Download or read book Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations written by Greg Hjorth and published by American Mathematical Soc.. This book was released on 2005 with total page 109 pages. Available in PDF, EPUB and Kindle. Book excerpt: This memoir is both a contribution to the theory of Borel equivalence relations, considered up to Borel reducibility, and measure preserving group actions considered up to orbit equivalence. Here $E$ is said to be Borel reducible to $F$ if there is a Borel function $f$ with $x E y$ if and only if $f(x) F f(y)$. Moreover, $E$ is orbit equivalent to $F$ if the respective measure spaces equipped with the extra structure provided by the equivalence relations are almost everywhere isomorphic. We consider product groups acting ergodically and by measure preserving transformations on standard Borel probability spaces.In general terms, the basic parts of the monograph show that if the groups involved have a suitable notion of 'boundary' (we make this precise with the definition of near hyperbolic), then one orbit equivalence relation can only be Borel reduced to another if there is some kind of algebraic resemblance between the product groups and coupling of the action. This also has consequence for orbit equivalence. In the case that the original equivalence relations do not have non-trivial almost invariant sets, the techniques lead to relative ergodicity results. An equivalence relation $E$ is said to be relatively ergodic to $F$ if any $f$ with $xEy \Rightarrow f(x) F f(y)$ has $[f(x)]_F$ constant almost everywhere.This underlying collection of lemmas and structural theorems is employed in a number of different ways. In the later parts of the paper, we give applications of the theory to specific cases of product groups. In particular, we catalog the actions of products of the free group and obtain additional rigidity theorems and relative ergodicity results in this context. There is a rather long series of appendices, whose primary goal is to give the reader a comprehensive account of the basic techniques. But included here are also some new results. For instance, we show that the Furstenberg-Zimmer lemma on cocycles from amenable groups fails with respect to Baire category, and use this to answer a question of Weiss. We also present a different proof that $F_2$ has the Haagerup approximation property.
Download or read book Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations written by Greg Hjorth Alexander S. Kechris and published by American Mathematical Soc.. This book was released on 2005-08-03 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt: This Memoir is both a contribution to the theory of Borel equivalence relations, considered up to Borel reducibility, and measure preserving group actions considered up to orbit equivalence. Here $E$ is said to be Borel reducible to $F$ if there is a Borel function $f$ with $x E y$ if and only if $f(x) F f(y)$. Moreover, $E$ is orbit equivalent to $F$ if the respective measure spaces equipped with the extra structure provided by the equivalence relations are almost everywhere isomorphic. We consider product groups acting ergodically and by measure preserving transformations on standard Borel probability spaces. In general terms, the basic parts of the monograph show that if the groups involved have a suitable notion of ``boundary'' (we make this precise with the definition of near hyperbolic), then one orbit equivalence relation can only be Borel reduced to another if there is some kind of algebraic resemblance between the product groups and coupling of the action. This also has consequence for orbit equivalence. In the case that the original equivalence relations do not have non-trivial almost invariant sets, the techniques lead to relative ergodicity results. An equivalence relation $E$ is said to be relatively ergodic to $F$ if any $f$ with $xEy \Rightarrow f(x) F f(y)$ has $[f(x)]_F$ constant almost everywhere. This underlying collection of lemmas and structural theorems is employed in a number of different ways. In the later parts of the paper we give applications of the theory to specific cases of product groups. In particular, we catalog the actions of products of the free group and obtain additional rigidity theorems and relative ergodicity results in this context. There is a rather long series of appendices, whose primary goal is to give the reader a comprehensive account of the basic techniques. But included here are also some new results. For instance, we show that the Furstenberg-Zimmer lemma on cocycles from amenable groups fails with respect to Baire category, and use this to answer a question of Weiss. We also present a different proof that $F_2$ has the Haagerup approximation property.
Download or read book Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations written by Greg Hjorth and published by American Mathematical Soc.. This book was released on 2005 with total page 126 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contributes to the theory of Borel equivalence relations, considered up to Borel reducibility, and measures preserving group actions considered up to orbit equivalence. This title catalogs the actions of products of the free group and obtains additional rigidity theorems and relative ergodicity results in this context.
