WWW.DISSERTATION.XLIBX.INFO
FREE ELECTRONIC LIBRARY - Dissertations, online materials
 
<< HOME
CONTACTS



«Zentrum Mathematik, Technische Universit¨t M¨nchen, a u John-von-Neumann Lecture, 2013 Sergiy Kolyada Topological Dynamics: Minimality, Entropy and ...»

Topological Dynamics:

Minimality, Entropy and Chaos.

Sergiy Kolyada

Institute of Mathematics, NAS of Ukraine, Kyiv

Zentrum Mathematik, Technische Universit¨t M¨nchen,

a u

John-von-Neumann Lecture, 2013

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps

6. Li-Yorke sensitivity and weakly mixing maps

Throughout this lecture a dynamical system (X, T ) is a pair where X is a nonvoid compact metric space with metric ρ and T : X → X is a surjective, continuous map.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps

6. Li-Yorke sensitivity and weakly mixing maps Throughout this lecture a dynamical system (X, T ) is a pair where X is a nonvoid compact metric space with metric ρ and T : X → X is a surjective, continuous map. The assumption of compactness of X is vital.

The assumption of surjectivity of T is convenient and can always be obtained, given compactness, by replacing X by its largest T invariant subset if necessary.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps The different notions of chaos at least share a common motivating intuition of unpredictability due to the divergence of nearby orbits.

Different definitions begin with different interpretations of this divergence. We consider two popular ideas and examine a concept which bridges them.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps The different notions of chaos at least share a common motivating intuition of unpredictability due to the divergence of nearby orbits.

Different definitions begin with different interpretations of this divergence. We consider two popular ideas and examine a concept which bridges them.

The idea of sensitivity was introduced in Auslander and Yorke (1980) and popularized in Devaney (1989).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps The different notions of chaos at least share a common motivating intuition of unpredictability due to the divergence of nearby orbits.

Different definitions begin with different interpretations of this divergence. We consider two popular ideas and examine a concept which bridges them.

The idea of sensitivity was introduced in Auslander and Yorke (1980) and popularized in Devaney (1989). For a positive ε we consider pairs of points (x, y ) whose orbits are frequently at least ε apart. That is,

–  –  –

The different notions of chaos at least share a common motivating intuition of unpredictability due to the divergence of nearby orbits.

Different definitions begin with different interpretations of this divergence. We consider two popular ideas and examine a concept which bridges them.

The idea of sensitivity was introduced in Auslander and Yorke (1980) and popularized in Devaney (1989). For a positive ε we consider pairs of points (x, y ) whose orbits are frequently at least ε apart. That is,

–  –  –

Recall that in studying maps of the interval, Li and Yorke (1975) suggested that the “divergent pairs” to consider are the pairs (x, y ) which are proximal but not asymptotic.

–  –  –

Recall that in studying maps of the interval, Li and Yorke (1975) suggested that the “divergent pairs” to consider are the pairs (x, y ) which are proximal but not asymptotic. That is,

–  –  –

Recall that in studying maps of the interval, Li and Yorke (1975) suggested that the “divergent pairs” to consider are the pairs (x, y ) which are proximal but not asymptotic. That is,

–  –  –

Li and Yorke call a system chaotic when it contains an uncountable scrambled set. A subset A ⊂ X is scrambled when any pair of distinct points in A satisfy this Li–Yorke condition.

–  –  –

Motivated by these results, we define the concept which links sensitivity with the Li–Yorke versions of chaos.

Definitions A dynamical system (X, T ) is called Li–Yorke sensitive if there exists a positive ε such that every x ∈ X is a limit of points in ProxT (x) \ AsymT,ε (x), i.e. x ∈ ProxT (x) \ AsymT,ε (x).

–  –  –

The proof of the following theorem is based on arguments in a paper by Huang and Ye (2001). We will refer to these as the Huang-Ye Equivalnces.

Theorem 6.1 For a dynamical system (X, T ) the following conditions are equivalent.

