A key tool in applying amenability to dynamics is the rokhlin lemma of. It was proven by vladimir abramovich rokhlin and independently by shizuo kakutani. With his characteristic breadth rokhlin studied examples of dynamical. Zapis ki nauqnyh sem inarov pomi t om g a sz ucs tw o theorems of r okhlin abstra ct tw o theorems due to v a rokhlin are pro v ed on the third stable homotop. In both the additive and subadditive cases, these maximal. Rokhlin s grandmother, klara levenson, was one of the first women doctors in russia. In mathematics, the rokhlin lemma, or kakutanirokhlin lemma is an important result in ergodic theory. Invariants of orbit equivalence relations and baumslagsolitar groups kida, yoshikata, tohoku mathematical journal, 2014.
The other partition result is a generalization of the rokhlin lemma, stating that the space can be partitioned into denumerably many columns and the measures of the columns can be prescribed in advance. In section 2 we provide the necessary background in ergodic theory for the proof of szemer edis theorem. Vladimir abramovich rokhlina biographical tribute 23. In this expository paper, we survey nowadays classical tools or criteria used in problems of convergence everywhere to build counterexamples.
It is well known that the rokhlin lemma is also related to amenability. On approximation of homeomorphisms of a cantor set arxiv. A generalized shannonmcmillan theorem for the action. Three theorems in ergodic theory, one variation on rokhlin s lemma. The multiple rokhlin tower theorem for t provides a simple con dition for the existence of a partition of the space x into denumerably many columns for t, with prescribed measures.
One of the most relevant results in the theory is the rokhlin lemma ro stating that the set of periodic automorphisms is dudense in autx,b. We show that three theorems about the measurable dynamics of a fixed aperiodic measure preserving transformation. For the proof we will need to use the rokhlin lemma. The overflow blog socializing with coworkers while social distancing. Ergodic theory lies in somewhere among measure theory, analysis, probability, dynamical systems, and di. Pugh, clark robinson skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. An extension of the ergodic closing lemma ergodic theory.
As an application, we show that, c 1 densely in the complement of the closure of morsesmale diffeomorphisms and those with a homoclinic tangency, there exists a weakly hyperbolic structure dominated splittings with average hyperbolicity at almost every point on hyperbolic parts, and onedimensional center. The other partition result is a generalization of the rokhlin lemma, stating. Thus the first two results are equivalent to this strengthening of rokhlin s lemma. In the 1970s, furstenberg showed how to translate questions in combinatorial number theory into ergodic theory. Shizuo kakutani, kakutani shizuo, august 28, 1911 august 17, 2004 was a japaneseamerican mathematician, best known for his eponymous fixedpoint theorem. Mackey, ergodic theory and its significance for statistical mechanics and probability theory adv. We will choose one specic point of view but there are many others.
Ams proceedings of the american mathematical society. Jan 17, 20 for the proof we will need to use the rokhlin lemma. Homeox the set of all homeomorphisms conjugate to t is dense in the set of aperiodic homeomorphisms. First midwestern conference, ergodic theory and probability, columbus 1970, pp. Browse other questions tagged measure theory ergodic theory or ask your own question. One of the most useful results in ergodic theory which has many important applications is the rokhlin lemma r. The other partition result is a generalization of the rokhlin lemma, stating that the space can be partitioned into denumerably many columns and the measures of the columns can be prescribed. In probability theory, a standard probability space, also called lebesgue rokhlin probability space or just lebesgue space the latter term is ambiguous is a probability space satisfying certain assumptions introduced by vladimir rokhlin in 1940. Pdf the rokhlin lemma for homeomorphisms of a cantor set. In particular, it is proved that for any aperiodic t.
Notes on group actions on subfactors masuda, toshihiko, journal of the mathematical society of japan, 2003. Rokhlin s mother graduated from a medical school in france and was a doctor in baku. Notes on ergodic theory hebrew university of jerusalem. The c1 closing lemma, including hamiltonians volume 3 issue 2 charles c. Spectral properties in ergodic theory 651 this survey primarily deals with certain aspects of ergodic theory, i. Applications of this point of view include the areas of statistical physics, classical mechanics, number theory, population dynam. The idea of investigation of transformation groups by means of introducing var. We dedicate this paper to the memory of shizuo kakutani. Embedding topological dynamical systems with periodic. Ergodic theory constantine caramanis may 6, 1999 1 introduction ergodic theory involves the study of transformations on measure spaces. General introduction to ergodic theory is presented in 8, section 3. Ergodic theory is often concerned with ergodic transformations. Citeseerx the rokhlin lemma for homeomorphisms of a.
Later together with hurewicz rokhlin generalized it to measurable flows 12. Ergodic theory of differentiable dynamical by david ruelle systems dedicated to the memory of rufus bowen abstract. We also use the generalized rokhlin lemma, this time combined with a subadditive version of kacs formula, to deduce a subadditive version of the maximal ergodic theorem due to silva and thieullen. It is known that for free ergodic actions of amenable groups this notion coincides with classical kolmogorovsinai entropy. In the 1970s, furstenberg showed how to translate questions of combinatorial number theory into ergodic theory.
