DOCSLIB.ORG

(Α > 0) Diffeomorphism on a Compact

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
(Α > 0) Diffeomorphism on a Compact JOURNALOF MODERN DYNAMICS doi: 10.3934/jmd.2011.5.593 VOLUME 5, NO. 3, 2011, 593–608 BERNOULLI EQUILIBRIUM STATES FOR SURFACE DIFFEOMORPHISMS OMRI M. SARIG (Communicated by Anatole Katok) 1 ABSTRACT. Suppose f : M M is a C ® (® 0) diffeomorphism on a com- ! Å È pact smooth orientable manifold M of dimension 2, and let ¹ª be an equi- librium measure for a Hölder-continuous potential ª: M R. We show that ! if ¹ª has positive measure-theoretic entropy, then f is measure-theoretically isomorphic mod ¹ª to the product of a Bernoulli scheme and a finite rota- tion. 1. STATEMENTS 1 ® Suppose f : M M is a C Å (® 0) diffeomorphism on a compact smooth ! È orientable manifold M of dimension two and ª: M R is Hölder-continuous. ! An invariant probability measure ¹ is called an equilibrium measure, if it max- imizes the quantity h (f ) R ªd¹, where h (f ) is the measure-theoretic en- ¹ Å ¹ tropy. Such measures always exist when f is C 1, because in this case the func- tion ¹ h (f ) is upper semicontinuous [12]. Let ¹ be an ergodic equilibrium 7! ¹ ª measure of ª. We prove: THEOREM 1.1. If h (f ) 0, then f is measure-theoretically isomorphic with ¹ª È respect to ¹ª to the product of a Bernoulli scheme (see §3.3) and a finite rotation (a map of the form x x 1 (mod p) on {0,1,...,p 1}). 7! Å ¡ In the particular case of the measure of maximal entropy (ª 0) we can say ´ more, see §5.1. The proof applies to certain potentials that are not necessarily Hölder-continuous, such as t log J (t R), where J is the unstable Jacobian, ¡ u 2 u see §5.2. The theorem is false in higher dimensions: Let f denote the product of a hyperbolic toral automorphism and an irrational rotation. This C 1 diffeomor- phism has many equilibrium measures of positive entropy. But f cannot satisfy the conclusion of the theorem with respect to any of these measures, because f has the irrational rotation as a factor, and therefore none of its powers can have ergodic components with the K property. Bowen [4] and Ratner [20] proved Theorem 1.1 for uniformly hyperbolic dif- feomorphisms. In the nonuniformly hyperbolic case, Pesin proved that any Received May 2, 2011; revised July 19, 2011. 2000 Mathematics Subject Classification: Primary: 37D35; Secondary: 37D25. Key words and phrases: Bernoulli, surface diffeomorphisms, equilibrium measures, count- able Markov partitions. This work was supported by ERC award ERC-2009-StG n± 239885. INTHEPUBLICDOMAINAFTER 2039 593 ©2011 AIMSCIENCES 594 OMRI M. SARIG absolutely continuous ergodic invariant measure all of whose Lyapunov expo- nents are nonzero is isomorphic to the product of a Bernoulli scheme and a finite rotation [19]. By Pesin’s Entropy Formula and Ruelle’s Entropy Inequal- ity, these measures are equilibrium measures of log J . Ledrappier extended ¡ u Pesin’s result to all equilibrium measures with nonzero exponents for the po- tential log J , including those which are not absolutely continuous [10]. These ¡ u results hold in any dimension. The work of Pesin and Ledrappier (see also [17]) uses the following prop- erty of equilibrium measures of log J : the conditional measures on unstable ¡ u manifolds are absolutely continuous [10]. This is false for general Hölder po- tentials [11]. Theorem 1.1 is proved in three steps: 1. Symbolic dynamics: Any ergodic equilibrium measure on M with pos- itive entropy is a finite-to-one Hölder factor of an ergodic equilibrium measure on a countable Markov shift. 