Download or read book Handbook of Set Theory written by Matthew Foreman and published by Springer Science & Business Media. This book was released on 2009-12-10 with total page 2200 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numbers imitate space, which is of such a di?erent nature —Blaise Pascal It is fair to date the study of the foundation of mathematics back to the ancient Greeks. The urge to understand and systematize the mathematics of the time led Euclid to postulate axioms in an early attempt to put geometry on a ?rm footing. With roots in the Elements, the distinctive methodology of mathematics has become proof. Inevitably two questions arise: What are proofs? and What assumptions are proofs based on? The ?rst question, traditionally an internal question of the ?eld of logic, was also wrestled with in antiquity. Aristotle gave his famous syllogistic s- tems, and the Stoics had a nascent propositional logic. This study continued with ?ts and starts, through Boethius, the Arabs and the medieval logicians in Paris and London. The early germs of logic emerged in the context of philosophy and theology. The development of analytic geometry, as exempli?ed by Descartes, ill- tratedoneofthedi?cultiesinherentinfoundingmathematics. Itisclassically phrased as the question ofhow one reconciles the arithmetic with the geom- ric. Arenumbers onetypeofthingand geometricobjectsanother? Whatare the relationships between these two types of objects? How can they interact? Discovery of new types of mathematical objects, such as imaginary numbers and, much later, formal objects such as free groups and formal power series make the problem of ?nding a common playing ?eld for all of mathematics importunate. Several pressures made foundational issues urgent in the 19th century.
Download or read book Global Aspects of Ergodic Group Actions written by A. S. Kechris and published by American Mathematical Soc.. This book was released on 2010 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: A study of ergodic, measure preserving actions of countable discrete groups on standard probability spaces. It explores a direction that emphasizes a global point of view, concentrating on the structure of the space of measure preserving actions of a given group and its associated cocycle spaces.
Download or read book Invariant Descriptive Set Theory written by Su Gao and published by CRC Press. This book was released on 2008-09-03 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents Results from a Very Active Area of ResearchExploring an active area of mathematics that studies the complexity of equivalence relations and classification problems, Invariant Descriptive Set Theory presents an introduction to the basic concepts, methods, and results of this theory. It brings together techniques from various areas of mathem
Download or read book The Mapping Class Group from the Viewpoint of Measure Equivalence Theory written by Yoshikata Kida and published by American Mathematical Soc.. This book was released on 2008 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author obtains some classification result for the mapping class groups of compact orientable surfaces in terms of measure equivalence. In particular, the mapping class groups of different closed surfaces cannot be measure equivalent. Moreover, the author gives various examples of discrete groups which are not measure equivalent to the mapping class groups. In the course of the proof, the author investigates amenability in a measurable sense for the actions of the mapping class group on the boundary at infinity of the curve complex and on the Thurston boundary and, using this investigation, proves that the mapping class group of a compact orientable surface is exact.
Download or read book Twisted Tensor Products Related to the Cohomology of the Classifying Spaces of Loop Groups written by Katsuhiko Kuribayashi and published by American Mathematical Soc.. This book was released on 2006 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: Let $G$ be a compact, simply connected, simple Lie group. By applying the notion of a twisted tensor product in the senses of Brown as well as of Hess, we construct an economical injective resolution to compute, as an algebra, the cotorsion product which is the $E_2$-term of the cobar type Eilenberg-Moore spectral sequence converging to the cohomology of classifying space of the loop group $LG$. As an application, the cohomology $H^*(BLSpin(10); \mathbb{Z}/2)$ is explicitly determined as an $H^*(BSpin(10); \mathbb{Z}/2)$-module by using effectively the cobar type spectral sequence and the Hochschild spectral sequence, and further, by analyzing the TV-model for $BSpin(10)$.
Download or read book Geometry Rigidity and Group Actions written by Robert J. Zimmer and published by University of Chicago Press. This book was released on 2011-04-15 with total page 659 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of group actions is more than 100 years old but remains a widely studied topic in a variety of mathematic fields. A central development in the last 50 years is the phenomenon of rigidity, whereby one can classify actions of certain groups. This book looks at rigidity.
