic realizations of some rotations of the circle – CORRIGENDUM
- Author
-
Shilpak Banerjee
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
We correct two technical errors in the original paper. The main result in the original paper remains valid without any changes.
- Published
- 2016
31. The K-property for some unique equilibrium states in flows and homeomorphisms
- Author
-
Benjamin Call
- Subjects
Pure mathematics ,Property (philosophy) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Decomposition theory ,Set (abstract data type) ,Flow (mathematics) ,0103 physical sciences ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,010307 mathematical physics ,0101 mathematics ,Orbit (control theory) ,Mathematics - Abstract
We set out some general criteria to prove the K-property, refining the assumptions used in an earlier paper for the flow case, and introducing the analogous discrete-time result. We also introduce one-sided $\lambda $ -decompositions, as well as multiple techniques for checking the pressure gap required to show the K-property. We apply our results to the family of Mañé diffeomorphisms and the Katok map. Our argument builds on the orbit decomposition theory of Climenhaga and Thompson.
- Published
- 2021
32. Limit theorems for numbers of multiple returns in non-conventional arrays
- Author
-
Yuri Kifer
- Subjects
Applied Mathematics ,General Mathematics ,Applied mathematics ,Limit (mathematics) ,Mathematics - Abstract
For a $\psi $ -mixing process $\xi _0,\xi _1,\xi _2,\ldots $ we consider the number ${\mathcal N}_N$ of multiple returns $\{\xi _{q_{i,N}(n)}\in {\Gamma }_N,\, i=1,\ldots ,\ell \}$ to a set ${\Gamma }_N$ for n until either a fixed number N or until the moment $\tau _N$ when another multiple return $\{\xi _{q_{i,N}(n)}\in {\Delta }_N,\, i=1,\ldots ,\ell \}$ , takes place for the first time where ${\Gamma }_N\cap {\Delta }_N=\emptyset $ and $q_{i,N}$ , $i=1,\ldots ,\ell $ are certain functions of n taking on non-negative integer values when n runs from 0 to N. The dependence of $q_{i,N}(n)$ on both n and N is the main novelty of the paper. Under some restrictions on the functions $q_{i,N}$ we obtain Poisson distributions limits of ${\mathcal N}_N$ when counting is until N as $N\to \infty $ and geometric distributions limits when counting is until $\tau _N$ as $N\to \infty $ . We obtain also similar results in the dynamical systems setup considering a $\psi $ -mixing shift T on a sequence space ${\Omega }$ and studying the number of multiple returns $\{ T^{q_{i,N}(n)}{\omega }\in A^a_n,\, i=1,\ldots ,\ell \}$ until the first occurrence of another multiple return $\{ T^{q_{i,N}(n)}{\omega }\in A^b_m,\, i=1,\ldots ,\ell \}$ where $A^a_n,\, A_m^b$ are cylinder sets of length n and m constructed by sequences $a,b\in {\Omega }$ , respectively, and chosen so that their probabilities have the same order.
- Published
- 2021
33. Effective equidistribution for generalized higher-step nilflows
- Author
-
Minsung Kim
- Subjects
Nilpotent ,Pure mathematics ,Polynomial ,Equidistributed sequence ,Mathematics::Dynamical Systems ,Flow (mathematics) ,Applied Mathematics ,General Mathematics ,Diophantine equation ,Ergodic theory ,Measure (mathematics) ,Projection (linear algebra) ,Mathematics - Abstract
In this paper we prove bounds for ergodic averages for nilflows on general higher-step nilmanifolds. Under Diophantine condition on the frequency of a toral projection of the flow, we prove that almost all orbits become equidistributed at polynomial speed. We analyze the rate of decay which is determined by the number of steps and structure of general nilpotent Lie algebras. Our main result follows from the technique of controlling scaling operators in irreducible representations and measure estimation on close return orbits on general nilmanifolds.