2. Ornstein Theory: Factors of equilibrium measures of Hölder potentials on topologically mixing countable Markov shifts are Bernoulli. 3. Spectral decomposition: The nonmixing case. 1 1 Notation. a M § b means M ¡ b a Mb. Æ · · 2. STEPONE: SYMBOLIC DYNAMICS Let G be a directed graph with a countable collection of vertices V such that every vertex has at least one edge coming in, and at least one edge coming out. The countable Markov shift associated to G is the set Z § §(G ) : {(vi )i Z V : vi vi 1 for all i}. Æ Æ 2 2 ! Å The natural metric d(u,v) : exp[ min{ i : u v }] turns § into a complete Æ ¡ j j i 6Æ i separable metric space. § is compact iff G is finite. § is locally compact iff every vertex of G has finite degree. The cylinder sets (2.1) m[am,...,an] : {(vi )i Z § : vi ai (i m,...,n)} Æ 2 2 Æ Æ form a basis for the topology, and they generate the Borel σ-algebra B(§). The left shift map σ: § § is defined by σ[(vi )i Z] (vi 1)i Z. Given a,b n ! 2 Æ Å 2 2 V , write a b when there is a path a v1 vn 1 b in G . The left shift ¡! ! !n ¢¢¢ ! ¡ ! n is topologically transitive iff a,b V n (a b). In this case gcd{n : a a} is 8 2 9 ¡! ¡! the same for all a V , and is called the period of σ. The left shift is topolog- 2 ically mixing iff it is topologically transitive and its period is equal to one. See [9]. Let # § : {(vi )i Z § : u,v V nk ,mk such that v m u,vn v}. Æ 2 2 9 2 9 " 1 ¡ k Æ k Æ Every σ-invariant probability measure gives §# full measure, because of Poin- caré’s Recurrence Theorem. JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 593–608 BERNOULLI EQUILIBRIUM STATES FOR SURFACE DIFFEOMORPHISMS 595 1 ® Suppose f : M M is a C Å -diffeomorphism of a compact, orientable and ! smooth manifold M such that dimM 2. If h (f ) 0, then every f -invariant Æ top Æ measure has zero entropy by the variational principle, and Theorem 1.1 holds trivially. So we assume without loss of generality that h (f ) 0. top È Fix 0  h (f ). A set ­ M is said to be Â-large, if ¹(­) 1 for every Ç Ç top ½ Æ ergodic invariant probability measure ¹ whose entropy is greater than Â. The following theorems are in [23]: THEOREM 2.1. There exists a locally compact countable Markov shift §Â and a # Hölder-continuous map ¼Â : §Â M such that ¼Â σ f ¼Â, ¼Â[§Â] is Â-large, # ! ± Æ ± and every point in ¼Â[§Â] has finitely many preimages. THEOREM 2.2. Denote the set of states of §Â by VÂ. There exists a function ' : V V N such that if x ¼Â[(vi )i Z] and vi u for infinitely many neg- £ ! Æ 2 Æ 1 ative i, and v v for infinitely many positive i, then ¼¡ (x) ' (u,v). i Æ j  j ·  THEOREM 2.3. Every ergodic f -invariant probability measure ¹ on M such that 1 h (f )  equals ¹ ¼¡ for some ergodic σ-invariant probability measure ¹ on ¹ È b ±  b §Â with the same entropy. We will use these results to reduce the problem of Bernoullicity for equilib- rium measures for f : M M and the potential ª, to the problem of Bernoul- ! licity for equilibrium measures for σ: § § and the potential à : ª ¼ .  ! Â Æ ±  3. STEP TWO:ORNSTEIN THEORY First we describe the structure of equilibrium measures of Hölder-continuous potentials on countable Markov shifts, and then we show how this structure forces, in the topologically mixing case, isomorphism to a Bernoulli scheme. 