Hyperbolic manifolds, discrete groups and ergodic theory

Hyperbolic manifolds , discrete groups and ergodic theory Course Notes
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Graev , I. Ergodic theory and topological dynamics of group actions on homogeneous spaces Mohammed El Bachir Bekka , Matthias Mayer. Oppenheim conjecture. Interactions between ergodic theory , Lie groups , number theory. On the space of ergodic invariant measures of unipotent flows Shahar Mozes , Nimish A. Discrete Groups. Amenable coverings of complex manifolds and holomorphic probability measures Curtis T.

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Feb 18, 14 Ergodic theory at infinity of hyperbolic manifolds. .. representations of discrete groups like SLn(Z), which is a facet of ergodic theory since it. 1 Ergodic theory References for this section: CFS]. 1. The basic setting of ergodic theory: a measure-preserving transformation T of a probability space (X; B; m).

It was answered in the negative by Ol'shanskii in the s. The measurable version formulated by Gaboriau-Lyons asks whether every non-amenable measured equivalence relation contains a non-amenable treeable subequivalence relation. This paper obtains a positive answer in the case of arbitrary Bernoulli shifts over a non-amenable group, extending work of Gaboriau-Lyons. The proof uses an approximation to the random interlacement process by random multistep of geometrically-killed random walk paths.

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There are two applications: 1 the Gaboriau-Lyons problem for actions with positive Rokhlin entropy admits a positive solution, 2 for any non-amenable group, all Bernoulli shifts factor onto each other. All properly ergodic Markov chains over a free group are orbit equivalent Previous work showed that all Bernoulli shifts over a free group are orbit-equivalent. This result is strengthened here by replacing Bernoulli shifts with the wider class of properly ergodic countable state Markov chains over a free group.

A list of related open problems is provided. Examples in the entropy theory of countable group actions Kolmogorov-Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups.

These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.

Ergodic Properties of the Horocycle Flow and Classification of Fuchsian Groups

Here this is extended to nonamenable groups. In fact, the proof shows that every action is a factor of a zero entropy action! This uses the strange phenomena that in the presence of nonamenability, entropy can increase under a factor map. The proof uses Seward's recent generalization of Sinai's Factor Theorem, the Gaboriau-Lyons result and my theorem that for every nonabelian free group, all Bernoulli shifts factor onto each other. Entropy 18 , no. Integrable orbit equivalence rigidity for free groups It is shown that every accessible group which is integrable orbit equivalent to a free group is virtually free.

Moreover, we also show that any integrable orbit-equivalence between finitely generated groups extends to their end compactifications. Israel J. Hyperbolic geometry and pointwise ergodic theorems We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real-rank-one, going well beyond the radial case considered previously.

Entropies for negatively curved compact manifolds (Lecture 3) by Lin Shu

The proof is based on a new approach to pointwise ergodic theorems, which is independent of spectral theory. Instead, the main new ingredient is the use of direct geometric arguments in hyperbolic space. Moreover, we provide conditions which imply that this holds for any non-trivial probability space K. Second, we use this result to prove that any non-amenable unimodular locally compact second countable group admits uncountably many free ergodic pmp actions which are pairwise not von Neumann equivalent hence, pairwise not orbit equivalent.

Groups Geom. Lastly, we provide a simple characterization of normality for subequivalence relations and an algebraic description of the quotient. Equivalence relations that act on bundles of hyperbolic spaces Consider a measured equivalence relation acting on a bundle of hyperbolic metric spaces by isometries.

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We classify elements of the full group according to their action on fields on boundary measures extending earlier results of Kaimanovich , study the existence and residuality of different types of elements and obtain an analogue of Tits' alternative. Ergodic Theory Dynam. Mean convergence of Markovian spherical averages for measure-preserving actions of the free group Mean convergence of Markovian spherical averages is established for a measure-preserving action of a finitely-generated free group on a probability space.

Equivalently, we establish the triviality of the tail sigma-algebra of the corresponding Markov operator. This convergence was previously known only for symmetric Markov chains, while the conditions ensuring convergence in our paper are inequalities rather than equalities, so mean convergence of spherical averages is established for a much larger class of Markov chains.

Dedicata , — Property T and the Furstenberg entropy of nonsingular actions We establish a new characterization of property T in terms of the Furstenberg entropy of nonsingular actions. We show that this is also a sufficient condition. We furthermore show that if the action of G on its Poisson boundary is essentially free then a generic measure is isomorphic to the Poisson boundary. Next, we consider the space of stationary actions, equipped with a standard topology known as the weak topology. Here we show that when G has property T , the ergodic actions are meager. Finally, for a weaker topology on the set of actions, which we call the very weak topology, we show that a dynamical property e.