(1) The system is sensitive.

(2) There exists a positive ε such that Asymε (T ) is a first category subset of X × X.

(3) There exists a positive ε such that for every x ∈ X AsymT,ε (x) is a first category subset of X.

(4) There exists a positive ε such that every x ∈ X is a limit point of the complement of AsymT,ε (x), i.e. x ∈ X \ AsymT,ε (x).

(5) There exists a positive ε such that the set of pairs

–  –  –





This condition is strictly stronger than sensitivity. For example, any minimal system which is distal but not equicontinuous is sensitive but not Li–Yorke sensitive. On the other hand, we prove that weak mixing systems are Li–Yorke sensitive.

–  –  –

This condition is strictly stronger than sensitivity. For example, any minimal system which is distal but not equicontinuous is sensitive but not Li–Yorke sensitive. On the other hand, we prove that weak mixing systems are Li–Yorke sensitive.

Theorem 6.2 Let (X, T ) be a dynamical system.

If (X, T ) is Li–Yorke sensitive then it is sensitive.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps This condition is strictly stronger than sensitivity. For example, any minimal system which is distal but not equicontinuous is sensitive but not Li–Yorke sensitive. On the other hand, we prove that weak mixing systems are Li–Yorke sensitive.

Theorem 6.2 Let (X, T ) be a dynamical system.

If (X, T ) is Li–Yorke sensitive then it is sensitive. If (X, T ) is sensitive and for every x ∈ X the proximal cell ProxT (x) is dense in X then (X, T ) is Li–Yorke sensitive.

–  –  –

Beweis.

By the Huang-Ye Equivalences, Li–Yorke sensitivity implies sensitivity. On the other hand, Prox(T ) is a Gδ subset of X × X and so each ProxT (x) is a Gδ subset of X.

–  –  –

Beweis.

By the Huang-Ye Equivalences, Li–Yorke sensitivity implies sensitivity. On the other hand, Prox(T ) is a Gδ subset of X × X and so each ProxT (x) is a Gδ subset of X. Hence, if ProxT (x) is dense and AsymT,ε (x) is of first category, then ProxT (x) \ AsymT,ε (x) is dense as well by the Baire Category Theorem.

–  –  –

The weakly mixing systems are a good class of test systems for any topological definition of chaos. The system (X, T ) is called weakly mixing when the product system (X × X, T × T ) is transitive.

–  –  –

The weakly mixing systems are a good class of test systems for any topological definition of chaos. The system (X, T ) is called weakly mixing when the product system (X × X, T × T ) is transitive.

The Furstenberg Intersection Lemma implies that for weakly mixing systems the collection of subsets of Z+ : {n(U, V ) ∩ [k, ∞) : U, V opene in X and k ≥ 0} generates a filter which we will denote F).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps The weakly mixing systems are a good class of test systems for any topological definition of chaos. The system (X, T ) is called weakly mixing when the product system (X × X, T × T ) is transitive.

The Furstenberg Intersection Lemma implies that for weakly mixing systems the collection of subsets of Z+ : {n(U, V ) ∩ [k, ∞) : U, V opene in X and k ≥ 0} generates a filter which we will denote F). The dual family kF = {A : A ∩ B = ∅ for all B ∈ F} satisfies what Furstenberg calls the Ramsey Property: if a finite union of subsets of Z+ lies in kF then one of the subsets is in kF (this condition is just dual to the filter property for F).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps The weakly mixing systems are a good class of test systems for any topological definition of chaos. The system (X, T ) is called weakly mixing when the product system (X × X, T × T ) is transitive.

The Furstenberg Intersection Lemma implies that for weakly mixing systems the collection of subsets of Z+ : {n(U, V ) ∩ [k, ∞) : U, V opene in X and k ≥ 0} generates a filter which we will denote F). The dual family kF = {A : A ∩ B = ∅ for all B ∈ F} satisfies what Furstenberg calls the Ramsey Property: if a finite union of subsets of Z+ lies in kF then one of the subsets is in kF (this condition is just dual to the filter property for F). Notice that if A ∈ kF and B ∈ F then A ∩ B is infinite because B ∩ [k, ∞) ∈ F for all k 0.