3.1. Equilibrium measures on one-sided countable Markov shifts [7]. Sup- pose G is a countable directed graph. The one-sided countable Markov shift associated to G is N {0} §Å §Å(G ) : {(vi )i 0 V [ : vi vi 1 for all i}. Æ Æ ¸ 2 ! Å Proceeding as in the two-sided case, we equip §Å with the metric d(u,v) : Æ exp[ min{i 0 : u v }]. The cylinder sets ¡ ¸ i 6Æ i (3.1) [a0,...,an 1] : {u §Å : ui ai (i 0,...,n 1)} ¡ Æ 2 Æ Æ ¡ form a basis for the topology of §Å. Notice that unlike the two-sided case (2.1), there is no left subscript; the cylinder starts at the zero coordinate. The left shift map σ: §Å §Å is given by σ :(v ,v ,...) (v ,v ,...). This ! 0 1 7! 1 2 map is not invertible. The natural extension of (§Å,σ) is conjugate to (§,σ). A function Á: §Å R is said to be weakly Hölder-continuous if there are con- ! stants C 0 and θ (0,1) such that var Á Cθn for all n 2, where È 2 n Ç ¸ var Á : sup{Á(u) Á(v): u v (i 0,...,n 1)}. n Æ ¡ i Æ i Æ ¡ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 593–608 596 OMRI M. SARIG The following inequality holds: Ãn 1 ! X¡ j X1 (3.2) varn m Á σ varj Á. Å j 0 ± · j m 1 Æ Æ Å If Á is bounded, weak Hölder-continuity is the same as Hölder-continuity. The equilibrium measures for weakly Hölder potentials were described by Ruelle [21] for finite graphs and by Buzzi and the author for countable graphs [7]. Make the following assumptions: (a) σ: § § is topologically mixing. ! (b) Á is weakly Hölder-continuous and supÁ . (This can be relaxed [7].) Ç 1 (c) P (Á) : sup{h (σ) R Ádm} , where the supremum ranges over all G Æ m Å Ç 1 shift-invariant measures m such that h (σ) R Ádm . The po- m Å 6Æ 1 ¡ 1 tentials we will study satisfy P (Á) h (f ) max ª .
Load more

Recommended publications
  • The Theory of Filtrations of Subalgebras, Standardness and Independence
    The theory of filtrations of subalgebras, standardness and independence Anatoly M. Vershik∗ 24.01.2017 In memory of my friend Kolya K. (1933{2014) \Independence is the best quality, the best word in all languages." J. Brodsky. From a personal letter. Abstract The survey is devoted to the combinatorial and metric theory of fil- trations, i. e., decreasing sequences of σ-algebras in measure spaces or decreasing sequences of subalgebras of certain algebras. One of the key notions, that of standardness, plays the role of a generalization of the no- tion of the independence of a sequence of random variables. We discuss the possibility of obtaining a classification of filtrations, their invariants, and various links to problems in algebra, dynamics, and combinatorics. Bibliography: 101 titles. Contents 1 Introduction 3 1.1 A simple example and a difficult question . .3 1.2 Three languages of measure theory . .5 1.3 Where do filtrations appear? . .6 1.4 Finite and infinite in classification problems; standardness as a generalization of independence . .9 arXiv:1705.06619v1 [math.DS] 18 May 2017 1.5 A summary of the paper . 12 2 The definition of filtrations in measure spaces 13 2.1 Measure theory: a Lebesgue space, basic facts . 13 2.2 Measurable partitions, filtrations . 14 2.3 Classification of measurable partitions . 16 ∗St. Petersburg Department of Steklov Mathematical Institute; St. Petersburg State Uni- versity Institute for Information Transmission Problems. E-mail: [email protected]. Par- tially supported by the Russian Science Foundation (grant No. 14-11-00581). 1 2.4 Classification of finite filtrations . 17 2.5 Filtrations we consider and how one can define them .
