Williams Ergodic Theory Conference

Williams College

Friday, July 27 - Sunday, July 29, 2012

All talks will be in room Bronfman 106 of the Bronfman Science Center on the Williams College campus.

Jon Aaronson (Tel Aviv University, Israel):
Rational weak mixing in infinite measure spaces

Abstract: Motivated by an example of E. Hopf (1937), K. Krickeberg (1967) introduced a topological ratio mixing property which is enjoyed by Markov shifts with the strong ratio limit property and many infinite dynamical systems "of non-uniformly hyperbolic, smooth origin".

The theory of weakly wandering sets as developed by Hajian and Kakutani (1964) shows that there is no measure theoretic version of this ratio mixing property.

Nevertheless, many of these systems are "rationally weak mixing" in a sense which implies that for every pair of measurable sets in a hereditary ring, ratio mixing is satisfied along a subsequence of density one.

The property is meagre in the group of measure preserving transformations of R, but the power version of the subsequence property is generic.

Consideration of rationally weakly mixing Markov shifts leads to some questions about aperiodic renewal sequences.

Ethan Akin (The City College, New York, USA):
Homeomorphisms of the Cantor Set

Abstract: Let X be the Cantor set and H be the group of homeomorphisms on X. H is a Polish group which acts on itself by inner automorphism. It is by now well known that this action is topologically transitive. In fact, there exists a dense G_delta orbit. While reviewing a paper by Shimomura I noticed that his results describe the two subsets of H which are minimal subsets for this action. One is the fixed point at the identity. The other is the set of homeomorphisms h such that the action of h on X is chain mixing and admits a fixed point in X.

Rachel Bayless (University of North Carolina, Chapel Hill, USA):
Negative Generalized Boole Transformations and Krengel Entropy

Abstract: The generalized Boole transformations have the form T(x) = x + B + sum_{n=0}^{N} p_{n}/(t_{n}-x), p_{n} >= 0, and all such T preserve Lebesgue measure. If B = 0, then T is ergodic (Li - Schweiger) and exact (Aaronson). We show that the negative function, -T, is exact for all real B. Furthermore, we compute the Krengel entropy of a specific parametrized family of quadratic rational maps which are negative generalized Boole transformations.

Darren Creutz (Vanderbilt University, USA):
Stabilizers of Ergodic Actions of Lattices and Commensurators

Abstract: The Margulis Normal Subgroup Theorem states that any normal subgroup of an irreducible lattice in a center-free higher-rank semisimple Lie group is of finite index. Stuck and Zimmer, expanding on Margulis' approach, showed that any properly ergodic probability-preserving ergodic action of such a lattice is essentially free.

I will present similar results: my work with Y. Shalom on normal subgroups of lattices in products of simple locally compact groups and normal subgroups of commensurators of lattices, and my work with J. Peterson generalizing this result to stabilizers of ergodic probability-preserving actions of such groups. As a consequence, S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) as do lattices in certain product groups. In particular, any nontrivial ergodic probability-preserving action of $\mathrm{PSL}_{n}(\mathbb{Q})$, for $n \geq 3$, is essentially free.

The key idea in the study of normal subgroups is considering nonsingular actions which are the extreme opposite of measure-preserving. Somewhat surprisingly, the key idea in understanding stabilizers of probability-preserving actions also involves studying such actions and the bulk of our work is directed towards properties of these contractive, or SAT, actions.

Alexandre Danilenko (Institute for Low Temperature Physics, Ukraine):
Odometer actions of Heisenberg group

Abstract: Odometer actions of a 3-dimensional real Heisenberg group are considered. It is shown that non-degenerated odometer actions are isospectral, i.e. such actions are isomorphic whenever the corresponding Koopman unitary representations are unitarily equivalent. The decomposition of Heisenberg odometers into irreducible unitary representations is explicitly described (in terms of the underlying nested sequence of lattices). (Joint work with M.Leman'czyk.)

Kelly Funk (University of Illinois, Urbana Champaign, USA):
Generic Homeomorphisms

Abstract: In this talk I will discuss how to produce generic homeomorphisms of the two torus and the Klein bottle that are weakly mixing and uniformly rigid. We will examine the different approaches for each compact manifold and why they are different.

Jane Hawkins (University of North Carolina, Chapel Hill, USA):
Complex Dynamics and Infinite Ergodic Theory

Abstract: In complex dynamics the focus is usually on the unique probability measure of maximal entropy. However this measure is almost always singular with respect to a more natural geometric measure (called a conformal measure) supported on the Julia set. It was shown by Aaronson, Denker, and Urbanski that in the presence of a rationally neutral fixed point there is often an invariant ergodic infinite sigma-finite measure equivalent to this conformal measure for rational maps. There is one family of such maps related to inner functions, but some have fractal Julia sets. Computing the Hausdorff dimension becomes important in determining the finiteness of the measure. We give an overview of this topic, mentioning several parametrized families of these maps and some of their properties.

Yuri Karlovich (Universidad Autónoma del Estado de Morelos, Mexico):
Ergodic number theory and AP factorization of almost periodic matrix functions

Abstract: The existence and explicit construction of canonical almost periodic ($AP$) factorizations for triangular almost periodic matrix functions is closely related to the location of the Fourier spectra and to the values of Fourier coefficients of the matrix entries [1]. Applying ergodic theorems for the fractional parts of irrational numbers, we study such factorization problems for some classes of triangular almost periodic matrix functions (cf. [2], [3]). These results are applied to studying the invertibility of corresponding Toeplitz and Wiener-Hopf operators (see [1]).

