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     

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.     

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.     

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.     

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.     

Applying projective geometry to transformations on rank one idempotents

We present a short proof of Molnár's characterization of bijective transformations on the set of all rank one idempotent operators on a Banach space which preserve zero products in both directions. An improvement in the finite-dimensional case is given. We apply these results to describe automorphisms of standard operator semigroups and to improve Uhlhorn's version of Wigner's theorem.     

Transformations preserving the Hausdorff-Besicovitch dimension

Continuous transformations preserving the Hausdorff-Besicovitch dimension (“DP-transformations”) of every subset of R ¹ resp. [0, 1] are studied. A class of distribution functions of random variables with independent s-adic digits is analyzed. Necessary and sufficient conditions for dimension preservation under functions which are distribution functions of random variables with independent s-adic digits are found. In particular, it is proven that any strictly increasing absolutely continuous distribution function from the above class is a DP-function. Relations between the entropy of probability distributions, their Hausdorff-Besicovitch dimension and their DP-properties are discussed. Examples are given of singular distribution functions preserving the fractal dimension and of strictly increasing absolutely continuous functions which do not belong to the DP-class.      

A large set containing few orbits of measure preserving transformations

Summary There exists a Borel set C of product Lebesgue measure one in the Hilbert cube having the property that, for every measure preserving transformationT of the unit interval, allT-orbits contained inC originate from a zero set. This settles an infinite dimensional version of a problem raised by Th. M. Rassias.     

Normality preserving multiplication operators

We show that a multiplication operator (T)=ATB is normality preserving if and only if it is hyponormality preserving, if and only if it is either of the formA=fg,B=h f, orA=D,B=D* for someC andD* D=I. Also we show that is (semi-) Fredholmness prserving if and only ifA andB are (semi-) Fredholm operators.Supported by the Science Foundation of Zhejiang Province and NSF.     

Multiple recurrence of Markov shifts and other infinite measure preserving transformations

We discuss the concept of multiple recurrence, considering an ergodic version of a conjecture of Erdős. This conjecture applies to infinite measure preserving transformations. We prove a result stronger than the ergodic conjecture for the class of Markov shifts and show by example that our stronger result is not true for all measure preserving transformations.