–  –  –

Beweis.

Given a point x ∈ X and an opene set U in X it suffices to find a point y ∈ U which is proximal to x.

For k = 1, 2,... let Gk be a cover by opene subsets of diameter less than 1/k.

–  –  –

Beweis.

Given a point x ∈ X and an opene set U in X it suffices to find a point y ∈ U which is proximal to x.

For k = 1, 2,... let Gk be a cover by opene subsets of diameter less than 1/k. Observe that the union of the hitting time sets n(x, G ) as G varies over Gk is all of Z+.

–  –  –

Beweis.

Given a point x ∈ X and an opene set U in X it suffices to find a point y ∈ U which is proximal to x.

For k = 1, 2,... let Gk be a cover by opene subsets of diameter less than 1/k. Observe that the union of the hitting time sets n(x, G ) as G varies over Gk is all of Z+. So by the Ramsey Property, there exists Gk ∈ Gk an opene set of diameter less than 1/k such that n(x, Gk ) ∈ kF.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps Beweis.

Given a point x ∈ X and an opene set U in X it suffices to find a point y ∈ U which is proximal to x.

For k = 1, 2,... let Gk be a cover by opene subsets of diameter less than 1/k. Observe that the union of the hitting time sets n(x, G ) as G varies over Gk is all of Z+. So by the Ramsey Property, there exists Gk ∈ Gk an opene set of diameter less than 1/k such that n(x, Gk ) ∈ kF.

Now let U0 = U and define inductively opene sets U1, U2,... and positive integers nk as follows.

–  –  –

Beweis (Cont.) Because n(Uk−1, Gk ) ∈ F, the intersection n(Uk−1, Gk ) ∩ n(x, Gk ) is infinite and so we can choose Uk an opene subset of Uk−1 and an integer nk k such that T nk (x) ∈ Gk and T nk (Uk ) ⊂ Gk.

–  –  –

Beweis (Cont.) Because n(Uk−1, Gk ) ∈ F, the intersection n(Uk−1, Gk ) ∩ n(x, Gk ) is infinite and so we can choose Uk an opene subset of Uk−1 and an integer nk k such that T nk (x) ∈ Gk and T nk (Uk ) ⊂ Gk. If y is a point of the nonempty intersection ∩k Uk then T nk (x), T nk (y ) ∈ Gk and so ρ(T nk (x), T nk (y )) 1/k. Thus, y is a point of U which is proximal to x.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.

Li-Yorke sensitivity and weakly mixing maps Beweis (Cont.) Because n(Uk−1, Gk ) ∈ F, the intersection n(Uk−1, Gk ) ∩ n(x, Gk ) is infinite and so we can choose Uk an opene subset of Uk−1 and an integer nk k such that T nk (x) ∈ Gk and T nk (Uk ) ⊂ Gk. If y is a point of the nonempty intersection ∩k Uk then T nk (x), T nk (y ) ∈ Gk and so ρ(T nk (x), T nk (y )) 1/k. Thus, y is a point of U which is proximal to x.

Corollary 6.1 If a nontrivial system (X, T ) is weakly mixing then it is Li–Yorke sensitive.

–  –  –

Beweis (Cont.) Because n(Uk−1, Gk ) ∈ F, the intersection n(Uk−1, Gk ) ∩ n(x, Gk ) is infinite and so we can choose Uk an opene subset of Uk−1 and an integer nk k such that T nk (x) ∈ Gk and T nk (Uk ) ⊂ Gk. If y is a point of the nonempty intersection ∩k Uk then T nk (x), T nk (y ) ∈ Gk and so ρ(T nk (x), T nk (y )) 1/k. Thus, y is a point of U which is proximal to x.

