DOCSLIB.ORG

A Blender? Ch

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Blender? Ch THE GRADUATE STUDENT SECTION WHAT IS… a Blender? Ch. Bonatti, S. Crovisier, L. J. Díaz, A. Wilkinson Communicated by Cesar E. Silva A blender is a compact, invariant set on which a diffeo- morphism has a certain behavior. This behavior forces topologically “thin” sets to intersect in a robust way, pro- ducing rich dynamics. The term “blender” describes its function: to blend together stable and unstable manifolds. Blenders have been used to construct diffeomorphisms with surprising properties and have played an important role in the classification of smooth dynamical systems. One of the original applications of blenders is also one of the more striking. A diffeomorphism 푔 of a compact manifold is robustly transitive if there exists a point 푥 whose orbit {푔푛(푥) ∶ 푛 ≥ 0} is dense in the manifold, and moreover this property persists when 푔 is slightly perturbed. Until the 1990s there were no known robustly transitive diffeomorphisms in the isotopy class of the identity map on any manifold. Bonatti and Díaz (Ann. of Math., 1996)1 used blenders to construct robustly transitive diffeomorphisms as perturbations of the identity map on certain 3-manifolds. To construct a blender one typically starts with a proto- blender; an example is the map 푓 pictured in Figure 1. The function 푓 maps each of the two rectangles, 푅1 and 푅2, affinely onto the square 푆 and has the property that the vertical projections of 푅1 and 푅2 onto the horizontal Figure 1. An example of a proto-blender. The map 푓 is direction overlap. Each rectangle contains a unique fixed defined on the union of the two rectangles, 푅1 and 푅2, point for 푓. in the square 푆; 푓 sends each 푅푖 onto the entire square 푆 affinely, respecting the horizontal and Ch. Bonatti is CNRS researcher in mathematics at the Université vertical directions, with the horizontal expansion de Bourgogne, Dijon, France. His email address is bonatti@u- factor less than 2. Note that 푓 fixes a unique point in bourgogne.fr. each rectangle 푅푖. S. Crovisier is CNRS researcher in mathematics at the Université Paris-Sud. His email address is [email protected] psud.fr. −푛 L. J. Díaz is professor of mathematics at the Pontifícia Universi- The compact set Ω = ⋂푛≥0 푓 푆 is 푓-invariant, meaning dade Católica in Rio de Janeiro. His email address is lodiaz@mat. 푓(Ω) = Ω, and is characterized as the set of points in 푆 on puc-rio.br. which 푓 can be iterated infinitely many times: 푥 ∈ Ω if and A. Wilkinson is professor of mathematics at the University of only if 푓푛(푥) ∈ 푆 for all 푛 ≥ 0. Ω is a Cantor set, obtained Chicago. Her email address is [email protected]. by intersecting all preimages 푓−푖(푆) of the square, which 1This is also where the term “blender” was coined. nest in a regular pattern, as in Figure 2. For permission to reprint this article, please contact: Any vertical line ℓ between the fixed points in 푅1 and in [email protected]. 푅2 will meet Ω. To prove this, it is enough to see that for DOI: http://dx.doi.org/10.1090/noti1438 every 푖 the vertical projection of the set 푓−푖(푆) (consisting November 2016 Notices of the AMS 1175 THE GRADUATE STUDENT SECTION metric point of view, the fractal set Ω, when viewed along nearly vertical directions, appears to be 1-dimensional, allowing Ω to intersect a vertical line robustly. If the rectangles 푅1 and 푅2 had disjoint projections, the proto- blender property would be destroyed. This type of picture is embedded in a variety of smooth dynamical systems, where it is a robust mechanism for chaos. The search for robust mechanisms for chaotic dy- namics has a long history, tracing back to Henri Poincaré’s discovery of chaotic motion in the three-body problem of celestial mechanics. Figure 3(a) depicts the mechanism behind Poincaré’s discovery, a local diffeomorphism of the plane with a saddle fixed point 푝 and another point 푥 whose orbit converges to 푝 both under forward and backward iterations (that is, under both the map and Figure 2. The invariant Cantor set Ω produced by the its inverse). Meeting at 푝 are two smooth curves 푊푠(푝) proto-blender 푓 is the nested intersection of and 푊푢(푝), the stable and unstable manifolds at 푝, respec- preimages of 푆 under 푓. Any vertical line segment ℓ tively. 