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Rokhlin Lemma
In mathematics, the Rokhlin lemma, or Kakutani–Rokhlin lemma is an important result in ergodic theory. 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. It was proven by Vladimir Abramovich Rokhlin and independently by Shizuo Kakutani. The lemma is used extensively in ergodic theory, for example in Ornstein theory and has many generalizations. Terminology Rokhlin lemma belongs to the group mathematical statements such as Zorn's lemma in set theory and Schwarz lemma in complex analysis which are traditionally called lemmas despite the fact that their roles in their respective fields are fundamental. Statement of the lemma Lemma: Let T\colon X\to X be an invertible measure-preserving transformation on a standard measure space \textstyle (X,\Sigma,\mu) with \textstyle \mu(X)=1. We assume \textstyle T is (measurably) aperiodic, that is, the set of periodic p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Publications Mathématiques De L'IHÉS
''Publications Mathématiques de l'IHÉS'' is a peer-reviewed mathematical journal. It is published by Springer Science+Business Media on behalf of the Institut des Hautes Études Scientifiques, with the help of the Centre National de la Recherche Scientifique. The journal was established in 1959 and was published at irregular intervals, from one to five volumes a year. It is now biannual. The editor-in-chief is Claire Voisin (Collège de France). See also *''Annals of Mathematics'' *'' Journal of the American Mathematical Society'' *''Inventiones Mathematicae ''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors ...'' External links * Back issues from 1959 to 2010 Mathematics journals Publications established in 1959 Springer Science+Business Media academic journals Biannual journal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. See also * ''Bulletin of the American Mathematical Society'' * '' Journal of the American Mathematical Society'' * ''Memoirs of the American Mathematical Society'' * ''Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of j ... American Mathematical Society academic journals Mathematics journals Publications established in 1900 {{math-journal-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal D'Analyse Mathématique
The ''Journal d'Analyse Mathématique'' is a triannual peer-reviewed scientific journal published by Magnes Press (Hebrew University of Jerusalem). It was established in 1951 by Binyamin Amirà. It covers research in mathematics, especially classical analysis and related areas such as complex function theory, ergodic theory, functional analysis, harmonic analysis, partial differential equations, and quasiconformal mapping. Abstracting and indexing The journal is abstracted and indexed in: *MathSciNet *Science Citation Index Expanded *Scopus *ZbMATH Open According to the ''Journal Citation Reports'', the journal has a 2021 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 1.132. References External links *{{Official website, 1=https://www.springer.com/mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amenable Group
In mathematics, an amenable group is a locally compact topological group ''G'' carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure (or mean) on subsets of ''G'', was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "''mean''". The amenability property has a large number of equivalent formulations. In the field of analysis, the definition is in terms of linear functionals. An intuitive way to understand this version is that the support of the regular representation is the whole space of irreducible representations. In discrete group theory, where ''G'' has the discrete topology, a simpler definition is used. In this setting, a group is amenable if one can say what proport ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Benjamin Weiss
Benjamin Weiss ( he, בנימין ווייס; born 1941) is an American-Israeli mathematician known for his contributions to ergodic theory, topological dynamics, probability theory, game theory, and descriptive set theory. Biography Benjamin ("Benjy") Weiss was born in New York City. In 1962 he received B.A. from Yeshiva University and M.A. from the Graduate School of Science, Yeshiva University. In 1965, he received his Ph.D. from Princeton under the supervision of William Feller. Academic career Between 1965 and 1967, Weiss worked at the IBM Research. In 1967, he joined the faculty of the Hebrew University of Jerusalem; and since 1990 occupied the Miriam and Julius Vinik Chair in Mathematics (Emeritus since 2009). Weiss held visiting positions at Stanford, MSRI, and IBM Research Center. Weiss published over 180 papers in ergodic theory, topological dynamics, orbit equivalence, probability, information theory, game theory, descriptive set theory; with notable contribution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald Samuel Ornstein