Download or read book A Categorical Approach to Imprimitivity Theorems for C Dynamical Systems written by Siegfried Echterhoff and published by American Mathematical Soc.. This book was released on 2006 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt: It has become apparent that studying the representation theory and structure of crossed-product C*-algebras requires imprimitivity theorems. This monograph shows that the imprimitivity theorem for reduced algebras, Green's imprimitivity theorem for actions of groups, and Mansfield's imprimitivity theorem for coactions of groups can all be understoo
Download or read book Borel Liftings of Borel Sets Some Decidable and Undecidable Statements written by Gabriel Debs and published by American Mathematical Soc.. This book was released on 2007 with total page 134 pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the aims of this work is to investigate some natural properties of Borel sets which are undecidable in $ZFC$. The authors' starting point is the following elementary, though non-trivial result: Consider $X \subset 2omega\times2omega$, set $Y=\pi(X)$, where $\pi$ denotes the canonical projection of $2omega\times2omega$ onto the first factor, and suppose that $(\star)$: Any compact subset of $Y$ is the projection of some compact subset of $X$. If moreover $X$ is $\mathbf{\Pi 0 2$ then $(\star\star)$: The restriction of $\pi$ to some relatively closed subset of $X$ is perfect onto $Y$ it follows that in the present case $Y$ is also $\mathbf{\Pi 0 2$. Notice that the reverse implication $(\star\star)\Rightarrow(\star)$ holds trivially for any $X$ and $Y$. But the implication $(\star)\Rightarrow (\star\star)$ for an arbitrary Borel set $X \subset 2omega\times2omega$ is equivalent to the statement $\forall \alpha\in \omegaomega, \, \aleph 1$ is inaccessible in $L(\alpha)$. More precisely The authors prove that the validity of $(\star)\Rightarrow(\star\star)$ for all $X \in \varSigma0 {1+\xi+1 $, is equivalent to $\aleph \xi \aleph 1$. $ZFC$, derive from $(\star)$ the weaker conclusion that $Y$ is also Borel and of the same Baire class as $X$. This last result solves an old problem about compact covering mappings. In fact these results are closely related to the following general boundedness principle Lift$(X, Y)$: If any compact subset of $Y$ admits a continuous lifting in $X$, then $Y$ admits a continuous lifting in $X$, where by a lifting of $Z\subset \pi(X)$ in $X$ we mean a mapping on $Z$ whose graph is contained in $X$. The main result of this work will give the exact set theoretical strength of this principle depending on the descriptive complexity of $X$ and $Y$. The authors also prove a similar result for a variation of Lift$(X, Y)$ in which continuous liftings are replaced by Borel liftings, and which answers a question of H. Friedman. Among other applications the authors obtain a complete solution to a problem which goes back to Lusin concerning the existence of $\mathbf{\Pi 1 1$ sets with all constituents in some given class $\mathbf{\Gamma $ of Borel sets, improving earlier results by J. Stern and R. Sami. Borel sets (in $ZFC$) of a new type, involving a large amount of abstract algebra. This representation was initially developed for the purposes of this proof, but has several other applications.
Download or read book On Maps from Loop Suspensions to Loop Spaces and the Shuffle Relations on the Cohen Groups written by Jie Wu and published by American Mathematical Soc.. This book was released on 2006 with total page 78 pages. Available in PDF, EPUB and Kindle. Book excerpt: The maps from loop suspensions to loop spaces are investigated using group representations in this article. The shuffle relations on the Cohen groups are given. By using these relations, a universal ring for functorial self maps of double loop spaces of double suspensions is given. Moreover the obstructions to the classical exponent problem in homotopy theory are displayed in the extension groups of the dual of the important symmetric group modules Lie$(n)$, as well as in the top cohomology of the Artin braid groups with coefficients in the top homology of the Artin pure braid groups.