    [Show full text]
  • Arxiv:1408.3427V2 [Math.DS] 29 Oct 2016 Parameterized Curve Γ(T) = Φ (P), 0 ≤ T ≤ ` S.T
    SYMBOLIC DYNAMICS FOR THREE DIMENSIONAL FLOWS WITH POSITIVE TOPOLOGICAL ENTROPY YURI LIMA AND OMRI M. SARIG Abstract. We construct symbolic dynamics on sets of full measure (with respect to an ergodic measure of positive entropy) for C1+" flows on closed smooth three dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a closed C1 surface has at least const ×(ehT =T ) simple closed orbits of period less than T , whenever the topological entropy h is positive { and without further assumptions on the curvature. 1. Introduction The aim of this paper is to develop symbolic dynamics for smooth flows with topological entropy h > 0 on three dimensional closed (compact and boundaryless) Riemannian manifolds. Earlier works treated geodesic flows on hyperbolic surfaces [Ser81, Ser87, KU07], geodesic flows on surfaces with variable negative curvature [Rat69], and uniformly hyperbolic flows in any dimension [Rat73, Bow73]. This work only assumes that h > 0 and that the flow has positive speed (i.e. the vector that generates the flow has no zeroes). This generality allows us to cover several cases of interest that could not be treated before, for example: (1) Geodesic flows with positive entropy in positive curvature: There are many Rie- mannian metrics with positive curvature somewhere (even everywhere) whose geodesic flow has positive topological entropy [Don88, BG89, KW02, CBP02]. (2) Reeb flows with positive entropy: These arise from Hamiltonian flows on sur- faces of constant energy, see [Hut10]. Examples with positive topological en- tropy are given in [MS11]. (This application was suggested to us by G. Forni.) (3) Abstract non-uniformly hyperbolic flows in three dimensions, see [BP07, Pes76].
    [Show full text]
  • The Entropy Function for Non Polynomial Problems and Its Applications for Turing Machines
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 January 2020 doi:10.20944/preprints202001.0360.v1 The entropy function for non polynomial problems and its applications for turing machines. Matheus Santana Harvard Extension School [email protected] Abstract We present a general process for the halting problem, valid regardless of the time and space computational complexity of the decision problem. It can be interpreted as the maximization of entropy for the utility function of a given Shannon-Kolmogorov-Bernoulli process. Applications to non-polynomials prob- lems are given. The new interpretation of information rate proposed in this work is a method that models the solution space boundaries of any decision problem (and non polynomial problems in general) as a communication channel by means of Information Theory. We described a sort method that order objects using the intrinsic information content distribution for the elements of a constrained solution space - modeled as messages transmitted through any communication systems. The limits of the search space are defined by the Kolmogorov-Chaitin complexity of the sequences encoded as Shannon-Bernoulli strings. We conclude with a discussion about the implications for general decision problems in Turing machines. Keywords: Computational Complexity, Information Theory, Machine Learning, Computational Statistics, Kolmogorov-Chaitin Complexity; Kelly criterion 1. Introduction Consider a Shanon-Bernoulli process P defined as a sequence of independent binary random variables X1, X2, X3 . Xn. For each element the value can be Preprint submitted to Journal of Information and Computation January 29, 2020 © 2020 by the author(s). Distributed under a Creative Commons CC BY license.
    [Show full text]
  • A Structural Approach to Solving the 6Th Hilbert Problem Udc 519.21
    Teor Imovr.taMatem.Statist. Theor. Probability and Math. Statist. Vip. 71, 2004 No. 71, 2005, Pages 165–179 S 0094-9000(05)00656-3 Article electronically published on December 30, 2005 A STRUCTURAL APPROACH TO SOLVING THE 6TH HILBERT PROBLEM UDC 519.21 YU. I. PETUNIN AND D. A. KLYUSHIN Abstract. The paper deals with an approach to solving the 6th Hilbert problem based on interpreting the field of random events as a partially ordered set endowed with a natural order of random events obtained by formalization and modification of the frequency definition of probability. It is shown that the field of events forms an atomic generated, complete, and completely distributive Boolean algebra. The probability distribution of the field of events generated by random variables is studied. It is proved that the probability distribution generated by random variables is not a measure but only a finitely additive function of events in the case of continuous random variables (both rational- and real-valued). Introduction In 1900 in his lecture [1] David Hilbert formulated the problem of axiomatization of probability theory. Here is the corresponding quote from his lecture: “The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases.” As is well known [2], there is no generally accepted solution of this problem so far.
