DOCSLIB.ORG

Set of Periods of a Subshift

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Set of Periods of a Subshift Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:63 https://doi.org/10.1007/s12044-018-0431-2 Set of periods of a subshift K ALI AKBAR1,∗ and V KANNAN2 1Department of Mathematics, Central University of Kerala, Kasaragod 671 316, India 2School of Mathematics and Statistics, University of Hyderabad, Hyderabad 500 046, India *Corresponding author. E-mail: [email protected]; [email protected] MS received 31 October 2016; revised 15 June 2017; accepted 12 October 2017; published online 25 October 2018 Abstract. In this article, subsets of N that can arise as sets of periods of the following subshifts are characterized: (i) subshifts of finite type, (ii) transitive subshifts of finite type, (iii) sofic shifts, (iv) transitive sofic shifts, and (v) arbitrary subshifts. Keywords. Subshift of finite type; transitive subshift; strongly connected digraph; sofic shift. 2010 Mathematics Subject Classification. Primary: 37B10; Secondary: 54H20. 1. Introduction and preliminaries The subshifts of finite type (SFT) form a very important class of dynamical systems. There are plenty of books that explain how their study would throw light on still larger classes of dynamical systems (see [7,8,11]). A dynamical system is a pair (X, f ), where X is a metric space and f is a continuous self map. For each dynamical system (X, f ), the set Per( f ) ={n ∈ N :∃x ∈ X such that f n(x) = x = f m(x) ∀m < n} consisting of the lengths of the cycles, is a subset of the set N of positive integers. Given a class of dynamical systems, we obtain in this way a family of subsets of N. There have been attempts to describe explicitly this family of subsets of N, for various classes of dynamical systems (see [2,3,5,6,8,9,12] and [13]). It is therefore natural to ask for a description of sets of periods for the class of all subshifts of finite type. In this paper, we answer this and two other related questions (see Theorems 1 and 3). Here, we concentrate on two-sided shifts. The case of one-sided shifts is similar. Let A be a non-empty finite set, called alphabet with discrete topology and consider the Z set A , which denotes the set of doubly-infinite sequences (xi )i∈Z where each xi ∈ A, with product topology. It is compact and metrizable. The shift is the homeomorphism Z Z Z σ : A → A given by σ(x)i = xi+1 for all i ∈ Z. The pair (A ,σ)forms a dynamical system called a full shift. We say that a point x ∈ AZ is periodic if σ n x = x for some n ∈ N, where σ n is the composition of σ with itself n times. The smallest such positive integer n is called the σ-period (simply period) of x. A subset A ⊂ AZ is called σ -invariant © Indian Academy of Sciences 1 63 Page 2 of 8 Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:63 if σ(A) ⊂ A. A subshift is a σ-invariant nonempty closed subset X of a full shift, together with the restriction of σ to X. We denote the set of all periodic points of σ in X by P(X) and the set of periods of all periodic points of σ in X by Per(X).Awordw on A is a concatenation w1w2 ...wk, whenever each wi ∈ A, w¯ is defined by (w)¯ n = wr where Z n ≡ r(mod k). A subshift X is said to be a subshift of finite type if X = XF ={x ∈ A : no word in F appears in x} for some finite set of words F. A subshift is said to be sofic if there exists a continuous surjection from an SFT to it. A subshift (X,σ)is said to be transitive if for any nonempty open sets U, V in X, there exists n ∈ N such that σ n(U) ∩ V =∅. An infinite subshift is said to be Devaney chaotic (simply we say chaotic), if it is transitive and has a dense set of periodic points. Our main results are: (1) period set of an SFT (or sofic shift) is a non empty set of the form FNG, where F and G are finite subsets of N, (2) period set of a transitive SFT (or a transitive sofic shift) is either a singleton subset of N or a set of the form kN \ F for some finite subset F of N and for some k ∈ N, (3) for any S ⊂ N, there exists a subshift