- Published
- 2021
34. The geometric index and attractors of homeomorphisms of
- Author
-
Héctor Barge and J. J. Sánchez-Gabites
- Subjects
Pure mathematics ,Index (economics) ,Applied Mathematics ,General Mathematics ,Attractor ,Mathematics - Abstract
In this paper we focus on compacta$K \subseteq \mathbb {R}^3$which possess a neighbourhood basis that consists of nested solid tori$T_i$. We call these sets toroidal. Making use of the classical notion of the geometric index of a curve inside a torus, we introduce the self-geometric index of a toroidal setK, which roughly captures how each torus$T_{i+1}$winds inside the previous$T_i$as$i \rightarrow +\infty $. We then use this index to obtain some results about the realizability of toroidal sets as attractors for homeomorphisms of$\mathbb {R}^3$.
- Published
- 2021
35. Tying hairs for structurally stable exponentials
- Author
-
Robert L. Devaney and Ranjit Bhattacharjee
- Subjects
Pure mathematics ,symbols.namesake ,Applied Mathematics ,General Mathematics ,Tying ,Euler's formula ,symbols ,Fixed point ,Invariant (mathematics) ,Topology ,Julia set ,Mathematics ,Exponential function - Abstract
Our goal in this paper is to describe the structure of the Julia set of complex exponential functions that possess an attracting cycle. When the cycle is a fixed point, it is known that the Julia set is a ‘Cantor bouquet’, a union of uncountably many distinct curves or ‘hairs’. When the period of the cycle is greater than one, infinitely many of the hairs in the bouquet become pinched or attached together. In this paper, we develop an algorithm to determine which of these hairs are attached. Of crucial importance in this construction is the kneading invariant, a sequence that is derived from the topology of the basins of attraction of the attracting cycle.
- Published
- 2000
36. Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows
- Author
-
Ørjan Johansen and Richard Gjerde
- Subjects
Class (set theory) ,Pure mathematics ,Mathematics::Dynamical Systems ,Mathematics::Operator Algebras ,Applied Mathematics ,General Mathematics ,Dimension (graph theory) ,Substitution (logic) ,Toeplitz matrix ,Prime (order theory) ,Flow (mathematics) ,Equivalence (measure theory) ,Group theory ,Mathematics - Abstract
We construct Bratteli–Vershik models for Toeplitz flows and characterize a class of properly ordered Bratteli diagrams corresponding to these flows. We use this result to extend by a novel approach—using basic theory of dimension groups—an interesting and non-trivial result about Toeplitz flows, first shown by Downarowicz. (Williams had previously obtained preliminary results in this direction.) The result states that to any Choquet simplex $K$, there exists a $0$–$1$ Toeplitz flow $(Y,\psi)$, so that the set of invariant probability measures of $(Y,\psi)$ is affinely homeomorphic to $K$. Not only do we give a conceptually new proof of this result, we also show that we may choose $(Y,\psi)$ to have zero entropy and to have full rational spectrum.Furthermore, our Bratteli–Vershik model for a given Toeplitz flow explicitly exhibits the factor map onto the maximal equicontinuous (odometer) factor. We utilize this to give a simple proof of the existence of a uniquely ergodic 0–1 Toeplitz flow of zero entropy having a given odometer as its maximal equicontinuous factor and being strongly orbit equivalent to this factor. By the same token, we show the existence of 0–1 Toeplitz flows having the 2-odometer as their maximal equicontinuous factor, being strong orbit equivalent to the same, and assuming any entropy value in $[0,\ln 2)$.Finally, we show by an explicit example, using Bratteli diagrams, that Toeplitz flows are not preserved under Kakutani equivalence (in fact, under inducing)—contrasting what is the case for substitution minimal systems. In fact, the example we exhibit is an induced system of a 0–1 Toeplitz flow which is conjugate to the Chacon substitution system, thus it is prime, i.e. it has no non-trivial factors.The thrust of our paper is to demonstrate the relevance and usefulness of Bratteli–Vershik models and dimension group theory for the study of minimal symbolic systems. This is also exemplified in recent papers by Forrest and by Durand, Host and Skau, treating substitution minimal systems, and by papers by Boyle, Handelman and by Ormes.