푊푠(푝) is the set of points whose forward orbit close to the center of the square intersects Ω in at converges to 푝, and 푊푢(푝) is the set of points whose least one point. The line segment can be replaced by backward orbit converges to 푝. a segment with nearly vertical slope or even a In Figure 3(a), the intersection of 푊푠(푝) and 푊푢(푝) is smooth curve nearly tangent to the vertical direction. transverse at 푥: the tangent directions to 푊푠(푝) and 푊푢(푝) at 푥 span the set of all directions emanating from 푥—the tangent space at 푥 to the ambient manifold, in this case of 2푖 horizontal rectangles) onto the horizontal is an the plane. The point 푥 is called a transverse homoclinic interval. This can be checked inductively, observing that point for 푝. In Figure 3(b) a slight variation is depicted: the projection of 푓−푖−1(푆) is the union of two rescaled here there are two periodic saddles, 푝 and 푞, such that copies of the projection of 푓−푖(푆), which overlap. 푊푠(푝) and 푊푢(푞) intersect transversely at a point 푥, and A more careful inspection of this proof reveals that 푊푢(푝) and 푊푠(푞) intersect transversely at another point the intersection is robust in two senses: First, the line ℓ 푦. The points 푥 and 푦 are called transverse heteroclinic can be replaced by a line whose slope is close to vertical points, and they are arranged in a transverse heteroclinic or even by a 퐶1 curve whose tangent vectors are close cycle. to vertical; second, the map 푓 can be replaced by any 퐶1 In the classification of the so-called Axiom A diffeo- map 푓̂ whose derivative is close to that of 푓. Such an 푓̂ is morphisms, carried out by Stephen Smale in the 1960s, called a perturbation of 푓. transverse homoclinic and heteroclinic points play a cen- The (topological) dimension of the Cantor set Ω is 0, tral role. Any transverse homoclinic point or heteroclinic the dimension of ℓ is 1, the dimension of the square is 2, cycle for a diffeomorphism is contained in a special Can- and 0+1 < 2. From a topological point of view, one would tor set Λ called a horseshoe, an invariant compact set with not expect these sets to intersect each other. But from a strongly chaotic (or unpredictable) dynamical properties (see [2] for a discussion). Two notable properties of a horseshoe Λ are: (1) Every point in Λ can be approximated arbitrarily well by a periodic point in Λ. (2) There is a point in Λ whose orbit is dense in Λ. Horseshoes and periodic saddles are both examples of hyperbolic sets: a compact invariant set Λ for a diffeo- morphism 푔 is hyperbolic if at every point in Λ there are transverse stable and unstable manifolds 푊푠(푥) and 푊푢(푥) with 푔(푊푠(푥)) = 푊푠(푔(푥)) and 푔(푊푢(푥)) = 푊푢(푔(푥)). For a large class of diffeomorphisms known as Axiom A sys- Figure 3. (a) A transverse homoclinic intersection of tems, Smale proved that the set of recurrent points can stable and unstable manifolds, first discovered by be decomposed into a disjoint union of finitely many Poincaré in his study of the 3-body problem. (b) A hyperbolic sets on which (1) and (2) above hold. This horseshoe Λ produced by a pair of transverse hetero- theory relies on the most basic property of transverse in- clinic points 푥 and 푦. Every point in the Cantor set Λ tersections, first investigated by René Thom: robustness. can be approximated arbitrarily well both by a peri- A transverse intersection of submanifolds cannot be de- odic point and by a point whose orbit is dense in Λ. stroyed by a small perturbation of the manifolds; in the dynamical setting, a transverse intersection of stable and 1176 Notices of the AMS Volume 63, Number 10 THE GRADUATE STUDENT SECTION the cube 푄 is stretched and folded across itself by a local diffeomorphism 푔. The horseshoe Λ in Figure 5 is precisely the set of points whose orbits remain in the future and in the past in 푄. The set 푊푢(Λ) of points in the cube that accumulate on Λ in the past is the cartesian product of the Cantor set Ω with segments parallel to the third, expanded direction. 푊푢(Λ) is the analogue of the unstable manifold of a saddle, but it is a fractal object rather than a smooth submanifold. The set Λ is an example of a blender, and its main geometric property is that any vertical curve crossing 푄 Figure 4. (a) Transverse cycle. (b) Nontransverse close enough to the center intersects 푊푢(Λ). In other cycle. words, this blender is a horseshoe whose unstable set behaves like a surface even though its topological dimen- sion is one. This property is robust. While the definition unstable manifolds of two saddles cannot be destroyed of blender is still evolving as new constructions arise, a by perturbing the diffeomorphism. working definition is: A blender is a compact hyperbolic Classifying Axiom A systems was just the beginning.