[1] A. B\"ottcher, Yu. I. Karlovich, and I. M. Spitkovsky: Convolution Operators and Factorization of Almost Periodic Matrix Functions. Birkh\"auser, Basel, 2002.

[2] M. C. C\^{a}mara, Yu. I. Karlovich, and I. M. Spitkovsky: Constructive almost periodic factorization of some triangular matrix functions. J. Math. Anal. Appl. {\bf 367} (2010), 416--433.

[3] M. A. Bastos, A. Bravo, Yu. I. Karlovich, and I.M. Spitkovsky: Constructive factorization of some almost periodic triangular matrix functions with a quadrinomial off diagonal entry. J. Math. Anal. Appl. {\bf 376} (2011), 625--640.

Jared Hallett* (Williams College):
On Li-Yorke Measurable Sensitivity

Abstract: The notion of Li-Yorke sensitivity has been studied extensively in the case of topological dynamical systems. We introduce a measurable version of Li-Yorke sensitivity, for nonsingular (and measure-preserving) dynamical systems, and compare it with various mixing notions. It is known that in the case of nonsingular dynamical systems, ergodic Cartesian square implies double ergodicity, which in turn implies weak mixing, but the converses do not hold in general, though they are all equivalent in the finite measure-preserving case. We show that for nonsingular systems, ergodic Cartesian square implies Li-Yorke measurable sensitivity, which in turn implies weak mixing. As a consequence we obtain that, in the finite measure-preserving case, Li-Yorke measurable sensitivity is equivalent to weak mixing. We also show that with respect to totally bounded metrics, double ergodicity implies Li-Yorke measurable sensitivity. Joint work with L. Manuelli and C. Silva.

Zemer Kosloff (Tel Aviv University, Israel):
Maharam extensions of nonsingular Bernoulli shifts

Abstract: The Maharam extension of a non singular transformation is a R- extension via the Radon Nykodym derivative cocycle. It plays a crucial role in the study of non singular measurable transformations which don't admit an a.c.i.m. (absolutely continuous invariant measure). We show that for a conservative, non singular, Bernoulli shift which satisfies the K-property the Maharam extension is a K-transformation. As a corollary we show that the only K-Bernoulli shifts with an a.c.i.m. are those with an equivalent stationary product measure.

Mariusz Lemanczyk (Nicolaus Copernicus University, Poland):
Self-joinings of higher order in the problem of non-reversibility of ergodic flows

Abstract: We will deal with the problem of isomorphism/non-isomorphism of a flow with the flow obtained by the time reversed. I will present basic examples illustrating both possibilities and some introductory criteria (including special flows, or more generally, Mackey actions) By studying the weak closure of multidimensional off-diagonal self-joinings we provide our main criteria for non-isomorphism of a flow with its inverse. This is applied to special flows over rigid automorphisms. In particular, our criteria are applied to large classes of special flows over irrational rotations (including those of smooth origin) A topological counterpart is also developed with the full solution of the problem of the topological self-similarity of continuous special flows over irrational rotations. This yields examples of continuous special flows over irrational rotations without topological self-similarities and having all non-zero real numbers as scales of measure-theoretic self-similarities. The talk is based on a my joint work with K. Fraczek and J. Kulaga.

Tudor Pădurarlu* (University of California, Los Angeles):
On Rationally Ergodic and Rationally Weakly Mixing Rank-One Transformations

Abstract: We study Aaronson's notions of weak rational ergodicity and rational weak mixing in the context of rank-one transformations. We prove that various families of infinite measure-preserving rank-one transformations possess (or do not posses) these properties, and also relate them to other notions of mixing in infinite measure. Joint work with I. Dai, X. Garcia, and C. Silva.

Kyewon Park (Ajou University, South Korea):
Entropy dimension: examples

Raj Prasad (University of Massachusetts Lowell, USA):
Tilings of the integers arising from a class of infinite measure preserving transformations

Abstract: There are many infinite sequences of integers associated to an infinite measure preserving transformation: weakly wandering sequences, exhaustive weakly wandering (EWW) sequences and strongly weakly wandering sequences. EWW sequences give rise to an infinite tiling of the integers. We consider a class of ergodic transformations of an infinite measure space with no recurrent sequences (this class includes simple random walk on the integers) and study their hereditary tilings of the integers. This is joint with Eigen, Hajian, and Keane.

Emmanuel Roy (University of Paris 13, France):
Prime Poisson suspensions

Abstract: A Poisson suspension is a probability measure preserving transformation built from an infinite measure preserving one. In this talk, we give a criterion for a Poisson suspension to be "prime" (i.e. whose the only factors are the trivial ones). We then show it applies to some non-singular compact group rotations and to some infinite measure preserving rank one "mixing" transformations, yielding prime Poisson suspensions that are, moreover, very different from prime transformations known to date. Joint work with François Parreau.

Ilya Vinogradov (University of Paris 13, France):
Effective bisector estimate for PSL(2,C) with applications to circle packings

Abstract: Effective bisector estimate for PSL(2,C) with applications to circle packings Abstract: Let Gamma be a non-elementary discrete geometrically finite subgroup of PSL(2,C). Under the assumption that the critical exponent of Gamma is greater than 1 we prove an effective bisector counting theorem for Gamma. We then apply this Theorem to the Apollonian circle packing problem to get power savings and to compute the overall constant. The proof relies on spectral theory of Gamma\PSL(2,C).

(* undergraduate student)