This is the new website for ergodic theory at Ohio State. It is under construction.
Our final two seminars for the Fall 2013 semester are:
14th November: Joe Rosenblatt (UIUC)
21st November: Ronnie Pavlov (Denver)
Feb 21 2013 – 3:00pm – 4:00 pm
Donald Robertson (Ohio State University)
A coarse description of sub-sigma-algebras that characterize the behavior of averages arising from commuting actions of an amenable group will be given. The averages generalize those used in the ergodic proof of Szemeredi’s theorem. The description will be used to prove an ergodic version of Roth’s theorem for amenable groups.
Nov 29 2012 – 3:00pm – 3:50 pm
Paul Reschke (UIC)
By a theorem due to Gromov and Yomdin, the entropy of an endomorphism of a compact Kahler manifold is determined by its cohomological actions. For the case of automorphisms of compact Kahler surfaces, I will explain how the cohomological interpretation of entropy leads to characterizations of Salem numbers and I will present a variety of examples of such characterizations. I will then discuss other applications of the cohomological interpretation of entropy to dynamical questions on Kahler manifolds.
Oct 18 2012 – 3:00pm – 3:50 pm
Joe Rosenblatt (University of Illinois, Department of Mathematics)
Directional ergodicity and directional weak mixing of the action of two commuting transformations S and T can be analyzed by looking at extensions in which S and T are embedded in a two real variable flow. For a suitable class of extensions, the directional behavior observed is determined not by the extension itself, but by intrinsic spectral properties of the original action by S and T.
Oct 25 2012 – 3:00pm – 3:50 pm
David Koslicki (Mathematical Biosciences Institute, OSU)
Inspired by concepts from ergodic theory, we give new insight into coding sequence (CDS) density estimation for the human genome. Our approach is based on the introduction and study of topological pressure: a numerical quantity assigned to any finite sequence based on an appropriate notion of ‘weighted information content’. For human DNA sequences, each codon is assigned a suitable weight, and using a window size of approximately 60,000bp, we obtain a very strong positive correlation between CDS density and topological pressure. The weights are selected by an optimization procedure, and can be interpreted as quantitative data on the relative importance of different codons for the density estimation of coding sequences. This gives new insight into codon usage bias which is an important subject where long standing questions remain open. Inspired again by ergodic theory, we use the weightings on the codons to define a probability measure on finite sequences. We demonstrate that this measure is effective in distinguishing between coding and non-coding human DNA sequences of lengths approximately 5,000bp. We emphasize that topological pressure is a flexible tool and we expect it to be useful for the investigation of many other features of DNA sequences such as interspecies comparison of codon usage bias. We give a first result in this direction, investigating CDS density in the mouse genome and comparing our results with those for the human genome. This is joint work with Dan Thompson (OSU).
Nov 8 2012 – 3:00pm – 3:50 pm
Yunping Jiang (CUNY)
In the study of modern dynamical systems, an invariant measure is an important topic and in the study of modern complex analysis, the quasisymmetric condition on a map is an important topic. In this talk, I will combine these two topics together introducing a new interesting topic, symmetric invariant measure. I will discuss the existence and the uniqueness of a symmetric invariant measure. I will also discuss some ergodicity problem for symmetric measures and a related topic, smooth rigidity and symmetric rigidity.
Nov 15 2012 – 3:00pm – 3:50 pm
Ryan Peckner (Princeton)
The squarefree flow is a natural dynamical system whose topological and ergodic properties are closely linked to the behavior of squarefree numbers. We prove that the squarefree flow carries a unique measure of maximal entropy and describe the structure of the associated measure-preserving dynamical system. Our method involves first studying approximations arising from finite collections of prime numbers, then taking a limit under Ornstein’s dˉ-metric in order to consider all primes simultaneously. This is accomplished by proving uniform Gibbs bounds for a sequence of sofic systems and constructing explicit joinings between them in order to directly estimate their dˉ-distances.
Dec 6 2012 – 2:30pm – 3:30 pm
Jayadev Athreya (University of Illinois, Urbana-Champaign)
The Farey sequence F(Q) is the collection of fractions between [0,1] whose denominator (when written in lowest terms) is at most Q. As Q grows, these points become uniformly distributed in the interval, so in some sense, look `random’. However, when you look at the gaps between them, they do not behave like those for uniformly distributed random variables, but instead follow an unusual law known as Hall’s Distribution. We will explain a proof of this result that uses horocycle flow on the space of lattices SL(2,R)/SL(2, Z), and discuss how this picture can be generalized to explicitly computing the gap distribution between directions of saddle connections on Veech surfaces, which we had used computer experiments to approximate. This talk will include elements from joint work with Y. Cheung, joint work with J. Chaika, and joint work with J. Chaika and S. Lelievre.
Feb 14 2013 – 3:00pm – 4:00 pm
Kenichiro Yamamoto (Tokyo Denki University)
Feb 28 2013 – 3:00pm – 4:00 pm
Manfred Denker (The Pennsylvania State University)
Let T be a measure preserving transformation on a probability space. I will present three theorems on the almost sure and weak convergence of sums of the form
The difficulty here arises from the fact that the summands are not well defined as random variables on the probability space. Therefore I will explain how to describe reasonable subspaces of L2 where these variables can be defined a.s.
As a result I will state new ergodic theorems and new central limit theorems obtained from a suitable martingale approximation in the sense of Gordin’s 1968 paper.
