Abelian cocycles for nonsingular ergodic transformations and the genericity of type III1 transformations

The authors prove that in the space of nonsingular transformations of a Lebesgue probability space the type III₁ ergodic transformations form a denseG set with respect to the coarse topology. They also prove that for any locally compact second countable abelian groupH, and any ergodic type III transformationT, it is generic in the space ofH-valued cocycles for the integer action given byT that the skew product ofT with the cocycle is orbit equivalent toT. Similar results are given for ergodic measure-preserving transformations as well.Research supported in part by: Nat. Sci. and Eng. Res. Council #A7163 and # U0080 F.C.A.C. Quebec, NSF Grants # MCS-8102399 and # DMS-8418431.     

Graded Steinberg algebras and partial actions

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space which is totally disconnected, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{?}$-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given.     

Contributions to a Rigidity Conjecture

Let p and q be two relatively prime positive integers and a Borel probability measure invariant and ergodic by the semigroup generated by the action of both zp and zq. We analyse sufficient conditions to guarantee that is either the Lebesgue measure or supported on a periodic orbit. And extend the results for general expanding differentiable maps of the circle.     

Skew products over rotations with exotic properties

We show that a $\text{ Z}_{2}$ skew product of a badly approximable rotation can be minimal and not uniquely ergodic. This construction is used to construct a Z skew product of a rotation where the orbit of a.e. point is dense but Lebesgue measure is not ergodic.     

Continuity and Representation of Gaussian Mehler Semigroups

We present sufficient conditions on a Gaussian Mehler semigroup on a reflexive Banach space Eto be induced by a single positive symmetric operator Q \in , and give a counterexample which shows that this representation theorem is false in every nonreflexive Banach space with a Schauder basis. We also show that the transition semigroup of a Gaussian Mehler semigroup on a separable Banach space Eacts in a pointwise continuous way, uniformly on compact subsets of E, in the space BUC(E) of bounded uniformly continuous real-valued funtions on E. The transition semigroup is shown to be strongly continuous on BUC(E) if and only if S(t) = Ifor all t 0     

Ergodic theorems for spatial processes

Summary We investigate the ergodic properties of spatial processes, i.e. stochastic processes with an index set of bounded Borel subsets in ^v, and prove mean and individual ergodic theorems for them. As important consequences we get a generalization of McMillan's theorem due to Fritz [4]; the existence of specific energy for a large class of interactions in the case of marked point processes in ^v and the existence of the specific Minkowski Quermaßintegrals for Boolean models in ^v with convex, compact grains.Dedicated to Klaus Krickeberg on the occasion of his 50th birthday     

Noncommutative maximal ergodic theorems

This paper is devoted to the study of various maximal ergodic theorems in noncommutative -spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on and the analogue of Stein's maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in von Neumann algebra theory and in quantum probability.

     

非Lipschitz仿射映射的半拓扑半群的强遍历定理(英)

本文在Banach空间中给出了非Lipschitzian仿信射拓扑半群的强遍历定理。     

On bounded solutions to convolution equations

Periodicity of bounded solutions for convolution equations on a separable abelian metric group is established, and related Liouville type theorems are obtained. A non-constant Borel and bounded harmonic function is constructed for an arbitrary convolution semigroup on any infinite-dimensional separable Hilbert space, generalizing a classical result by Goodman (1973).

     

Ergodic theorems for noncommutative dynamical systems

Recently, E.C. Lance extended the pointwise ergodic theorem to actions of the group of integers on von Neumann algebras. Our purpose is to extend other pointwise ergodic theorems to von Neumann algebra context: the Dunford-Schwartz-Zygmund pointwise ergodic theorem, the pointwise ergodic theorem for connected amenable locally compact groups, the Wiener's local ergodic theorem for ₊ ^d and for general Lie groups.     

Ergodic properties of skew products in infinite measure

Let (Ω, µ) be a shift of finite type with a Markov probability, and (Y, ν) a non-atomic standard measure space. For each symbol i of the symbolic space, let Φ_i be a non-singular automorphism of (Y, ν). We study skew products of the form (ω, y) ? (σω, Φ_ω0 (y)), where σ is the shift map on (Ω, µ). We prove that, when the skew product is recurrent, it is ergodic if and only if the Φ_i’s have no common non-trivial invariant set.     