Load more

Recommended publications
  • Amie Wilkinson 1
    Amie Wilkinson 1 Amie Wilkinson Department of Mathematics, University of Chicago 5734 S. University Avenue Chicago, Illinois 60637 (773) 702-7337 [email protected] ACADEMIC POSITIONS • Professor of Mathematics, University of Chicago, 2012 { present. • Professor of Mathematics, Northwestern University, 2005 { 2012. • Associate Professor of Mathematics, Northwestern University, 2002 { 2005. • Assistant Professor of Mathematics, Northwestern University, 1999 { 2002. • Boas Assistant Professor of Mathematics, Northwestern University, 1996 { 1999. • Benjamin Peirce Instructor, Harvard University, 1995 { 1996. Visiting Positions • Visiting Professor, University of Chicago, Fall and Spring Quarters 2003{2004, Fall Quarter 2011. • Professeur Invit´e, Universit´ede Bourgogne, May 2002 and Sept 2003. • Member, Institut des Hautes Etudes Scientifiques (IHES), Summer 1993, 1996, 1998. • Visitor, IBM T.J. Watson Labs, Yorktown NY, Winter 1992 and 1994, Summer 1997, 1998, 2000, and 2001. • Graduate Research Assistant, The Center for Nonlinear Studies, Los Alamos National Laboratories, Summer, 1992. EDUCATION • Ph.D. in Mathematics, University of California, Berkeley, May 1995 • A.B. in Mathematics, Harvard University, June 1989 DATE OF BIRTH April, 1968. U.S. citizen. Amie Wilkinson 2 RESEARCH INTERESTS • Ergodic theory and smooth dynamical systems • Geometry and regularity of foliations • Actions of discrete groups on manifolds • Dynamical systems of geometric origin GRANTS, FELLOWSHIPS AND AWARDS • Levi L. Conant Prize, 2020. • Foreign Member, Academia Europaea, 2019. • Fellow of the American Mathematical Society, 2013. • Ruth Lyttle Satter Prize, 2011. • NSF Grant \Ergodicity, Rigidity, and the Interplay Between Chaotic and Regular Dynamics" $758,242, 2018{2021. • NSF Grant \Innovations in Bright Beam Science" (co-PI) $680,000, 2015{2018. • NSF Grant \RTG: Geometry and topology at the University of Chicago" (co-PI) $1,377,340, 2014{2019.