X such that Per(X) = S. 2. Main results 2.1 Subshifts of finite type Let G = (V, E) be any directed graph(digraph) with vertex set V and edge set E.A subgraph G = (V , E ) of G is said to be a full subgraph if E = E ∩{(v1,v2) : v1,v2 ∈ V }. A digraph is said to be simple if from every vertex v to a vertex w there is atmost one edge and it is said to be strongly connected if between any two vertices there exists a directed path. It is to be noted that a connected digraph may not be strongly connected. For every subshift of finite type, there is an associated digraph and an associated matrix as described in [7]. This may or may not be strongly connected. But, the period set of a strongly connected simple digraph can be described easily. The subshift of finite type associated with a simple digraph G is denoted as XG . Note that Per(XG ) = Per(XG ), G is a finite union of such strongly connected simple digraphs. A cycle C is said to be simple if XC contains a periodic point whose period is exactly the length of C. For a finite subset A of N, we denote A ⊂⊂ N. PROPOSITION 1 (see [7], p. 58) Every SFT is isomorphic to one in which every forbidden word has length 2. Lemma 1[see 4]. Let a, b ∈ N. (i) If gcd(a, b) = 1, then for every n > ab there exist positive integers x and y such that n = ax + by. (ii) If gcd(a, b) = 1 and n = ab, then there are no positive integers x and y such that n = ax + by. Proof. If gcd(a, b) = 1, there exist positive integers x and y such that ax − by = 1. For every n > ab, write n = m(ab) + k,1≤ k ≤ ab. Hence the lemma follows. Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:63 Page 3 of 8 63 The following remarks easily follows from Lemma 1. Remark 1. If ai ∈ N, i = 1, 2,...,n and gcd(a1,...,an) = 1, then for every positive integer n > a1a2 ···an there exist positive integers, xi , i = 1, 2,...,n such that n = a1x1 + a2x2 +···+an xn. Remark 2. If ai ∈ N, i = 1, 2,...,n and (a1, a2,...,an) = k, then for every n > ... a1a2 an = , ,..., = + + kn there exist positive integers xi , i 1 2 n such that kn a1x1 a2x2 ···+an xn. Theorem 1. The following are equivalent for a subset S of N: (1) S = Per(XG ) for some strongly connected simple digraph G, (2) either S is singleton or S = kN \ Fforsomek∈ N and for some F ⊂⊂ N. Proof. 1 ⇒ 2. Let S be a nonempty, non-singleton subset of N such that S = Per(XG ). Consider k = gcd(S). Claim. kN \ S is finite. Case 1. When k ∈ S.Ifm and l are distinct and are lengths two cycles having one vertex common, then al + bm ∈ S for all a, b ∈ N.IfG is a strongly connected digraph, then for all m ∈ S, there exists l ∈ S such that al + bm ∈ S for all a, b ≥ 1. Let k = gcd(l, m). Then all but finitely many elements of k N are in S and hence mN\ S is finite for all m ∈ S. In particular, kN \ S is finite. Case 2. When k is not in S. Weclaim that there exists a finite subset S of S such that k is equal to the gcd of all its elements. Note that k < s for all s ∈ S.Lets ∈ S. Since k = gcd(S), there exists t1 ∈ S such that k ≤ gcd(t1, s)<s. If gcd(t1, s) = k, then take S ={t1, s}. Else there exists t2 ∈ S such that k ≤ gcd(t1, t2, s)<gcd(t1, s).Ifk = gcd(t1, t2, s), then take S ={t1, t2, s}. After a finite number of steps we will get S ,say{s1, s2,...,sr },as we claimed. Let Ci be a simple cycle having length si and xi ∈ Ci for i = 1, 2,...,r. Then there exists a path from xi(mod r) to xi+1(mod r) which does not contain any cycle. Let C be a simple cycle of length s obtained from the cycles Ci s and the above paths. Then si N + sN ⊂ S for i = 1, 2,...,r. Hence s1N + sN + s2N + sN +···+sr N + sN ⊂ S. ie., s1N + s2N +···+snN + rsN ⊂ S.Buts1N + s2N +···+snN + rsN ⊂ S contains all but finitely many elements of kN. Therefore kN \ S is finite. Hence from Case 1 and Case 2, we have S = kN \ F, where F is the finite set kN \ S. 2 ⇒ 1. Now, let S be a subset of N such that it is either singleton or it is equal to kN \ F for some k ∈ N and F ⊂⊂ N. Case 1. When S ={k} for some k ∈ N. Consider a strongly connected simple digraph G with exactly one cycle of length k.