- Published
- 2000
37. On logarithmically small errors in the lattice point problem
- Author
-
M. M. Skriganov and A. N. Starkov
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mathematics - Abstract
In the present paper we give an improvement of a previous result of the paper [M. M. Skriganov. Ergodic theory on $SL(n)$, diophantine approximations and anomalies in the lattice point problem. Inv. Math.132(1), (1998), 1–72, Theorem 2.2] on logarithmically small errors in the lattice point problem for polyhedra. This improvement is based on an analysis of hidden symmetries of the problem generated by the Weyl group for $SL(n,\mathbb{B})$. Let $UP$ denote a rotation of a given compact polyhedron $P\subset\mathbb{B}^n$ by an orthogonal matrix $U\in SO(n)$, $tUP$ a dilation of $UP$ by a parameter $t>0$ and $N(tUP)$ the number of integer points $\gamma\in\mathbb{Z}^n$ which fall into the polyhedron $tUP$. We show that for almost all rotations $U$ (in the sense of the Haar measure on $SO(n)$) the following asymptotic formula \[ N(t\UP)=t^n{\rm vol} P+ O((\log t)^{n-1+\varepsilon}),\quad t\to\infty, \] holds with arbitrarily small $\varepsilon>0$.
- Published
- 2000
38. Sojourn times in small neighborhoods of indifferent fixed points of one-dimensional dynamical systems
- Author
-
Tomoki Inoue
- Subjects
Pure mathematics ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,Fixed point ,Topology ,Mathematics - Abstract
We study one-dimensional dynamical systems with indifferent fixed points $p$ and $q$. The dynamical systems we study in this paper have ergodic, infinite, invariant measures. We consider the limit of the ratio of the sojourn time of the trajectory in a small neighborhood of $p$ to that in a small neighborhood of $q$. We show that the limit does not exist under some conditions, which has been announced in a previous paper. In fact we prove that the limit supreme of the ratio is $\infty$ and that the limit infimum is 0.
- Published
- 2000
39. Stationary solutions of non-autonomous Kolmogorov–Petrovsky–Piskunov equations
- Author
-
Yu. M. Suhov and Vitaly Volpert
- Subjects
Kolmogorov equations (Markov jump process) ,Applied Mathematics ,General Mathematics ,Applied mathematics ,Mathematics - Abstract
The paper is devoted to the following problem: \[ w'' (x) + c w'(x)+ F(w(x),x) = 0, \quad x\in{\mathbb R}^1,\quad w(\pm \infty) = w_{\pm}, \] where the non-linear term $F$ depends on the space variable $x$. A classification of non-linearities is given according to the behaviour of the function $F(w,x)$ in a neighbourhood of the points $w_+$ and $w_-$. The classical approach used in the Kolmogorov–Petrovsky–Piskunov paper [10] for an autonomous equation (where $F=F(u)$ does not explicitly depend on $x$), which is based on the geometric analysis on the $(w,w')$-plane, is extended and new methods are developed to analyse the existence and uniqueness of solutions in the non-autonomous case. In particular, we study the case where the function $F(w,x)$ does not have limits as $x \rightarrow \pm \infty$.
- Published
- 1999
40. Smooth unimodal maps in the 1990s
- Author
-
Grzegorz Świa¸Tek and Jacek Graczyk
- Subjects
Information retrieval ,Development (topology) ,Point (typography) ,Process (engineering) ,Applied Mathematics ,General Mathematics ,Key (cryptography) ,Mathematics - Abstract
The purpose of this paper is to survey the key ideas which have played a role in the development of the theory of smooth unimodal maps in the 1990s so as to provide a starting point for more detailed study. Most papers underlying this survey are distinguished by great technical complexity; we have therefore attempted to extract their essence and present it in a way accessible to a non-specialist. For further study we compiled a long list of original references. Another goal of this paper is to present the multi-directional process in which ideas flowed and developed, placed in a historical order.
- Published
- 1999
41. Entropy and ${\bi r}$ equivalence
- Author
-
Deborah Heicklen
- Subjects
Entropy (classical thermodynamics) ,Applied Mathematics ,General Mathematics ,Statistical physics ,Mathematics - Abstract
In this paper, the structure of $r$ equivalence, which was introduced by Vershik and which classifies group actions of the group $G=\sum_{n=1}^\infty{\Bbb Z}\slash r_n{\Bbb Z}$, $r_n\in{\Bbb N}\setminus\{1\}$, is examined. This is an equivalence relation that naturally arises from looking at certain sequences of $\sigma$-algebras. Vershik proved that if a sequence $r=(r_1,r_2,\ldots)$ does not satisfy a super-rapid growth rate, then entropy is an invariant for $r$ equivalence. In this paper, a strong converse of this is proven: for any $r$ which does satisfy this super-rapid growth rate, we can find a zero entropy action in every $r$ equivalence class.