There we also show a Glasner-King type law stating that every dynamical property is either meager or residual. We also provide a complete classification of characteristic random subgroups of free abelian groups of countably infinite rank and elementary p-groups of countably infinite rank.

L1-measure equivalence and group growth Appendix to: Integrable measure equivalence for groups of polynomial growth by Tim Austin We prove that group growth is an L1-measure equivalence invariant. Weak density of orbit equivalence classes of free group actions It is proven that the orbit-equivalence class of any essentially free probability-measure-preserving action of a free group G is weakly dense in the space of actions of G. Duke Math. Under an additional condition, satisfied for example by all groups acting isometrically and properly discontinuously on CAT -1 spaces, we prove a pointwise ergodic theorem with respect to a sequence of probability measures supported on concentric spherical shells.

Amenable equivalence relations and the construction of ergodic averages for group actions We present a general new method for constructing pointwise ergodic sequences on countable groups, which is applicable to amenable as well as to non-amenable groups and treats both cases on an equal footing.

The principle underlying the method is that both cases can be viewed as instances of the general ergodic theory of amenable equivalence relations. Entropy theory for sofic groupoids I: the foundations This is the first part in a series in which sofic entropy theory is generalized to class-bijective extensions of sofic groupoids. Here we define topological and measure entropy and prove invariance. We also establish the variational principle, compute the entropy of Bernoulli shift actions and answer a question of Benjy Weiss pertaining to the isomorphism problem for non-free Bernoulli shifts.

The proofs are independent of previous literature. This concept is crucially important for the identification of the limit in pointwise ergodic theorems established by the author and Amos Nevo. This is applied to show that the action of a non-elementary Gromov hyperbolic group on its boundary with respect to a quasi-conformal measure is not type III0 and, if it is weakly mixing, then it is not stable type III0. Geometriae Dedicata, October , Volume , Issue 1, pp A horospherical ratio ergodic theorem for actions of free groups We prove a ratio ergodic theorem for amenable equivalence relations satisfying a strong form of the Besicovich covering property.

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We then use this result to study general non-singular actions of non-abelian free groups and establish a ratio ergodic theorem for averages along horospheres. We show that this simplex has a canonical Poulsen subsimplex whose complement has only a countable number of extreme points. Invariant random subgroups of the free group Let G be a locally compact group. We show that each nonabelian free group has a large "zoo" of IRS's.

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This contrasts with results of Stuck-Zimmer which show that there are no non-obvious IRS's of higher rank semisimple Lie groups with property T. Thomas generalized the theorem to a skew-product setting. Using L. Bowen's f-invariant we prove the addition theorem for actions of finitely generated free groups on skew-products with compact totally disconnected groups or compact Lie groups correcting an error from [Bo10c] and discuss examples. Systems 34 , no. Harmonic models and spanning forests of residually finite groups We prove a number of identities relating the sofic entropy of a certain class of non-expansive algebraic dynamical systems, the sofic entropy of the Wired Spanning Forest and the tree entropy of Cayley graphs of residually finite groups.

We also show that homoclinic points and periodic points in harmonic models are dense under general conditions. March , Volume , Issue 1, pp Random walks on coset spaces with applications to Furstenberg entropy We determine the range of Furstenberg entropy for stationary ergodic actions of nonabelian free groups by an explicit construction involving random walks on random coset spaces. Volume , Issue 2 , Page Pointwise ergodic theorems beyond amenable groups We prove pointwise and maximal ergodic theorems for probability measure preserving p.

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We show that this class contains all irreducible lattices in connected semisimple Lie groups without compact factors. Our approach is based on the following two principles. First, we show that it is possible to generalize the ergodic theory of p. Second, we show that it is possible to reduce the proof of ergodic theorems for p.

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An important partial result solving those conjectures with an extra assumption of positive entropy was proved by Elon Lindenstrauss , and he was awarded the Fields medal in for this result. Karlin and J. In general the time average and space average may be different. Burger , Horocycle flow on geometrically finite surfaces, Duke Math. In my talk I will describe and prove a property of the entropy of higher rank actions on homogenous spaces. Since the circle is symmetric around 0, it makes sense that the averages of the powers of U will converge to 0.

Geometric covering arguments and ergodic theorems for free groups We present a new approach to the proof of ergodic theorems for actions of free groups based on geometric covering and asymptotic invariance arguments. Our approach can be viewed as a direct generalization of the classical geometric covering and asymptotic invariance arguments used in the ergodic theory of amenable groups.

We use this approach to generalize the existing maximal and pointwise ergodic theorems for free group actions to a large class of geometric averages which were not accessible by previous techniques.

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Some applications of our approach to other groups and other problems in ergodic theory are also briefly discussed. Dynamical systems and group actions, 67—78, Contemp.