Mar 7 2013 – 3:00pm – 4:00 pm
William Mance (University of North Texas)
We show that the set of numbers that are Q-distribution normal but not simply Q-ratio normal has full Hausdorff dimension. We further show under some conditions that countable intersections of sets of this form still have full Hausdorff dimension even though they are not winning sets (in the sense of W. Schmidt). As a consequence of this, we construct many explicit examples of numbers that are simultaneously distribution normal but not simply ratio normal with respect to certain countable families of basic sequences. Additionally, we prove that some related sets are either winning sets or sets of the first category.
Mar 21 2013 – 3:00pm – 4:00 pm
Joseph Rosenblatt (University of Illinois at Urbana-Champaign)
Classical ergodic averages give good norm approximations, but these averages are not necessarily giving the best norm approximation
among all possible averages. We consider 1) what the optimal Cesaro norm approximation can be in terms of the transformation and the function, 2) when these optimal Cesaro norm approximations are comparable to the norm of the usual ergodic average, and 3) oscillatory behavior of these norm approximations.
Apr 9 2013 – 1:50pm – 2:50 pm
Jon Chaika (University of Chicago)
Recently Eskin-Mirzakhani-Mohammadi have proven a number of
powerful results about the SL_2(R) orbits of abelian differentials and
the SL_2(R) ergodic measures on the stratum. We discuss some results
motivated and enabled by this work. One result is that for
every abelian differential there is a measure on the stratum, such
that after rotating in almost every direction, the geodesic flow
equidistributes for this measure on the stratum. Another result is
that for any surface the conclusion of Oseledets multiplicative
ergodic theorem applies for the Kontsevich-Zorich cocycle. This has an
application, being explored by others, to the windtree model. This is
joint work with Alex Eskin.
Apr 18 2013 – 3:00pm – 3:50 pm
Younghwan Son (Ohio State)
Mixing properties are important invariants in ergodic theory. In recent decades the theory of quasicrystals has facilitated the study of mixing properties in tiling dynamical systems. In this talk we will survey some results and problems regarding weakly, mildly, and strongly mixing tiling dynamical systems.
May 16 2013 – 3:00pm – 4:00 pm
Karl Petersen (University of North Carolina at Chapel Hill)
In joint work with Kathleen Carroll and Benjamin Wilson, we explore two ways to study properties of topological dynamical systems, especially subshifts. The first is by constructing Markov diagrams, following the ideas of Weiss, Fischer, Krieger, Hofbauer, and Buzzi. It turns out that interesting such diagrams can be displayed even for highly structured, infinite memory, fundamentally non-Markovian systems, such as Sturmian and substitution subshifts. The second is a variation on entropy. Edelman, Sporns, and Tononi proposed a concept called “intricacy” to measure complexity or interconnectivity of neural networks, and Buzzi and Zambotti studied it for families of random variables. We define a version for topological dynamical systems and examine some of its properties, including comparison with topological entropy.
Sep 12 2013 – 4:30pm – 5:30 pm
Keith Burns (Northwestern)
The lecture will describe the Weil-Petersson metric on the moduli
space of a surface and (at least some of ) the ideas that go into the
proof that its geodesic flow is ergodic. This is joint work with Howard Masur and Amie Wilkinson.
Sep 12 2013 – 3:00pm
Vaughn Climenhaga (University of Houston)
When confronted with a smooth dynamical system that appears to possess some sort of non-uniform hyperbolicity, it is useful to find an invariant measure that controls the asymptotic properties of points chosen at random with respect to the natural volume on the phase space. Such SRB measures have been constructed for systems where it is possible to relate the dynamics to a symbolic system via a Markov partition or Young tower, and also for certain systems with a dominated splitting. We present a new approach that does not require any Markov structure or uniform geometric structure. The key is a notion of “effective hyperbolicity”, which can be used to prove a non-uniform version of the Hadamard-Perron theorem on stable and unstable manifolds. This is joint work with Dmitry Dolgopyat and Yakov Pesin.
Sep 26 2013 – 3:00pm – 4:00 pm
Daniel J Thompson (The Ohio State University)
For a broad class of symbolic dynamical systems without the Markov property, including the coding spaces of many piecewise continuous interval maps, we show how to approximate an arbitrary ergodic measure with a measure of almost the same entropy supported on a sofic subshift. This is interpreted as a symbolic analogue of a `hyperbolic horseshoe’ theorem. In addition to the intrinsic interest of this result as a structure theorem, it can be a useful tool in large deviations theory and multifractal analysis. I will discuss two ways to establish this result, both based on surgery on a single generic orbit. One proof is based on Ornstein’s d bar metric, and the other is based on the theory of Kolmogorov complexity. Both techniques can be explained in a simple and intuitive way.
Oct 3 2013 – 3:00pm – 4:00 pm
Joel Moreira (Ohio State)
Oct 10 2013 – 3:00pm – 4:00 pm
Tamara Kucherenko (CUNY)
We introduce the notion of localized topological pressure for continuous maps on compact metric spaces and establish a local version of the variational principle for several classes of dynamical systems and potentials. We also construct examples showing that the assumptions in the localized variational principle are fairly sharp. Next, we study localized equilibrium states and show that even in the case of subshifts of finite type and Holder continuous potentials, there are several new phenomena that do not occur in the theory of classical equilibrium states. In particular, ergodic localized equilibrium states for Holder continuous potentials are in general not unique. (joint work with C.Wolf)