    [Show full text]
  • The Ergodic Theorem
    The Ergodic Theorem 1 Introduction Ergodic theory is a branch of mathematics which uses measure theory to study the long term be- haviour of dynamic systems. The central object of consideration is known as a measure-preserving system, a type of dynamic system where the evolution of the system preserves a measure. Definition 1: Let (X; M; µ) be a finite measure space and let T map X to itself. We say that T is a measure-preserving transformation if for every A 2 M, we have µ(A) = µ(T −1(A)). The quadruple (X; M; µ, T ) is called a measure-preserving system (m.p.s.). Measure-preserving systems arise in a variety of contexts, such as probability theory, informa- tion theory, and of course in the study of dynamical systems. However, ergodic theory originated from statistical mechanics. In this setting, T represents the evolution of the system through time. Given a measurable function f : X ! R, the series of values f(x); f(T x); f(T 2x)::: are the values of a physical observable at certain time intervals. Of importance in statistical mechanics is the long-term average of these observables: N−1 1 X f (x) = f(T kx) N N k=0 The Ergodic Theorem (also known as the Pointwise or Birkhoff Ergodic Theorem) is central to the study of averages such as fN in the limit as N ! 1. In this paper we aim to prove the theorem, and then discuss a few of its applications. Before we can state the theorem, we need another definition.
    [Show full text]
  • Spectral Theory
    Chapter 3 Spectral Theory 3.1 The spectral approach to ergodic theory A basic problem in ergodic theory is to determine whether two ppt are measure theoretically isomorphic. This is done by studying invariants: properties, quantities, or objects which are equal for any two isomorphic systems. The idea is that if two ppt have different invariants, then they cannot be isomorphic. Ergodicity and mixing are examples of invariants for measure theoretic isomorphism. An effective method for inventing invariants is to look for a weaker equivalence relation, which is better understood. Any invariant for the weaker equivalence rela- tion is automatically an invariant for measure theoretic isomorphism. The spectral point of view is based on this approach. 2 The idea is to associate to the ppt (X,B, µ,T) the operator UT : L (X,B, µ) 2 2 → L (X,B, µ), Ut f = f T. This is an isometry of L (i.e. UT f 2 = f 2 and ◦ 2 k k k k UT f ,UT g = f ,g ). It is useful here to think of L as a Hilbert space over C. h i h i Definition 3.1. Two ppt (X,B, µ,T), (Y,C ,ν,S) are called spectrally isomorphic, 2 if their associated L –isometries UT and US are unitarily equivalent, namely if there exists a linear operator W : L2(X,B, µ) L2(Y,C ,ν) s.t. → 1. W is invertible; 2. W f ,Wg = f ,g for all f ,g L2(X,B, µ); h i h i ∈ 3. WUT = USW.
    [Show full text]
  • Finitary Codes, a Short Survey
    IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 262–273 c Institute of Mathematical Statistics, 2006 DOI: 10.1214/074921706000000284 Finitary Codes, a short survey Jacek Serafin1 Wroc law University of Technology Abstract: In this note we recall the importance of the notion of a finitary isomorphism in the classification problem of dynamical systems. 1. Introduction The notion of isomorphism of dynamical systems has been an object of an exten- sive research from the very beginning of the modern study of dynamics. Various invariants of isomorphism have been introduced, like ergodicity, mixing of different kinds, or entropy. Still, the task of deciding whether two dynamical systems are isomorphic remains a notoriously difficult one. Independent processes, referred here to as Bernoulli schemes (defined below), are most easily described examples of dynamical systems. However, even in this simplest situation, the isomorphism problem remained a mystery from around 1930, when it was stated, until late 1950’s, when Kolmogorov implemented Shannon’s ideas and introduced mean entropy into ergodic theory; later Sinai was able to compute entropy for Bernoulli schemes and show that Bernoulli schemes of different entropy were not isomorphic. In his ground-breaking work Ornstein (see [20]) produced metric isomorphism between Bernoulli schemes of the same entropy, and later he and his co-workers discovered a number of conditions which turned out to be very useful in studying the isomorphism problem. Among those conditions are notions of Weak Bernoulli, Very Weak Bernoulli and Finitely Determined. As those will not be of our interest in this note, we shall not go into the details, and the interested reader should for example consult [20] or [24].