Donald Samuel Ornstein (born July 30, 1934, New York) is an American mathematician working in the area of ergodic theory. He received a Ph.D. from the University of Chicago in 1957 under the guidance of Irving Kaplansky. During his career at Stanford University he supervised the Ph. D. thesis of twenty three students, including David H. Bailey, Bob Burton, Doug Lind, Ami Radunskaya, Dan Rudolph, and Jeff Steif. He is most famous for his work on the isomorphism of Bernoulli shifts for which he won the 1974 Bôcher Prize. He has been a member of the National Academy of Sciences since 1981. In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, .... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbit Capacity
In mathematics, the orbit capacity of a subset of a topological dynamical system may be thought of heuristically as a “topological dynamical probability measure” of the subset. More precisely, its value for a set is a tight upper bound for the normalized number of visits of orbits in this set. Definition A topological dynamical system consists of a compact Hausdorff topological space ''X'' and a homeomorphism T:X\rightarrow X. Let E\subset X be a set. Lindenstrauss introduced the definition of orbit capacity: :\operatorname(E)=\lim_\sup_ \frac 1 n \sum_^ \chi_E (T^k x) Here, \chi_E(x) is the membership function for the set E. That is \chi_E(x)=1 if x\in E and is zero otherwise. Properties One has 0\le\operatorname(E)\le 1. By convention, topological dynamical systems do not come equipped with a measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Boundary Property
In mathematics, the small boundary property is a property of certain topological dynamical systems. It is dynamical analog of the inductive definition of Lebesgue covering dimension zero. Definition Consider the category of topological dynamical system (''system'' in short) consisting of a compact metric space X and a homeomorphism T:X\rightarrow X. A set E\subset X is called small if it has vanishing orbit capacity, i.e., \operatorname(E) = 0. This is equivalent to: \forall\mu\in M_(X),\ \mu(E)=0 where M_T(X) denotes the collection of T-invariant measures on X. The system (X,T) is said to have the small boundary property (SBP) if X has a basis of open sets \_^\infty whose boundaries are small, i.e., \operatorname(\partial O_i)=0 for all i. Can one always lower topological entropy? Small sets were introduced by Michael Shub and Benjamin Weiss while investigating the question "can one always lower topological entropy?" Quoting from their article: "For measure theoretic e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Theory And Dynamical Systems
'' Ergodic Theory and Dynamical Systems'' is a peer-reviewed mathematics journal published by Cambridge University Press. Established in 1981, the journal publishes articles on dynamical systems. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... was 0.822. External links * Mathematics journals Academic journals established in 1981 English-language journals Cambridge University Press academic journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elon Lindenstrauss
Elon Lindenstrauss ( he, אילון לינדנשטראוס, born August 1, 1970) is an Israeli mathematician, and a winner of the 2010 Fields Medal. Since 2004, he has been a professor at Princeton University. In 2009, he was appointed to Professor at the Mathematics Institute at the Hebrew University. Biography Lindenstrauss was born into an Israeli-Jewish family with German Jewish origins. He was also born into a mathematical family, the son of the mathematician Joram Lindenstrauss, the namesake of the Johnson–Lindenstrauss lemma, and computer scientist Naomi Lindenstrauss, both professors at the Hebrew University of Jerusalem. His sister Ayelet Lindenstrauss is also a mathematician. He attended the Hebrew University Secondary School. In 1988 he was awarded a bronze medal at the International Mathematical Olympiad. He enlisted to the IDF's Talpiot program, and studied at the Hebrew University of Jerusalem, where he earned his BSc in Mathematics and Physics in 1991 and his m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Preserving Dynamical System
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of conservative systems. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from classical mechanics (in particular, most non-dissipative systems) as well as systems in thermodynamic equilibrium. Definition A measure-preserving dynamical system is defined as a probability space and a Invariant measure, measure-preserving transformation on it. In more detail, it is a system :(X, \mathcal, \mu, T) with the following structure: *X is a set, *\mathcal B is a sigma-algebra, σ-algebra over X, *\mu:\mathcal\rightarrow[0,1] is a probability measure, so that \mu (X) = 1, and \mu(\varnothing) = 0, * T:X \rightarrow X is a measurable function, measurable transformation whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