Load more

Recommended publications
  • Denseness of Intermediate Β-Shifts of Finite Type
    DENSENESS OF INTERMEDIATE β-SHIFTS OF FINITE TYPE BING LI, TUOMAS SAHLSTEN, TONY SAMUEL, AND WOLFGANG STEINER Abstract. We determine the structure of the set of intermediate β-shifts of finite type. Specifically, we show that this set is dense in the parameter space ∆ = f(β,α) 2 R2 : β 2 (1;2) and 0 ≤ α ≤ 2 − βg. This generalises the classical result of Parry from 1960 for greedy and (normalised) lazy β-shifts. 1. Introduction and statement of main results Since the pioneering work of Renyi´ [35] and Parry [31, 32, 33], intermediate β-shifts have been well-studied by many authors, and have been shown to have important connections with ergodic theory, fractal geometry and number theory. We refer the reader to [9, 23, 36] for survey articles concerning this topic. For β > 1 and x 2 [0;1=(β − 1)], a word (!n)n2N with letters in the alphabet f0;1g is called a β-expansion of x if 1 X −k x = !k β : k=1 When β is a natural number, Lebesgue almost all numbers x have a unique β-expansion. On the other hand, in [37, Theorem 1] it was shown that if β is not a natural number, then, for almost all x, the cardinality of the set of β-expansions of x is equal to the cardinality of the continuum. In [3, 4, 38] the set of points which have countable number of β-expansions has also been studied. Through iterating the maps Gβ : x 7! βx mod 1 on [0;1) and Lβ : x 7! β(x − 1) mod 1 on (0;1], see Figure1, for β 2 (1;2], one obtains subsets of f0;1gN known as the greedy and (normalised) lazy β-shifts, respectively, where each point !+ of the greedy β-shift is a β-expansion, and corresponds to a unique point in [0;1], and each point !− of the lazy β-shift is a β-expansion, and corresponds to a unique point in [(2 − β)=(β − 1);1=(β − 1)].
    [Show full text]
  • Dynamical Zeta Functions and the Distribution of Orbits
    Dynamical zeta functions and the distribution of orbits Mark Pollicott Warwick University Abstract In this survey we will consider various counting and equidistribution results associated to orbits of dynamical systems, particularly geodesic and Anosov flows. Key tools in this analysis are appropriate complex functions, such as the zeta functions of Selberg and Ruelle, and Poincar´eseries. To help place these definitions and results into a broader context, we first describe the more familiar Riemann zeta function in number theory, the Ihara zeta function for graphs and the Artin-Mazur zeta function for diffeomorphisms. Contents 1 Introduction 3 1.1 Zeta functions as complex functions . 3 1.2 Different types of zeta functions . 3 2 Zeta functions in Number Theory 3 2.1 Riemann zeta function . 4 2.2 Other types . 7 2.2.1 L-functions . 7 2.2.2 Dedekind Zeta functions . 7 2.2.3 Finite Field Zeta functions . 8 3 Zeta functions for graphs 8 3.1 Directed graphs . 8 3.1.1 The zeta function for directed graphs . 8 3.1.2 The prime graph theorem (for directed graphs) . 11 3.1.3 A dynamical viewpoint . 12 3.2 Undirected graphs: Ihara zeta function . 13 Key words: Dynamical zeta function, closed orbits, Selberg zeta function, Ihara zeta function MSC : 37C30, 37C27, 11M36 Acknowledgements: I am grateful to the organisers and participants of the Summer school \Number Theory and Dynamics" in Grenoble for their questions and comments and, in particular, to Mike Boyle for his detailed comments on the draft version of these notes. I am particularly grateful to Athanase Papadopoulos for his help with the final version.