Strong ergodic theorem for commutative semigroup of non-Lipschitzian mappings in multi-Banach space

Let C be a bounded closed convex subset of a uniformly convex multi-Banach space X and let \({\mathfrak {I}}_{j} = \{T_j(t) : t\in G\}\) be a commutative semigroup of asymptotically nonexpansive in the intermediate mapping from C into itself. In this paper, we prove the strong mean ergodic convergence theorem for the almost-orbit of \(\mathfrak {I}\). Our results extend and unify many previously known results especially (Dong et al. On the strong ergodic theorem for commutative semigroup of non-Lipschitzian mappings in Banach space, preprint).     

An Ergodic Decomposition Defined by Regular Jointly Measurable Markov Semigroups on Polish Spaces

For a regular jointly measurable Markov semigroup on the space of finite Borel measures on a Polish space we give a Yosida-type decomposition of the state space, which yields a parametrisation of the ergodic probability measures associated to this semigroup in terms of subsets of the state space. In this way we extend results by Costa and Dufour (J. Appl. Probab. 43:767?C781, 2006). As a consequence we obtain an integral decomposition of every invariant probability measure in terms of the ergodic probability measures. Our approach is completely centered around the reduction to and relationship with the case of a single regular Markov operator associated to the Markov semigroup, the resolvent operator, which enables us to fully exploit results in that situation (Worm and Hille in Ergod. Theory Dyn. Syst. 31(2):571?C597, 2011).     

Measure Perturbations of Markovian Semigroups

The perturbation of the semigroup of a Borel right process by a class of signed measures on the state space of the process is studied. The perturbation is defined by a Feynman–Kac functional associated with the measure. Under appropriate conditions the perturbed semigroup is strongly continuous in L^p(m), 1 p< where m is a fixed excessive measure. Both existence and uniqueness of the associated Schrödinger type equation are investigated.     

Spectral flow and Dixmier traces

We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the heat operator in a general semi-finite von Neumann algebra. Our results have several applications. We deduce a formula for the Chern character of an odd -summable Breuer-Fredholm module in terms of a Hochschild 1-cycle. We explain how to derive a Wodzicki residue for pseudo-differential operators along the orbits of an ergodic action on a compact space X. Finally, we give a short proof of an index theorem of Lesch for generalised Toeplitz operators.     

A converse to Dye's theorem

Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not universal for treeable in the ordering.

     

On submultiplicativity of spectral radius and transitivity of semigroups

It is shown that a transitive, closed, homogeneous semigroup of linear transformations on a finite-dimensional space either has zero divisors or is simultaneously similar to a group consisting of scalar multiples of unitary transformations. The proof begins with the result that for each closed homogeneous semigroup with no zero divisors there is a such that the spectral radius satisfies for all and in the semigroup.

It is also shown that the spectral radius is not -submultiplicative on any transitive semigroup of compact operators.

     

Ergodic Theorems and Ergodic Decomposition for Markov Chains

This paper considers Markov chains on a locally compact separable metricspace, which have an invariant probability measure but with no otherassumption on the transition kernel. Within this context, the limit providedby several ergodic theorems is explicitly identified in terms of the limitof the expected occupation measures. We also extend Yosidasergodic decomposition for Feller-like kernels to arbitrarykernels, and present ergodic results for empirical occupation measures, aswell as for additive-noise systems.     

The Perron-Frobenius theory for almost periodic representations of semigroups in the spaces Lp

The classical Perron-Frobenius theory of nonnegative matrices is generalized to nonnegative almost periodic representations of topological semigroups in the spaces L_p(, , ), where (, , ) is a space with a -finite measure, 1p<. With each such representation one connects the associated action of its Sushkevich kernel onto some naturally arising space with measure; this allows that the investigation of the spectral properties of the representation be reduced to the investigation of the ergodic properties of the corresponding action. In particular, it is established, that the boundary spectrum of an indecomposable representation is a subgroup of the dual group of the Sushkevich kernel (coincides with it if the considered semigroup is Abelian). In the general case the boundary spectrum is cyclic (i.e., the union of the subgroups of the dual group of the Sushkevich kernel). The results of the paper are new even at the consideration of the semigroups of degree one of an operator (if other words, the representations of the semigroup Z₊); this yields the generalized Perron-Frobenius theory for nonnegative a. p. operators.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 49, pp. 35–42, 1988.I express my gratitude to Yu. I. Lyubich for valuable discussions of the paper.     

Invariant cocycles, random tilings and the super- and strong Markov properties

We consider -cocycles with values in locally compact, second countable abelian groups on discrete, nonsingular, ergodic equivalence relations. If such a cocycle is invariant under certain automorphisms of these relations, we show that the skew product extension defined by the cocycle is ergodic. As an application we obtain an extension of many recent results of the author and K. Petersen to higher-dimensional shifts of finite type, and prove a transitivity result concerning rearrangements of certain random tilings.