    [Show full text]
  • Markov Chain 1 Markov Chain
    Markov chain 1 Markov chain A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property. Markov chains have many applications as statistical models of real-world processes. A simple two-state Markov chain Introduction Formally, a Markov chain is a random process with the Markov property. Often, the term "Markov chain" is used to mean a Markov process which has a discrete (finite or countable) state-space. Usually a Markov chain is defined for a discrete set of times (i.e., a discrete-time Markov chain)[1] although some authors use the same terminology where "time" can take continuous values.[2][3] The use of the term in Markov chain Monte Carlo methodology covers cases where the process is in discrete time (discrete algorithm steps) with a continuous state space. The following concentrates on the discrete-time discrete-state-space case. A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. The steps are often thought of as moments in time, but they can equally well refer to physical distance or any other discrete measurement; formally, the steps are the integers or natural numbers, and the random process is a mapping of these to states.
    [Show full text]
  • Isomorphisms of Β-Automorphisms to Markov Automorphisms
    Title Isomorphisms of β-automorphisms to Markov automorphisms Author(s) Takahashi, Yōichirō Citation Osaka Journal of Mathematics. 10(1) P.175-P.184 Issue Date 1973 Text Version publisher URL https://doi.org/10.18910/6877 DOI 10.18910/6877 rights Note Osaka University Knowledge Archive : OUKA https://ir.library.osaka-u.ac.jp/ Osaka University Takahashi, Y. Osaka J. Math. 10 (1973), 175-184 ISOMORPHISMS OF , -AUTOMORPHISMS TO MARKOV AUTOMORPHISMS YδiCHiRό TAKAHASHI (Received February 9, 1972) (Revised July 21, 1972) 0. Introduction The purpose of the present paper is to construct an isomorphism which shows the following: Theorem. A β-automorphίsm is isomorphίc to a mixing simple Markov automorphism in such a way that their futures are mutually isomorphίc. Though the state of this Markov automorphism is countable and not finite, we obtain immediately from the proof of the theorem: Corollary 1. The invariant probability measure of β-transformatίon is unique under the condition that its metrical entropy coincides with topological entropy log β. An extention of Ornstein's isomorphism theorem for countable generating partitions ([2]) shows the following known result (Smorodinsky [5], Ito- Takahashi [3]): Corollary 2. A β-automorphism is Bernoulli. We now give the definition of /3-automorphism and auxiliary notions. Let β be a real number >1. DEFINITION. A β-transformation is a transformation Tβ of the unit interval [0, 1] into itself defined by the relation (1) Tβt = βt (modi) (0^f<l) n n and by T βl=\im T βt. /->! This transformation has been studied by A. Renyi, W. Parry, Ito-Takahashi et al.
    [Show full text]
  • Combinatorial Encoding of Bernoulli Schemes and the Asymptotic
    Combinatorial encoding of Bernoulli schemes and the asymptotic behavior of Young tableaux A. M. Vershik∗ Abstract We consider two examples of a fully decodable combinatorial encoding of Bernoulli schemes: the encoding via Weyl simplices and the much more complicated encoding via the RSK (Robinson–Schensted–Knuth) corre- spondence. In the first case, the decodability is a quite simple fact, while in the second case, this is a nontrivial result obtained by D. Romik and P. Sniady´ and based on the papers [2], [12], and others. We comment on the proofs from the viewpoint of the theory of measurable partitions; another proof, using representation theory and generalized Schur–Weyl duality, will be presented elsewhere. We also study a new dynamics of Bernoulli variables on P -tableaux and find the limit 3D-shape of these tableaux. 1 Introduction This paper deals with common statistical and asymptotic properties of the clas- sical Bernoulli scheme (i.e., a sequence of i.i.d. variables) and a generalized RSK (Robinson–Schensted–Knuth) algorithm. Simultaneously, we study the so-called combinatorial encoding of Bernoulli schemes. The combinatorial method of en- coding continuous signals has certain advantages compared with the traditional invariant continual encoding; instead of functions with values in a continual space, it uses simple combinatorial relations between the coordinates of the message to be encoded. The main problem is to establish whether decoding is arXiv:2006.07923v1 [math.CO] 14 Jun 2020 possible and how difficult it is. The first example, originally considered in [1], is the encoding via Weyl simplices. It is typical, albeit quite simple; here we explain it in detail in terms of sequences of σ-algebras, or measurable partitions.
