Multiple recurrence and infinite measure preserving odometers

A family of infinite measure preserving odometers is presented which exhibit examples ofp-recurrent but notp+1-recurrent ergodic transformations for everyp>1.     

Spectral multiplicities of infinite measure preserving transformations

Each set E ⊂ ℕ is realized as the set of essential values of the multiplicity function of the Koopman operator for an ergodic conservative infinite measure preserving transformation.     

Measure preserving transformations similar to Markov shifts

Similarity, that is, the existence of joint common extensions, defines an interesting equivalence relation for infinite measure preserving transformations T. We provide a sufficient condition, given in terms of return processes to reference sets of finite measure, for T to be similar to a Markov shift. This is then shown to apply to various piecewise smooth dynamical systems, including weakly hyperbolic transformations with indifferent periodic points or flat critical points, and discrete random walks driven by (weakly) hyperbolic maps.     

Isometric extensions and multiple recurrence of infinite measure preserving systems

Furstenberg’s multiple recurrence theorem is investigated for infinite measure preserving systems arising from isometric extensions which are 1-factor maps. It will be shown that any isometric extension preservesd-recurrence for alld≥1 and multiple recurrence.     

On the asymptotics of a 1-parameter family of infinite measure preserving transformations

We estimate various aspects of the growth rates of ergodic sums for some infinite measure preserving transformations which are not rationally ergodic.Dedicated to R. Mañé     

Models for measure preserving transformations

Many recent results about the classification problem for ergodic measure preserving transformations involve global considerations about spaces of measure preserving transformations. This paper surveys recent joint work with Dan Rudolph and Benjamin Weiss in determining when various spaces of measure preserving transformations are equivalent in the sense of conjugacy preserving Borel isomorphism and in having the same generic dynamical properties.     

Hyperbolic structure preserving isomorphisms of Markov shifts

In this paper we consider metric isomorphisms of Markov shifts which are also isomorphisms of the hyperbolic structures of the shift spaces. We prove that such isomorphisms need not be finitary, and that finitary isomorphisms need not preserve the hyperbolic structures unless they have finite expected code lengths. In particular we show that certain explicity computable invariants previously associated with finitary isomorphisms with finite expected code lengths are, in fact, invariants of the hyperbolic structure of the Markov shifts.     

Infinite measure preserving transformations with Radon MSJ

We introduce concepts of Radon MSJ and Radon disjointness for infinite Radon measure preserving homeomorphisms of the locally compact Cantor space. We construct an uncountable family of pairwise Radon disjoint infinite Chacon like transformations. Every such transformation is Radon strictly ergodic, totally ergodic, asymmetric (not isomorphic to its inverse), has Radon MSJ and possesses Radon joinings whose ergodic components are not joinings.     

Generic distributional limits for measure preserving transformations

We construct a conservative ergodic transformation of the real line whose normalised Birkhoff sums are distributionally generic. This proves that distributional genericity is itself generic.     

The intrinsic normalising constants of transformations preserving infinite measures

We consider the collection of normalisations of a c.e.m.p.t. inside other c.e.m.p.t.s of which it is a factor. This forms an analytic, multiplicative subgroup ofR ₊. The groups corresponding to similar c.e.m.p.t.s coincide. “Usually” this group is {1}. Examples are given where the group is:R ₊, any countable subgroup ofR ₊, and also an uncountable subgroup ofR ₊ of any Haussdorff dimension. These latter groups are achieved by c.e.m.p.t.s which are not similar to their inverses. Research partially supported by NSERC grants A 8815 and A 3974.     

The ergodic infinite measure preserving transformation of boole

G. Boole proved that the transformation φ of the real line, defined by φ(x)=x−1/x, preserves Lebesgue measure. A general method is applied to proving that φ is ergodic. Some further applications of the method are also indicated.     

Hyperbolic structure preserving isomorphisms of Markov shifts. II

This paper is a continuation of [14] and deals with metric isomorphisms of Markov shifts which are finitary and hyperbolic structure preserving. We prove that theβ-function introduced by S. Tuncel in [15] is an invariant of such isomorphisms. Following [5] this result is extended to Gibbs measures arising from functions with summable variation. Finally we prove that, for anyC ² Axiom A diffeomorphism on a basic set Ω, and for any equilibrium state associated with a Hölder continuous function on Ω, the Markov shifts arising from different Markov partitions of Ω are isomorphic via a finitary, hyperbolic structure preserving isomorphism. This fact leads to a rich class of examples of such isomorphisms (other examples are provided by finitary isomorphisms of Markov shifts with finite expected code lengths — cf. [14]).     

Infinite measure preserving transformations with compact first regeneration

We study ergodic infinite measure preserving transformations T possessing reference sets of finite measure for which the set of densities of the conditional distributions given a first return (or entrance) at time n is precompact in a suitable function space. Assuming regular variation of wandering rates, we establish versions of the Darling-Kac theorem and the arcsine laws for waiting times and for occupation times which apply to transformations with indifferent orbits and to random walks driven by Gibbs-Markov maps. This research was supported by an APART [Austrian programme for advanced research and technology] fellowship of the Austrian Academy of Sciences. Much of this work was done at the Mathematics Department of Imperial College London. I also benefitted from a JRF at the ESI in Vienna.     

On the pointwise ergodic behaviour of transformations preserving infinite measures

We consider situations in which the asymptotic type of a measure preserving transformation manifests itself in a pointwise manner.     

The Dynkin-Lamperti arc-sine laws for measure preserving transformations

Arc-sine laws in the sense of renewal theory are proved for return time processes generated by transformations with infinite invariant measure on sets satisfying a type of Darling-Kac condition, and an application to real transformations with indifferent fixed points is discussed.