    [Show full text]
  • Invariant Measures for Hyperbolic Dynamical Systems
    Invariant measures for hyperbolic dynamical systems N. Chernov September 14, 2006 Contents 1 Markov partitions 3 2 Gibbs Measures 17 2.1 Gibbs states . 17 2.2 Gibbs measures . 33 2.3 Properties of Gibbs measures . 47 3 Sinai-Ruelle-Bowen measures 56 4 Gibbs measures for Anosov and Axiom A flows 67 5 Volume compression 86 6 SRB measures for general diffeomorphisms 92 This survey is devoted to smooth dynamical systems, which are in our context diffeomorphisms and smooth flows on compact manifolds. We call a flow or a diffeomorphism hyperbolic if all of its Lyapunov exponents are different from zero (except the one in the flow direction, which has to be 1 zero). This means that the tangent vectors asymptotically expand or con- tract exponentially fast in time. For many reasons, it is convenient to assume more than just asymptotic expansion or contraction, namely that the expan- sion and contraction of tangent vectors happens uniformly in time. Such hyperbolic systems are said to be uniformly hyperbolic. Historically, uniformly hyperbolic flows and diffeomorphisms were stud- ied as early as in mid-sixties: it was done by D. Anosov [2] and S. Smale [77], who introduced his Axiom A. In the seventies, Anosov and Axiom A dif- feomorphisms and flows attracted much attention from different directions: physics, topology, and geometry. This actually started in 1968 when Ya. Sinai constructed Markov partitions [74, 75] that allowed a symbolic representa- tion of the dynamics, which matched the existing lattice models in statistical mechanics. As a result, the theory of Gibbs measures for one-dimensional lat- tices was carried over to Anosov and Axiom A dynamical systems.
    [Show full text]
  • CURRENT EVENTS BULLETIN Friday, January 8, 2016, 1:00 PM to 5:00 PM Room 4C-3 Washington State Convention Center Joint Mathematics Meetings, Seattle, WA
    A MERICAN M ATHEMATICAL S OCIETY CURRENT EVENTS BULLETIN Friday, January 8, 2016, 1:00 PM to 5:00 PM Room 4C-3 Washington State Convention Center Joint Mathematics Meetings, Seattle, WA 1:00 PM Carina Curto, Pennsylvania State University What can topology tell us about the neural code? Surprising new applications of what used to be thought of as “pure” mathematics. 2:00 PM Yuval Peres, Microsoft Research and University of California, Berkeley, and Lionel Levine, Cornell University Laplacian growth, sandpiles and scaling limits Striking large-scale structure arising from simple cellular automata. 3:00 PM Timothy Gowers, Cambridge University Probabilistic combinatorics and the recent work of Peter Keevash The major existence conjecture for combinatorial designs has been proven! 4:00 PM Amie Wilkinson, University of Chicago What are Lyapunov exponents, and why are they interesting? A basic tool in understanding the predictability of physical systems, explained. Organized by David Eisenbud, Mathematical Sciences Research Institute Introduction to the Current Events Bulletin Will the Riemann Hypothesis be proved this week? What is the Geometric Langlands Conjecture about? How could you best exploit a stream of data flowing by too fast to capture? I think we mathematicians are provoked to ask such questions by our sense that underneath the vastness of mathematics is a fundamental unity allowing us to look into many different corners -- though we couldn't possibly work in all of them. I love the idea of having an expert explain such things to me in a brief, accessible way. And I, like most of us, love common-room gossip.
    [Show full text]
  • Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms
    Rufus Bowen Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms New edition of Lect. Notes in Math. 470, Springer, 1975. April 14, 2013 Springer Preface The Greek and Roman gods, supposedly, resented those mortals endowed with superlative gifts and happiness, and punished them. The life and achievements of Rufus Bowen (1947{1978) remind us of this belief of the ancients. When Rufus died unexpectedly, at age thirty-one, from brain hemorrhage, he was a very happy and successful man. He had great charm, that he did not misuse, and superlative mathematical talent. His mathematical legacy is important, and will not be forgotten, but one wonders what he would have achieved if he had lived longer. Bowen chose to be simple rather than brilliant. This was the hard choice, especially in a messy subject like smooth dynamics in which he worked. Simplicity had also been the style of Steve Smale, from whom Bowen learned dynamical systems theory. Rufus Bowen has left us a masterpiece of mathematical exposition: the slim volume Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Springer Lecture Notes in Mathematics 470 (1975)). Here a number of results which were new at the time are presented in such a clear and lucid style that Bowen's monograph immediately became a classic. More than thirty years later, many new results have been proved in this area, but the volume is as useful as ever because it remains the best introduction to the basics of the ergodic theory of hyperbolic systems. The area discussed by Bowen came into existence through the merging of two apparently unrelated theories.