- Published
- 1998
42. Quasisymmetric orbit-flexibility of multicritical circle maps
- Author
-
Edson de Faria and Pablo Guarino
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Gauss map ,Lebesgue measure ,Primary 37E10, Secondary 37E20, 37C40 ,Applied Mathematics ,General Mathematics ,Diophantine equation ,Dynamical Systems (math.DS) ,Bounded type ,Homeomorphism ,FOS: Mathematics ,SISTEMAS DINÂMICOS ,Uncountable set ,Diffeomorphism ,Mathematics - Dynamical Systems ,Rotation number ,Mathematics - Abstract
Two given orbits of a minimal circle homeomorphism $f$ are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with $f$. By a well-known theorem due to Herman and Yoccoz, if $f$ is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. As it follows from the a-priori bounds of Herman and Swiatek, the same holds if $f$ is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if $f$ is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$, then the number of equivalence classes is uncountable (Theorem A). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems B and C)., Comment: 38 pages, 5 figures. To appear in Ergodic Theory and Dynamical Systems
- Published
- 2021
43. Exponential mixing property for Hénon–Sibony maps of
- Author
-
Hao Wu
- Subjects
Property (philosophy) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mixing (physics) ,Exponential function ,Mathematics - Abstract
Let f be a Hénon–Sibony map, also known as a regular polynomial automorphism of $\mathbb {C}^k$ , and let $\mu $ be the equilibrium measure of f. In this paper we prove that $\mu $ is exponentially mixing for plurisubharmonic observables.
- Published
- 2021
44. Hyperbolicity of renormalization for dissipative gap mappings
- Author
-
Márcio R. A. Gouveia, Trevor Clark, Imperial College, and Universidade Estadual Paulista (UNESP)
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,gap mappings ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dynamical Systems (math.DS) ,Lorenz and Cherry flows ,Lorenz mappings ,01 natural sciences ,Primary 37E05, Secondary 37E20, 37E10 ,Renormalization ,Flow (mathematics) ,0103 physical sciences ,FOS: Mathematics ,Dissipative system ,Interval (graph theory) ,hyperbolicity of renormalization ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Topological conjugacy ,Mathematics - Abstract
A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have a gap between their ranges. They are one-dimensional dynamical systems, which arise in the study of certain higher dimensional flows, for example the Lorenz flow and the Cherry flow. In this paper, we prove hyperbolicity of renormalization acting on $C^3$ dissipative gap mappings, and show that the topological conjugacy classes of infinitely renormalizable gap mappings are $C^1$ manifolds.
- Published
- 2021
45. adic characterization of minimal ternary dendric shifts
- Author
-
FRANCE GHEERAERT, MARIE LEJEUNE, and JULIEN LEROY
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Characterization (mathematics) ,Ternary operation ,Mathematics - Abstract
Dendric shifts are defined by combinatorial restrictions of the extensions of the words in their languages. This family generalizes well-known families of shifts such as Sturmian shifts, Arnoux–Rauzy shifts and codings of interval exchange transformations. It is known that any minimal dendric shift has a primitive$\mathcal {S}$-adic representation where the morphisms in$\mathcal {S}$are positive tame automorphisms of the free group generated by the alphabet. In this paper, we investigate those$\mathcal {S}$-adic representations, heading towards an$\mathcal {S}$-adic characterization of this family. We obtain such a characterization in the ternary case, involving a directed graph with two vertices.
- Published
- 2021
46. Markovian random iterations of homeomorphisms of the circle
- Author
-
Edgar Matias
- Subjects
Pure mathematics ,symbols.namesake ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,symbols ,Markov process ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper we prove a local exponential synchronization for Markovian random iterations of homeomorphisms of the circle $S^{1}$ , providing a new result on stochastic circle dynamics even for $C^1$ -diffeomorphisms. This result is obtained by combining an invariance principle for stationary random iterations of homeomorphisms of the circle with a Krylov–Bogolyubov-type result for homogeneous Markov chains.