    [Show full text]
  • L'institut Fourier
    R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Jérôme BUZZI Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps Tome 60, no 3 (2010), p. 801-852. <http://aif.cedram.org/item?id=AIF_2010__60_3_801_0> © Association des Annales de l’institut Fourier, 2010, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 60, 3 (2010) 801-852 PUZZLES OF QUASI-FINITE TYPE, ZETA FUNCTIONS AND SYMBOLIC DYNAMICS FOR MULTI-DIMENSIONAL MAPS by Jérôme BUZZI Abstract. — Entropy-expanding transformations define a class of smooth dy- namics generalizing interval maps with positive entropy and expanding maps. In this work, we build a symbolic representation of those dynamics in terms of puz- zles (in Yoccoz’s sense), thus avoiding a connectedness condition, hard to satisfy in higher dimensions. Those puzzles are controled by a «constraint entropy» bounded by the hypersurface entropy of the aforementioned transformations.
    [Show full text]
  • On Moebius Orthogonality for Subshifts of Finite Type with Positive
    ON MOEBIUS ORTHOGONALITY FOR SUBSHIFTS OF FINITE TYPE WITH POSITIVE TOPOLOGICAL ENTROPY. D. KARAGULYAN Abstract. In this note we prove that Moebius orthogonality does not hold for subshifts of finite type with positive topological entropy. This, in particular, shows that all C1+α surface diffeomorphisms with positive entropy correlate with the Moebius function. 1. Introduction Let µ denote the M¨obius function, i.e. k (−1) if n = p1p2 ··· p for distinct primes p , µ(n)= k i (0 otherwise. In [11], [12] Sarnak introduced the following related conjecture. Recall that a topological dynamical system (Y, T ) is a compact metric space Y with a homeomorphism T : Y → Y , and the topological entropy h(Y, T ) of such a system is defined as 1 h(Y, T ) = lim lim log N(ǫ, n), ǫ→0 n→∞ n where N(ǫ, n) is the largest number of ǫ-separated points in Y using the metric dn : Y × Y → (0, ∞) defined by i i dn(x, y) = max d(T x, T y). 0≤i≤n A sequence f : Z → C is said to be deterministic if it is of the form arXiv:1701.01968v1 [math.DS] 8 Jan 2017 f(n)= F (T nx), for all n and some topological dynamical system (Y, T ) with zero topological entropy h(Y, T ) = 0, a base point x ∈ Y , and a continuous function F : Y → C. Conjecture 1. (P. Sarnak) Let f : N → C be a deterministic sequence. Then 1 n (1.1) S (T (x), f)= µ(k)f(k)= o(1).
    [Show full text]
  • An Introduction to C∗-Algebras from Symbolic Dynamics
    AN INTRODUCTION TO C∗-ALGEBRAS FROM SYMBOLIC DYNAMICS FRANCESCA ARICI Abstract. This is a crash course into the realm of operator algebraic tecniques in the study of symbolic dynamics, focusing on a pedagogical example, that of Cuntz and Cuntz-Krieger algebras. Contents 1. Introduction 1 2. Shifts and subshifts of finite type 1 2.1. A class of Markov chains 1 2.2. The space of infinite words and the shift operator 2 3. Groupoids for (sub)shifts of finite type and their C∗-algebras 3 3.1. Etale´ groupoids 3 4. The Renault groupoid and Cuntz-Krieger algebras 5 4.1. Cuntz-Krieger algebras 5 References 6 1. Introduction The paradigm/philosophy of noncommutative geometry is the idea of using the language of operator algebras to study objects coming from different fields in mathematics hoping to get some insight from invariants coming from their associated C∗-algebras. One starts from an input, typically a dynamical object, like a dynamical system or a graph, or a geometric object, like a foliation or a singular space, and associates to this object a C∗-algebra that encodes all the relevant information. By looking at the properties of this algebra and its invariants, e.g. its K-theory, we want to get information that we can relate back to the input 1. Very often, the first passage in this receipe involves passing through an intermediate topological object, a groupoid. This talk will hopefully give you a hint of what happens in the first passage in the case of dynamical systems. We will consider a particular class of dynamical systems, namely shifts and subshifts of finite type, and we will construct, using a groupoid model, their corresponding C∗-algebras, that go under the name of Cuntz [1] and Cuntz-Krieger [2] algebras.