    [Show full text]
  • President's Report
    Newsletter Volume 43, No. 3 • mAY–JuNe 2013 PRESIDENT’S REPORT Greetings, once again, from 35,000 feet, returning home from a major AWM conference in Santa Clara, California. Many of you will recall the AWM 40th Anniversary conference held in 2011 at Brown University. The enthusiasm generat- The purpose of the Association ed by that conference gave rise to a plan to hold a series of biennial AWM Research for Women in Mathematics is Symposia around the country. The first of these, the AWM Research Symposium 2013, took place this weekend on the beautiful Santa Clara University campus. • to encourage women and girls to study and to have active careers The symposium attracted close to 150 participants. The program included 3 plenary in the mathematical sciences, and talks, 10 special sessions on a wide variety of topics, a contributed paper session, • to promote equal opportunity and poster sessions, a panel, and a banquet. The Santa Clara campus was in full bloom the equal treatment of women and and the weather was spectacular. Thankfully, the poster sessions and coffee breaks girls in the mathematical sciences. were held outside in a courtyard or those of us from more frigid climates might have been tempted to play hooky! The event opened with a plenary talk by Maryam Mirzakhani. Mirzakhani is a professor at Stanford and the recipient of multiple awards including the 2013 Ruth Lyttle Satter Prize. Her talk was entitled “On Random Hyperbolic Manifolds of Large Genus.” She began by describing how to associate a hyperbolic surface to a graph, then proceeded with a fascinating discussion of the metric properties of surfaces associated to random graphs.
    [Show full text]
  • What Are Lyapunov Exponents, and Why Are They Interesting?
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 1, January 2017, Pages 79–105 http://dx.doi.org/10.1090/bull/1552 Article electronically published on September 6, 2016 WHAT ARE LYAPUNOV EXPONENTS, AND WHY ARE THEY INTERESTING? AMIE WILKINSON Introduction At the 2014 International Congress of Mathematicians in Seoul, South Korea, Franco-Brazilian mathematician Artur Avila was awarded the Fields Medal for “his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalization as a unifying principle.”1 Although it is not explicitly mentioned in this citation, there is a second unify- ing concept in Avila’s work that is closely tied with renormalization: Lyapunov (or characteristic) exponents. Lyapunov exponents play a key role in three areas of Avila’s research: smooth ergodic theory, billiards and translation surfaces, and the spectral theory of 1-dimensional Schr¨odinger operators. Here we take the op- portunity to explore these areas and reveal some underlying themes connecting exponents, chaotic dynamics and renormalization. But first, what are Lyapunov exponents? Let’s begin by viewing them in one of their natural habitats: the iterated barycentric subdivision of a triangle. When the midpoint of each side of a triangle is connected to its opposite vertex by a line segment, the three resulting segments meet in a point in the interior of the triangle. The barycentric subdivision of a triangle is the collection of 6 smaller triangles determined by these segments and the edges of the original triangle: Figure 1. Barycentric subdivision. Received by the editors August 2, 2016.
    [Show full text]
  • Stable Accessibility Is C1 Dense
    Stable accessibility is C1 dense Dmitry Dolgopyat and Amie Wilkinson August 27, 2003 Abstract We prove that in the space of all C r (r ≥ 1) partially hyperbolic diffeomorphisms, there is a C 1 open and dense set of accessible dif- feomorphisms. This settles the C 1 case of a conjecture of Pugh and Shub. The same result holds in the space of volume preserving or symplectic partially hyperbolic diffeomorphisms. Combining this theorem with results in [Br], [Ar] and [PugSh3], we obtain several corollaries. The first states that in the space of volume preserving or symplectic partially hyperbolic diffeomorphisms, topological transitivity holds on an open and dense set. Further, on a symplectic n-manifold (n ≤ 4) the C 1-closure of the stably transitive symplectomorphisms is precisely the closure of the partially hyperbolic symplectomorphisms. Finally, stable ergodicity is C 1 open and dense among the volume preserving, partially hyperbolic diffeomorphisms satisfying the additional technical hypotheses of [PugSh3] . Introduction This paper is about the accessibility property of partially hyperbolic diffeo- morphisms. We show that accessibility holds for a C1 open and dense set in the space of all partially hyperbolic diffeomorphisms, thus settling the C1 version of a conjecture of Pugh and Shub [PugSh1]. Partially hyper- bolic diffeomorphisms are similar to Anosov diffeomorphisms, in that they possess invariant hyperbolic directions, but unlike Anosov diffeomorphisms, they can also possess invariant directions of non-hyperbolic behavior. Ac- cessibility means that the hyperbolic directions fill up the manifold on a macroscopic scale. Accessibility often provides enough hyperbolicity for a 1 variety of chaotic properties, such as topological transitivity [Br] and ergod- icity [PugSh3], to hold.