    [Show full text]
  • 34 Lecture 34 KS Entropy As Isomorphism Invari- Ant
    34. LECTURE 34 KS ENTROPY AS ISOMORPHISM INVARIANT 355 34 Lecture 34 KS entropy as isomorphism invari- ant 34.1 Kolmogorov-Sinai entropy as isomorphism invariant The Kolmogorov-Sinai entropy was originally proposed as an invariant under iso- morphism of measure theoretical dynamical systems. Suppose there are two measure theoretical dynamical systems (T; 휇, Γ) and (T 0; 휇0; Γ0). Let 휑 :Γ ! Γ0 be a one-to- one (except for measure zero sets) correspondent and measure-preserving: 휇 = 휇0 ∘ 휑 (that is, the measure of any measurable set measured by the measure 휇 on Γ and measured by the measure 휇0 on Γ0 after mapping from Γ by 휑 are identical).379 If the following diagram: T ?Γ −! ?Γ ?휑 ?휑 y 0 y Γ0 −!T Γ0 is commutative (except on measure zero sets of Γ and Γ0), that is, if T = 휑−1 ∘ T 0 ∘ 휑, these two measure-theoretical dynamical systems are said to be isomorphic. It is almost obvious that the Kolmogorov-Sinai entropies of isomorphic measure- theoretical dynamical systems are identical, because for a partition A of Γ H(A) = H(휑A) and 휑 ∘ T = T 0 ∘ 휑. We say that the Kolmogorov-Sinai entropy is an isomor- phism invariant. If two dynamical systems have different Kolmogorov-Sinai entropies, then they cannot be isomorphic. For example, as can be seen from (34.2) B(1=2; 1=2) and B(1=3; 2=3) cannot be isomorphic. An isomorphism invariant that takes the same value if and only if dynamical systems are isomorphic is called a complete isomorphism invariant.
    [Show full text]
  • The Asymptotics of the Partition of the Cube Into Weyl Simplices, and an Encoding of a Bernoulli Scheme∗
    The asymptotics of the partition of the cube into Weyl simplices, and an encoding of a Bernoulli scheme∗ A. M. Vershiky 02.02.2019 Abstract We suggest a combinatorial method of encoding continuous symbolic dynamical systems. A con- tinuous phase space, the infinite-dimensional cube, turns into the path space of a tree, and the shift is mapped to a transformation which was called a \transfer." The central problem is that of dis- tinguishability: does the encoding separate almost all points of the space? The main result says that the partition of the cube into Weyl simplices satisfies this property.1 Contents 1 Introduction 2 2 Classical and combinatorial encoding of transformations 4 2.1 Classical encoding and generators . 4 2.2 Combinatorial encoding . 4 2.3 The frame as a combinatorial invariant of an encoding . 5 2.4 Distinguishability problem . 6 2.5 Entropy estimates . 8 3 The main example: encoding a Bernoulli scheme by Weyl simplices and the trian- gular compactum M 9 3.1 The partition of the cube into Weyl simplices . 9 3.2 The triangular compactum of paths in the tree of i-permutations . 11 arXiv:1904.02924v1 [math.CO] 5 Apr 2019 3.3 An isomorphism between the cube I1 and the triangular compactum. The positive answer to the distinguishability problem . 12 3.4 Transfer for the tree of i-permutations and the triangular compactum . 14 ∗Partially supported by the RFBR grant 17-01-00433. ySt. Petersburg Department of Steklov Institute of Mathematics, St. Petersburg State University, and Institute for Information Transmission Problems. 1Keywords: combinatorial encoding, transfer, Bernoulli scheme, graded graph.
    [Show full text]