- Published
- 2021
47. Topological entropy of semi-dispersing billiards
- Author
-
S. Ferleger, Dmitri Burago, and A. Kononenko
- Subjects
Applied Mathematics ,General Mathematics ,Lorentz transformation ,Mathematical analysis ,Boundary (topology) ,Topological entropy ,Riemannian geometry ,Manifold ,Nonlinear Sciences::Chaotic Dynamics ,symbols.namesake ,Flow (mathematics) ,symbols ,Sectional curvature ,Dynamical billiards ,Mathematics - Abstract
In this paper we continue to explore the applications of the connections between singular Riemannian geometry and billiard systems that were first used in [6] to prove estimates on the number of collisions in non-degenerate semi-dispersing billiards.In this paper we show that the topological entropy of a compact non-degenerate semi-dispersing billiard on any manifold of non-positive sectional curvature is finite. Also, we prove exponential estimates on the number of periodic points (for the first return map to the boundary of a simple-connected billiard table) and the number of periodic trajectories (for the billiard flow). In \S5 we prove some estimates for the topological entropy of Lorentz gas.
- Published
- 1998
48. A rigidity theorem for manifolds without conjugate points
- Author
-
Bruce Kleiner and Christopher B. Croke
- Subjects
Pure mathematics ,Rigidity (electromagnetism) ,Applied Mathematics ,General Mathematics ,Conjugate points ,Geometry ,Mathematics - Abstract
In this paper we show that some nonsimply connected manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, section I.5]. This is a companion paper to [Cr-Kl1] that deals with the simply connected case. In particular, we show that one cannot make a nontrivial, compactly supported, change to a complete flat metric without introducing conjugate points.
- Published
- 1998
49. Inverse problems and rigidity questions in billiard dynamics
- Author
-
Vadim Kaloshin and Alfonso Sorrentino
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dynamics (mechanics) ,Inverse problem ,01 natural sciences ,Rigidity (electromagnetism) ,Classical mechanics ,Settore MAT/05 ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Dynamical billiards ,Mathematics - Abstract
A Birkhoff billiard is a system describing the inertial motion of a point mass inside a strictly convex planar domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain: while it is evident how the shape determines the dynamics, a more subtle and difficult question is the extent to which the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing inverse problems and unanswered rigidity questions, which have been the focus of very active research in recent decades. In this paper we describe some of these questions, along with their connection to other problems in analysis and geometry, with particular emphasis on recent results obtained by the authors and their collaborators.
- Published
- 2021
50. Lyapunov exponent of random dynamical systems on the circle
- Author
-
Dominique Malicet
- Subjects
Sequence ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,Diophantine equation ,010102 general mathematics ,Dynamical Systems (math.DS) ,State (functional analysis) ,Lyapunov exponent ,Computer Science::Computational Geometry ,Lambda ,01 natural sciences ,Combinatorics ,Orientation (vector space) ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,symbols ,Taylor series ,Computer Science::Symbolic Computation ,010307 mathematical physics ,Diffeomorphism ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Abstract
We consider products of an independent and identically distributed sequence in a set $\{f_1,\ldots ,f_m\}$ of orientation-preserving diffeomorphisms of the circle. We can naturally associate a Lyapunov exponent $\lambda $ . Under few assumptions, it is known that $\lambda \leq 0$ and that the equality holds if and only if $f_1,\ldots ,f_m$ are simultaneously conjugated to rotations. In this paper, we state a quantitative version of this fact in the case where $f_1,\ldots ,f_m$ are $C^k$ perturbations of rotations with rotation numbers $\rho (f_1),\ldots ,\rho (f_m)$ satisfying a simultaneous diophantine condition in the sense of Moser [On commuting circle mappings and simultaneous diophantine approximations. Math. Z.205(1) (1990), 105–121]: we give a precise estimate of $\lambda $ (Taylor expansion) and we prove that there exist a diffeomorphism g and rotations $r_i$ such that $\mbox {dist}(gf_ig^{-1},r_i)\ll |\lambda |^{{1}/{2}}$ for $i=1,\ldots , m$ . We also state analogous results for random products of $2\times 2$ matrices, without any diophantine condition.
- Published
- 2021
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