    [Show full text]
  • Read Press Release
    The Work of Artur Avila Artur Avila has made outstanding contributions to dynamical systems, analysis, and other areas, in many cases proving decisive results that solved long-standing open problems. A native of Brazil who spends part of his time there and part in France, he combines the strong mathematical cultures and traditions of both countries. Nearly all his work has been done through collaborations with some 30 mathematicians around the world. To these collaborations Avila brings formidable technical power, the ingenuity and tenacity of a master problem-solver, and an unerring sense for deep and significant questions. Avila's achievements are many and span a broad range of topics; here we focus on only a few highlights. One of his early significant results closes a chapter on a long story that started in the 1970s. At that time, physicists, most notably Mitchell Feigenbaum, began trying to understand how chaos can arise out of very simple systems. Some of the systems they looked at were based on iterating a mathematical rule such as 3x(1−x). Starting with a given point, one can watch the trajectory of the point under repeated applications of the rule; one can think of the rule as moving the starting point around over time. For some maps, the trajectories eventually settle into stable orbits, while for other maps the trajectories become chaotic. Out of the drive to understand such phenomena grew the subject of discrete dynamical systems, to which scores of mathematicians contributed in the ensuing decades. Among the central aims was to develop ways to predict long-time behavior.
    [Show full text]
  • Uniformly Hyperbolic Control Theory Christoph Kawan
    HYPERBOLIC CONTROL THEORY 1 Uniformly hyperbolic control theory Christoph Kawan Abstract—This paper gives a summary of a body of work at the of control-affine systems with a compact and convex control intersection of control theory and smooth nonlinear dynamics. range. The main idea is to transfer the concept of uniform hyperbolicity, These results are grounded on the topological theory of central to the theory of smooth dynamical systems, to control- affine systems. Combining the strength of geometric control Colonius-Kliemann [7] which provides an approach to under- theory and the hyperbolic theory of dynamical systems, it is standing the global controllability structure of control systems. possible to deduce control-theoretic results of non-local nature Two central notions of this theory are control and chain that reveal remarkable analogies to the classical hyperbolic the- control sets. Control sets are the maximal regions of complete ory of dynamical systems. This includes results on controllability, approximate controllability in the state space. The definition of robustness, and practical stabilizability in a networked control framework. chain control sets involves the concept of "-chains (also called "-pseudo-orbits) from the theory of dynamical systems. The Index Terms—Control-affine system; uniform hyperbolicity; main motivation for this concept comes from the facts that (i) chain control set; controllability; robustness; networked control; invariance entropy chain control sets are outer approximations of control sets and (ii) chain control sets in general are easier to determine than control sets (both analytically and numerically). I. INTRODUCTION As examples show, chain control sets can support uniformly hyperbolic and, more generally, partially hyperbolic structures.