    [Show full text]
  • A Characterization of the Entropies of Multidimensional Shifts of Finite Type
    ANNALS OF MATHEMATICS A characterization of the entropies of multidimensional shifts of finite type By Michael Hochman and Tom Meyerovitch SECOND SERIES, VOL. 171, NO. 3 May, 2010 anmaah Annals of Mathematics, 171 (2010), 2011–2038 A characterization of the entropies of multidimensional shifts of finite type By MICHAEL HOCHMAN and TOM MEYEROVITCH Abstract We show that the values of entropies of multidimensional shifts of finite type (SFTs) are characterized by a certain computation-theoretic property: a real num- ber h 0 is the entropy of such an SFT if and only if it is right recursively enu- merable, i.e. there is a computable sequence of rational numbers converging to h from above. The same characterization holds for the entropies of sofic shifts. On the other hand, the entropy of strongly irreducible SFTs is computable. 1. Introduction A shift of finite type (SFT) is an ensemble of colorings of Z (a one-dimensional SFT) or Zd for d > 1 (a multidimensional SFT) defined by local rules. SFTs are one of the fundamental objects of study in symbolic dynamics, and their most significant invariant is their (topological) entropy, which measures the asymptotic growth of the number of legal colorings of finite regions (see 2 for definitions). Besides having been studied extensively from a dynamical perspective as topolog- ical analogs of Markov chains[21],[28],[27], SFTs appear naturally in a wide range of other disciplines. In information theory, SFTs were used by Shannon as models for discrete communication channels[24], for which entropy describes the capacity; similarly, SFTs model “two-dimensional” channels[8].
    [Show full text]
  • Computation in Sofic Quantum Dynamical Systems
    Computation in Sofic Quantum Dynamical Systems 1, 1, Karoline Wiesner ∗ and James P. Crutchfield † 1Center for Computational Science & Engineering and Physics Department, University of California Davis, One Shields Avenue, Davis, CA 95616 We analyze how measured quantum dynamical systems store and process information, introduc- ing sofic quantum dynamical systems. Using recently introduced information-theoretic measures for quantum processes, we quantify their information storage and processing in terms of entropy rate and excess entropy, giving closed-form expressions where possible. To illustrate the impact of measurement on information storage in quantum processes, we analyze two spin-1 sofic quantum systems that differ only in how they are measured. I. INTRODUCTION shifts in sequence space. Definitions of entropy followed from this. However, no connection to quantum finite- Extending concepts from symbolic dynamics to the state automata was established there. quantum setting, we forge a link between quantum dy- In the following we develop an approach that differs namical systems and quantum computation, in general, from this and other previous attempts since it explic- and with quantum automata, in particular. itly accounts for observed sequences, in contrast to se- quences of (unobservable) quantum objects, such as spin Symbolic dynamics originated as a method to study chains. The recently introduced computational model general dynamical systems when, nearly 100 years ago, class of quantum finite-state generators provides the re- Hadamard used infinite sequences of symbols to analyze quired link between quantum dynamical systems and the the structure of geodesics on manifolds of negative cur- theory of automata and formal languages [11]. It gives vature; see Ref.