Download or read book Foliations Geometry and Topology written by Nicolau Corção Saldanha and published by American Mathematical Soc.. This book was released on 2009 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents the proceedings of the conference on Foliations, Geometry, and Topology, held August 6-10, 2007, in Rio de Janeiro, Brazil, in honor of the 70th birthday of Paul Schweitzer. The papers focus on the theory of foliations and related areas such as dynamical systems, group actions on low dimensional manifolds, and geometry of hypersurfaces.
Download or read book KAM Stability and Celestial Mechanics written by Alessandra Celletti and published by American Mathematical Soc.. This book was released on 2007 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: KAM theory is a powerful tool apt to prove perpetual stability in Hamiltonian systems, which are a perturbation of integrable ones. The smallness requirements for its applicability are well known to be extremely stringent. A long standing problem, in this context, is the application of KAM theory to ``physical systems'' for ``observable'' values of the perturbation parameters. The authors consider the Restricted, Circular, Planar, Three-Body Problem (RCP3BP), i.e., the problem of studying the planar motions of a small body subject to the gravitational attraction of two primary bodies revolving on circular Keplerian orbits (which are assumed not to be influenced by the small body). When the mass ratio of the two primary bodies is small, the RCP3BP is described by a nearly-integrable Hamiltonian system with two degrees of freedom; in a region of phase space corresponding to nearly elliptical motions with non-small eccentricities, the system is well described by Delaunay variables. The Sun-Jupiter observed motion is nearly circular and an asteroid of the Asteroidal belt may be assumed not to influence the Sun-Jupiter motion. The Jupiter-Sun mass ratio is slightly less than 1/1000. The authors consider the motion of the asteroid 12 Victoria taking into account only the Sun-Jupiter gravitational attraction regarding such a system as a prototype of a RCP3BP. for values of mass ratios up to 1/1000, they prove the existence of two-dimensional KAM tori on a fixed three-dimensional energy level corresponding to the observed energy of the Sun-Jupiter-Victoria system. Such tori trap the evolution of phase points ``close'' to the observed physical data of the Sun-Jupiter-Victoria system. As a consequence, in the RCP3BP description, the motion of Victoria is proven to be forever close to an elliptical motion. The proof is based on: 1) a new iso-energetic KAM theory; 2) an algorithm for computing iso-energetic, approximate Lindstedt series; 3) a computer-aided application of 1)+2) to the Sun-Jupiter-Victoria system. The paper is self-contained but does not include the ($\sim$ 12000 lines) computer programs, which may be obtained by sending an e-mail to one of the authors.
Download or read book Tangential Boundary Stabilization of Navier Stokes Equations written by Viorel Barbu and published by American Mathematical Soc.. This book was released on 2006 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: In order to inject dissipation as to force local exponential stabilization of the steady-state solutions, an Optimal Control Problem (OCP) with a quadratic cost functional over an infinite time-horizon is introduced for the linearized N-S equations. As a result, the same Riccati-based, optimal boundary feedback controller which is obtained in the linearized OCP is then selected and implemented also on the full N-S system. For $d=3$, the OCP falls definitely outside the boundaries of established optimal control theory for parabolic systems with boundary controls, in that the combined index of unboundedness--between the unboundedness of the boundary control operator and the unboundedness of the penalization or observation operator--is strictly larger than $\tfrac{3}{2}$, as expressed in terms of fractional powers of the free-dynamics operator. In contrast, established (and rich) optimal control theory [L-T.2] of boundary control parabolic problems and corresponding algebraic Riccati theory requires a combined index of unboundedness strictly less than 1. An additional preliminary serious difficulty to overcome lies at the outset of the program, in establishing that the present highly non-standard OCP--with the aforementioned high level of unboundedness in control and observation operators and subject, moreover, to the additional constraint that the controllers be pointwise tangential--be non-empty; that is, it satisfies the so-called Finite Cost Condition [L-T.2].
Download or read book Stability of Spherically Symmetric Wave Maps written by Joachim Krieger and published by American Mathematical Soc.. This book was released on 2006 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents a study of Wave Maps from ${\mathbf{R}}^{2+1}$ to the hyperbolic plane ${\mathbf{H}}^{2}$ with smooth compactly supported initial data which are close to smooth spherically symmetric initial data with respect to some $H^{1+\mu}$, $\mu>0$.