Spectral properties and combinatorial constructions in. Rokhlins lemma in the remainder of the course, our most substantial results are about the existence of factor maps between various pairs of mpss or sources, sometimes with special extra properties. The rokhlin lemma for homeomorphisms of a cantor set. We miss his kind manner, gentle presence and keen insight. The rokhlin lemma for homeomorphisms of a cantor set core. Pages in category ergodic theory the following 49 pages are in this category, out of 49 total. If this is the first time you use this feature, you will be asked to authorise cambridge core to connect with your. The intuition behind such transformations, which act on a given set, is that they do a thorough job stirring the elements of that set e.
The c1 closing lemma, including hamiltonians ergodic. Three theorems in ergodic theory, one variation on rokhlin s lemma we dedicate this paper to. Mrt is a generalization of one of the basic constructions in ergodic theory, rokhlin s lemma see for example kornfelds survey 18. Lemma 6 rokhlin lemma let be an invertible ergodic m. Alperns multiple rokhlin tower theorem in this paper we show that kakutanis proof of rokhlin s lemma hal56 can be used to give a short, elementary proof of the following multiple rokhlin tower theorem of alperns alp79, cor 2. Interchanging the words \measurable function and \ probability density function translates many results from real analysis to results in probability theory. Lecture notes introduction to ergodic theory tiago pereira department of mathematics imperial college london our course consists of. First, we state a l1version of the continuity principle and give an example of its usefulness. The rokhlin lemma for homeomorphisms of a cantor set arxiv. The c1 closing lemma, including hamiltonians ergodic theory. This brings us to the domain of ergodic theory, which is the study of recurrence phenomena in these types of dynamical systems, known as measure preserving systems.
Alperns multiple rokhlin tower theorem in this paper we show that kakutanis proof of rokhlin s lemma hal56 can be used to give a short, elementary proof of the following multiple rokhlin. It states that an aperiodic measure preserving dynamical system can be decomposed to an arbitrary high tower of measurable sets and a remainder of arbitrarily small measure. Alperns multiple rokhlin tower theorem in this paper we show that kakutanis proof of rokhlins lemma hal56 can be used to give a short, elementary proof of the following multiple rokhlin tower theorem of alperns alp79, cor 2. The basic ergodic theorems, yet again cubo, a mathematical. Rokhlin s mother was the sister of the wellknown literary figure and childrens writer kornei chukovsky. A generalization of rokhlins tower lemma is presented. Krengel, u transformations without finite invariant measure have finite strong generators. This statement asserts that given an aperiodic nonsingular automorphism t of a standard measure space x,b. The rokhlin lemma for homeomorphisms of a cantor set, proc.
In order to aid the reading and understanding of the material presented we have tried to keep technical complexity to a minimum. For each, let large enough so that, and let be the set provided by the rokhlin lemma with and. In simple terms, ergodic theory studies dynamics systems that preserve a probability measure. Iff is a g tm diffeomorphism of a compact manifold m, we prove the existence of stable manifolds, almost everywhere with respect to every finvariant probability measure on m. A nice feature of the notion of finite rokhlin dimension is that, although it has weaker requirements compared to other adaptations of the rokhlin. Alperns multitower theorem has also played an important role in the proof of many recent important results in ergodic theory see 1 for the original application. From uniform laws of large numbers to uniform ergodic. Mane, ergodic theory and differentiable dynamics, springer 1987 translated from the portuguese mr0889254 zbl 0616. First, we state a l1version of the continuity principle and give an. The main result of this pap er is the rokhlin lemma prov ed for an arbitrar y ap e rio dic homeomorphism of can tor dynamics.
Alpern, steve and prasad, vidhu 2008 multitowers, conjugacies and codes. Mrt is a generalization of one of the basic constructions in ergodic theory, rokhlins lemma see for example kornfelds survey 18. To start with, we need to consider a topology on homeox analogous. The invariance of means that we are in an equilibrium situation, but not necessarily a static one. Towers, conjugacy and coding london school of economics. Informally, it is a probability space consisting of an interval andor a finite or countable number of atoms. Finite group actions on calgebras with the rohlin property, i izumi, masaki, duke mathematical journal, 2004. Then there exists some set such that are disjoint and. Foundations of ergodic theory rich with examples and applications, this textbook provides a coherent and selfcontained introduction to ergodic theory suitable for a variety of one or twosemester courses. The other partition result is a generalization of the rokhlin lemma. Our goal is to prove a version of the rokhlin lemma in the context of cantor dynamics. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
We show that a finite alpern multitower may always be constructed whose base is independent of p. The maximal ergodic theorem is then obtained as a corollary. Criteria of divergence almost everywhere in ergodic theory. Moreover, new classes of systems are found to have the marker property, in particular, extensions of aperiodic systems with a. It is w ell kno wn that the rokhlin lemma is also related to amenabilit y. A generalization of rokhlin s tower lemma is presented. In mathematics, the rokhlin lemma, or kakutani rokhlin lemma is an important result in ergodic theory. Rokhlin lemma w as generalized in v arious directions see, for example, ap, ep, fl, l w, o w. Let t be any invertible, ergodic, aperiodic measurepreserving transformation of a lebesgue probability space x,b. Rokhlin s lemma in the remainder of the course, our most substantial results are about the existence of factor maps between various pairs of mpss or sources, sometimes with special extra properties.