Corollary 6.1 If a nontrivial system (X, T ) is weakly mixing then it is Li–Yorke sensitive.

The following problem is still open – Are all Li–Yorke sensitive systems Li–Yorke chaotic?

–  –  –

6. Li-Yorke sensitivity and weakly mixing maps General references

1. F. Blanchard, E. Glasner, S. Kolyada, and A. Maass, On LiYorke pairs, J. Reine Angew. Math., 547, 51–68 (2002).

2. Wen Huang and Xiangdong Ye, Devaney’s chaos or 2–scattering implies Li–Yorke chaos, Topology Appl., 117, 259–272.

3. E. Akin and S. Kolyada, Li-Yorke sensitivity, Nonlinearity, 16 (2003), 1421–1433.

4. E. Akin, Recurrence in Topological Dynamics: Furstenberg Families and Ellis Actions, Plenum, New York, 1997.

–  –  –

6. Li-Yorke sensitivity and weakly mixing maps HOMEWORK The system is called proximal when every pair is proximal.

Exercise 6.1 To prove that a proximal dynamical system is Li–Yorke sensitive iff it is sensitive.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.



Similar works:

«Pour citer cet article : LE BRETON, D. « Concepts et significations majeures des conduites à risque », Journal des socio-anthropologues de l'adolescence et de la jeunesse, Revue en-ligne. Date de publication : janvier 2012. [http://anthropoado.com/le-journal-des-socio-anthropologues-de-l-adolescence-et-de-lajeunesse-textes-en-ligne/] Concepts et significations majeures des conduites à risque Par David Le Breton Significations des conduites à risque Paradoxalement les conduites à risque...»

«14. CONGRESS OF THE INTERNATIONAL SOCIETY FOR PHOTOGRAMMETRY HAMBURG 1980 COMMISSION IV, WORKING GROUP 1 PRESENTED PAPER HEINRICH EBNER, BERNHARD HOFMANN-WELLENHOF, PETER REISS, FRANZ STEIDLER LEHRSTUHL FOR PHOTOGRAMMETRIE TECHNISCHE UNIVERSIT~T MONCHEN ARCISSTR. 21 POSTFACH 202420 08000 MONCHEN 2 HIFI A ~1INICOMPUTER PROGRAM PACKAGE FOR HEIGHT INTERPOLATION BY FINITE ELEMENTS *) ABSTRACT A general minicomputer program package for height interpolation is presented. It is written in FORTRAN...»

«1 PRINCIPLES & TECHNIQUES USED TO CREATE A NATURAL HAIRLINE IN SURGICAL HAIR RESTORATION Ronald Shapiro Md, Facial Plastic Surgery Clinics of North America, 2004, Volume 12, Number 2 :201-218 INTRODUCTION Creating a natural hairline has always been one of the most important elements of a successful hair transplant. Our ability to create a natural hairline has dramatically increased over the years. Many of us promise full and undetectable hairlines in our promotional materials. Not unexpectedly...»

«EXPERIMENTAL AND THEORETICAL ASSESSMENT OF THIN GLASS PANELS AS INTERPOSERS FOR MICROELECTRONIC PACKAGES A Thesis Presented to The Academic Faculty by Scott R. McCann In Partial Fulfillment of the Requirements for the Degree Master's of Science in the School of Mechanical Engineering Georgia Institute of Technology May 2014 COPYRIGHT 2014 BY SCOTT MCCANN EXPERIMENTAL AND THEORETICAL ASSESSMENT OF THIN GLASS PANELS AS INTERPOSERS FOR MICROELECTRONIC PACKAGES Approved by: Dr. Suresh K. Sitaraman,...»

«ECNDT 2006 Tu.2.1.1 In-Service Inspection Concept for GLARE® – An Example for the Use of New UT Array Inspection Systems Wolfgang BISLE, Theodor MEIER, Sascha MUELLER, Sylvia RUECKERT Airbus, Bremen Abstract. UT phased arrays and array transducers for FIT (Field Inspection Technology) open new chances for NDT of complex structure materials in aeronautics. GLARE® is such a new material which will widely be used on the new Airbus A380. Even as GLARE® in the A380 needs no scheduled NDT...»