    [Show full text]
  • Periodic Points for Onto Cellular Automata
    PERIODIC POINTS FOR ONTO CELLULAR AUTOMATA MIKE BOYLE AND BRUCE KITCHENS Summary. Let ' be a one-dimensional surjective cellular automaton map. We prove that if ' is a closing map, then the con¯gurations which are both spatially and temporally periodic are dense. (If ' is not a closing map, then we do not know whether the temporally periodic con¯gurations must be dense.) The results are special cases of results for shifts of ¯nite type, and the proofs use symbolic dynamical techniques. 1. Introduction and sermon Let ' be a surjective one-dimensional cellular automaton map (in the language of symbolic dynamics, ' is a surjective endomorphism of a full shift). Must the set of '-periodic points be dense? This is a basic question for understanding the topological dynamics of ', and we are unable to resolve it. However, if ' is a closing endomorphism of a mixing subshift of ¯nite type σA, then we can show the points which are periodic for both ' and σA are dense (Theorem 4.4). This is our main result, and of course it answers our question in the case that the c.a. map is closing. We give a separate proof for the special case that ' is an algebraic map (Proposition 3.2). The proofs are completely di®erent and both have ingredients which might be useful in more general settings. The paper is organized so that a reader with a little background can go directly to Sections 3 and 4 and quickly understand our results. We work in the setting of subshifts of ¯nite type, and to explain this to some c.a.
    [Show full text]
  • Lecture 4. Entropy and Markov Chains
    preliminary version : Not for di®usion Lecture 4. Entropy and Markov Chains The most important numerical invariant related to the orbit growth in topological dynamical systems is topological entropy.1 It represents the exponential growth rate of the number of orbit segments which are distinguishable with an arbitrarily high but ¯nite precision. Of course, topological entropy is invariant by topological conjugacy. For measurable dynamical systems, an entropy can be de¯ned using the invariant measure. It gives an indication of the amount of randomness or complexity of the system. The relation between measure{theoretical entropy and topological entropy is given by a variational principle. 4.1 Topological Entropy We will follow the de¯nition given by Rufus Bowen in [Bo, Chapter 4]. Let X be a compact metric space. De¯nition 4.1 Let S ½ X, n 2 N and " > 0. S is a (n; "){spanning set if for every x 2 X there exists y 2 S such that d(f j(x); f j(y)) · " for all 0 · j · n. It is immediate to check that the compactness of X implies the existence of ¯nite spanning sets. Let r(n; ") be the least number of points in an (n; "){spanning set. If we bound the time of observation of our dynamical system by n and our precision in making measurements is " we will see at most r(n; ") orbits. Exercise 4.2 Show that if X admits a cover by m sets of diameter · " then r(n; ") · mn+1. De¯nition 4.3 The topological entropy htop(f) of f is given by 1 htop(f) = lim lim sup log r(n; ") : [4:1] "!0 n!+1 n In the previous de¯nition one cannot replace lim sup with lim since there exist examples of maps for which the limit does not exist.
    [Show full text]
  • Shifts of Finite Type As Fundamental Objects in the Theory of Shadowing
    Invent. math. https://doi.org/10.1007/s00222-019-00936-8 Shifts of finite type as fundamental objects in the theory of shadowing Chris Good1 · Jonathan Meddaugh2 Received: 17 September 2017 / Accepted: 26 November 2019 © The Author(s) 2019 Abstract Shifts of finite type and the notion of shadowing, or pseudo-orbit tracing, are powerful tools in the study of dynamical systems. In this paper we prove that there is a deep and fundamental relationship between these two concepts. Let X be a compact totally disconnected space and f : X → X a continuous map. We demonstrate that f has shadowing if and only if the system ( f, X) is (conjugate to) the inverse limit of a directed system satisfying the Mittag-Leffler condition and consisting of shifts of finite type. In particular, this implies that, in the case that X is the Cantor set, f has shadowing if and only if ( f, X) is the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type. Moreover, in the general compact metric case, where X is not necessarily totally disconnected, we prove that f has shadowing if ( f, X) is a factor of the inverse limit of a sequence satisfying the Mittag-Leffler condition and consisting of shifts of finite type by a quotient that almost lifts pseudo-orbits. The authors gratefully acknowledge support from the European Union through funding the H2020-MSCA-IF-2014 project ShadOmIC (SEP-210195797). The first author gratefully acknowledges the support of the Institut Mittag-Leffler, through the workshop Thermodynamic Formalism – Applications to Geometry, Number Theory, and Stochastics.