    [Show full text]
  • Transformations)
    TRANSFORMACJE (TRANSFORMATIONS) Transformacje (Transformations) is an interdisciplinary refereed, reviewed journal, published since 1992. The journal is devoted to i.a.: civilizational and cultural transformations, information (knowledge) societies, global problematique, sustainable development, political philosophy and values, future studies. The journal's quasi-paradigm is TRANSFORMATION - as a present stage and form of development of technology, society, culture, civilization, values, mindsets etc. Impacts and potentialities of change and transition need new methodological tools, new visions and innovation for theoretical and practical capacity-building. The journal aims to promote inter-, multi- and transdisci- plinary approach, future orientation and strategic and global thinking. Transformacje (Transformations) are internationally available – since 2012 we have a licence agrement with the global database: EBSCO Publishing (Ipswich, MA, USA) We are listed by INDEX COPERNICUS since 2013 I TRANSFORMACJE(TRANSFORMATIONS) 3-4 (78-79) 2013 ISSN 1230-0292 Reviewed journal Published twice a year (double issues) in Polish and English (separate papers) Editorial Staff: Prof. Lech W. ZACHER, Center of Impact Assessment Studies and Forecasting, Kozminski University, Warsaw, Poland ([email protected]) – Editor-in-Chief Prof. Dora MARINOVA, Sustainability Policy Institute, Curtin University, Perth, Australia ([email protected]) – Deputy Editor-in-Chief Prof. Tadeusz MICZKA, Institute of Cultural and Interdisciplinary Studies, University of Silesia, Katowice, Poland ([email protected]) – Deputy Editor-in-Chief Dr Małgorzata SKÓRZEWSKA-AMBERG, School of Law, Kozminski University, Warsaw, Poland ([email protected]) – Coordinator Dr Alina BETLEJ, Institute of Sociology, John Paul II Catholic University of Lublin, Poland Dr Mirosław GEISE, Institute of Political Sciences, Kazimierz Wielki University, Bydgoszcz, Poland (also statistical editor) Prof.
    [Show full text]
  • Symbolic Dynamics and Markov Partitions
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 35, Number 1, January 1998, Pages 1–56 S 0273-0979(98)00737-X SYMBOLIC DYNAMICS AND MARKOV PARTITIONS ROY L. ADLER Abstract. The decimal expansion of real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by sym- bolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a general theory of representing dynamical systems by sym- bolic systems by means of so-called Markov partitions. We apply the results to one of the more tractable examples: namely, hyperbolic automorphisms of the two dimensional torus. While there are some results in higher dimensions, this area remains a fertile one for research. 1. Introduction We address the question: how and to what extent can a dynamical system be represented by a symbolic one? The first use of infinite sequences of symbols to describe orbits is attributed to a nineteenth century work of Hadamard [H]. How- ever the present work is rooted in something very much older and familiar to us all: namely, the representation of real numbers by infinite binary expansions. As the example of Section 3.2 shows, such a representation is related to the behavior of a special partition under the action of a map. Partitions of this sort have been linked to the name Markov because of their connection to discrete time Markov processes.
    [Show full text]
  • On the Ergodicity of Partially Hyperbolic Systems
    ANNALS OF MATHEMATICS On the ergodicity of partially hyperbolic systems By Keith Burns and Amie Wilkinson SECOND SERIES, VOL. 171, NO. 1 January, 2010 anmaah Annals of Mathematics, 171 (2010), 451–489 On the ergodicity of partially hyperbolic systems By KEITH BURNS and AMIE WILKINSON Abstract Pugh and Shub have conjectured that essential accessibility implies ergodicity for a C 2, partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satisfied in particular by all partially hyperbolic systems with 1-dimensional center bundle. We also 1 ı obtain ergodicity results for C C partially hyperbolic systems. Introduction In[Hop39], Eberhard Hopf introduced a simple argument that proved the ergodicity (with respect to Liouville measure) of the geodesic flow of a compact, negatively curved surface. The argument has since been applied to increasingly general classes of dynamical systems. The key feature that these systems possess is hyperbolicity. The strongest form of hyperbolicity is uniform hyperbolicity. A diffeomorphism f M M of a compact manifold M is uniformly hyperbolic or W ! Anosov if there exists a splitting of the tangent bundle into Tf -invariant subbundles, given by TM Es Eu; D ˚ and a continuous Riemannian metric such that, for every unit vector v TM , 2 Tf v < 1 if v Es; k k 2 Tf v > 1 if v Eu: k k 2 Anosov flows are defined similarly, with Es Eu complementary to the bundle ˚ E0 that is tangent to the flow direction. The bundles Es and Eu of an Anosov system are tangent to the stable and unstable foliations ᐃs and ᐃu, respectively.
    [Show full text]