«Defect detection in textured surfaces using color ring-projection correlation D. M. Tsai and Y. H. Tsai Machine Vision Lab. Department of Industrial Engineering and Management Yuan-Ze University, Chung-Li, Taiwan, R.O.C. E-mail: iedmtsai@saturn.yzu.edu.tw 1. INTRODUCTION Image analysis techniques are being increasingly used to automate industrial inspection. For defect inspection in complicated material surfaces, color and texture are two of the most important properties. Detecting an entire...»

«PUTTING WOODWORKING TOOLS OF THE MIDDLE BRONZE AGE TO THE TEST Building a gateway at the Terramare settlement in Montale, Italy by Wolfgang F. A. Lobisser, Wien Introduction The VIAS – Vienna Institute for Archaeological Science – is an interdisciplinary research unit of the University of Vienna. For the past 20 years employees of the experimental archaeological working group of the VIAS have been dealing with the evaluation of archaeological findings, implementation of prehistoric...»

«Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2015 The human brain response to dental pain relief Meier, M L; Widmayer, S; Abazi, J; Brügger, M; Lukic, N; Lüchinger, R; Ettlin, D A Abstract: Local anesthesia has made dental treatment more comfortable since 1884, but little is known about associated brain mechanisms. Functional magnetic resonance imaging is a modern neuroimaging tool widely used for investigating...»

«1 Michael N Shadlen CURRICULUM VITAE FIELD OF SPECIALIZATION: I am a neuroscientist and a neurologist with an abiding interest in fundamental, basic, neuroscience of vision and cognition. I have been studying the neurobiology of decision making for over 20 years, and I have made fundamental contributions to this field by combining behavior, electrophysiology and computational methods. My research serves to elucidate the neural mechanisms that support normal cognitive operations that underlie...»

«Behaviour 152 (2015) 335–357 brill.com/beh Non-reciprocal but peaceful fruit sharing in wild bonobos in Wamba Shinya Yamamoto a,b,∗ a Graduate School of Intercultural Studies, Kobe University, 1-2-1 Tsurukabuto, Nada-ku, 657-8501 Kobe, Japan b Wildlife Research Center, Kyoto University, Yoshida-honmachi, Sakyo-ku, 606-8501 Kyoto, Japan * Author’s e-mail address: shinyayamamoto1981@gmail.com Accepted 30 December 2014; published online 29 January 2015 Abstract Food sharing is considered to...»

«About the Authors Christina Amcoff Nyström is presently an Assistant Professor in Informatics at Mid Sweden University, Sweden. She is sharing her professional interests between tutoring master courses and researching technical communication, including communication in organizations and virtual collaboration among project members. Hans Andersin was born in 1930. In 1954-1955 he took part in building the first electronic computer in Finland. In 1956 he joined IBM and worked in various sales,...»

«TWO-PHASE FLOW AND HEAT TRANSFER IN PIN-FIN ENHANCED MICRO-GAPS A Thesis Presented to The Academic Faculty by Steven A. Isaacs In Partial Fulfillment of the Requirements for the Degree Master of Science in the School of Mechanical Engineering Georgia Institute of Technology December 2013 COPYRIGHT 2013 BY STEVEN ISAACS TWO-PHASE FLOW AND HEAT TRANSFER IN PIN-FIN ENHANCED MICRO-GAPS Approved by: Dr. Yogendra Joshi, Advisor School of Mechanical Engineering Georgia Institute of Technology Dr....»





 
<<  HOME   |    CONTACTS
2016 www.dissertation.xlibx.info - Dissertations, online materials

Materials of this site are available for review, all rights belong to their respective owners.
If you do not agree with the fact that your material is placed on this site, please, email us, we will within 1-2 business days delete him.