    [Show full text]
  • INTERMEDIATE Β-SHIFTS of FINITE TYPE Bing Li Tuomas Sahlsten Tony Samuel
    DISCRETE AND CONTINUOUS doi:10.3934/dcds.2016.36.323 DYNAMICAL SYSTEMS Volume 36, Number 1, January 2016 pp. 323{344 INTERMEDIATE β-SHIFTS OF FINITE TYPE Bing Li Department of Mathematics South China University of Technology Guangzhou, 510641, China Tuomas Sahlsten Einstein Institute of Mathematics The Hebrew University of Jerusalem Givat Ram, Jerusalem 91904, Israel Tony Samuel Fachbereich 3 Mathematik Universit¨atBremen 28359 Bremen, Germany (Communicated by Sylvain Crovisier) Abstract. An aim of this article is to highlight dynamical differences between the greedy, and hence the lazy, β-shift (transformation) and an intermediate β-shift (transformation), for a fixed β 2 (1; 2). Specifically, a classification in terms of the kneading invariants of the linear maps Tβ,α : x 7! βx + α mod 1 for which the corresponding intermediate β-shift is of finite type is given. This characterisation is then employed to construct a class of pairs (β; α) such that the intermediate β-shift associated with Tβ,α is a subshift of finite type. It is also proved that these maps Tβ,α are not transitive. This is in contrast to the situation for the corresponding greedy and lazy β-shifts and β-transformations, for which both of the two properties do not hold. 1. Introduction, motivation and statement of main results. 1.1. Introduction and motivation. For a given real number β 2 (1; 2) and a real number x 2 [0; 1=(β − 1)], an infinite word (!n)n2N over the alphabet f0; 1g is called a β-expansion of the point x if 1 X −k x = !kβ : k=1 If β is a natural number, then the β-expansions of a point x correspond to the β-adic expansions of x.
    [Show full text]
  • Dynamical Systems and Ergodic Theory Phd Exam – Spring 2012
    Dynamical Systems and Ergodic Theory PhD Exam { Spring 2012 Introduction: This is the first year for the Dynamical Systems and Ergodic Theory PhD exam. As such the material on the exam will conform fairly closely to the topics in this year's MTG6401 and MTG6402. In future years a standard syllabus will be developed and the exam and courses will cover that material. As such, the sample problems listed below probably give you the best idea of what material to expect on the Spring, 2012 exam. Topics: • Topological Dynamics { Definitions and basic results about the following for maps and flows. ◦ periodic points and orbits ◦ alpha and omega limit sets ◦ recurrence, recurrent set ◦ nonwandering set ◦ transitivity (both forward orbit and full orbit) ◦ topologically mixing ◦ minimal sets ◦ sensitivity on initial data ◦ conjugacy and semicongugacy ◦ stability of fixed and periodic points • Symbolic Dynamics { Definitions and basic results about the following ◦ cylinder sets ◦ metrics ◦ subshifts of finite type ◦ The Perron-Fr¨obenius Theorem and its consequences. • Differential Equations and flows { Definitions and basic results about the following ◦ stability of rest points ◦ stability of periodic orbits and Floquet multipliers 1 • Ergodic theory { Definitions and basic results about the following ◦ von Neumann's L2 ergodic theorem ◦ Birkhoffs pointwise ergodic theorem ◦ ergodicity and strong mixing ◦ Bernoulli and Markov measures on subshifts of finite type. • Area preserving dynamics { Definitions and basic results about the following ◦ Poincar´eRecurrence Theorem ◦ Connections of ergodicity and topological dynamics ◦ existence of invariant measures ◦ The space of invariant measures Book sections: These sections give you some idea of the material. More will be added.
    [Show full text]