Department of

# Mathematics

Seminar Calendar
for Ergodic Theory events the year of Tuesday, December 24, 2013.

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events for the
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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, January 22, 2013

1:00 pm in 345 Altgeld Hall,Tuesday, January 22, 2013

#### Norm approximation in Ergodic Theory

###### Joe Rosenblatt (UIUC)

Abstract: 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.

Monday, January 28, 2013

3:00 pm in 145 Altgeld Hall,Monday, January 28, 2013

#### Tower multiplexing and slow weak mixing

###### Terrence M Adams (U.S. Government)

Abstract: A new technique is presented for multiplexing two ergodic measure preserving transformations together to derive a third limiting transformation. This technique is used to show the following: given any rigidity sequence for an ergodic measure preserving transformation, there exists a weak mixing transformation which is rigid along the same sequence. The proof is constructive, and should contain plenty of pictures (i.e. Rohklin towers).

Monday, February 4, 2013

4:00 pm in 241 Altgeld Hall,Monday, February 4, 2013

#### Low complexity Alpern Lemmas for $\mathbb R^d$ actions.

###### Ayşe Şahin (DePaul University)

Abstract: In recent work with B. Kra and A. Quas we showed that every ergodic, measure preserving $\mathbb R^d$ action has a section where the return times consist of $d+1$ rectangular times. This answers a question of Rudolph about the optimal complexity of return times. For $d=2$ using a detailed analysis of the set of invariant measures on tilings of $\mathbb R^2$ by two rectangles we show that this bound is optimal for actions with completely positive entropy. In the zero entropy category we show that there exist mixing $\mathbb R^2$ actions whose orbits can be tiled by 2 tiles. The talk will start by explaining how to interpret return times to a section in terms of tilings of the acting group.

Monday, February 11, 2013

4:00 pm in 241 Altgeld Hall,Monday, February 11, 2013

#### Ergodic theorems and fractal return-time structure for infinite measure.

###### Albert Fisher (University of São Paulo, Brazil)

Abstract: Certain infinite measure-preserving transformations and flows exhibit fractal-like return times; evidence for this is existence of a dimension for return times, and of an order-two ergodic theorem. Examples include horocycle flows for finitely generated Fuchsian groups of second type, and certain adic transformations and maps of the interval with an indifferent fixed point. We explain the overall framework and then focus on recent results with Marina Talet on renewal processes and indifferent fixed points.

Thursday, February 14, 2013

1:00 pm in 241 Altgeld Hall,Thursday, February 14, 2013

#### Computability and uniformity in analysis

###### Jeremy Avigad (Carnegie Mellon University)

Abstract: Countless theorems of analysis assert the convergence of sequences of numbers, functions, or elements of an abstract space. Classical proofs often establish such results without providing explicit rates of convergence, and, in fact, it is often impossible to compute the limiting object or a rate of convergence from the given data. This results in the curious situation that a theorem may tell us that a sequence converges, but we have no way of knowing how fast it converges, or what it converges to. On the positive side, it is often possible to "mine" quantitative and computational information from a convergence theorem, even when a rate of convergence is generally unavailable. Moreover, such information can often be surprisingly uniform in the data. In this talk, I will discuss examples that illustrate the kinds of information that can and cannot be obtained, focusing on results in ergodic theory.

Monday, February 18, 2013

4:00 pm in 241 Altgeld Hall,Monday, February 18, 2013

#### Fine Inducing and Equilibrium Measures for Rational Functions of the Riemann Sphere

###### Mariusz Urbanski (University of North Texas)

Abstract: Let $f:\mathbb{C}\to \mathbb{C}$ be an arbitrary holomorphic endomorphism of degree larger than $1$ of the Riemann sphere $\mathbb{C}$.. Denote by $J(f)$ its Julia set. Let $\varphi:J(f) \to\mathbb{R}$ be a H\"older continuous function whose topological pressure exceeds its supremum. It is known that then there exists a unique equilibrium measure $\mu_\varphi$ for this potential. I will discuss a special inducing scheme with fine recurrence properties. This construction allows us to prove three results. Dimension rigidity, i.e. a characterization of all maps and potentials for which $HD(\mu_\varphi)=HD(J(f))$. Real analyticity of topological pressure $P(t\varphi)$ as a function of $t$. Exponential decay of correlations, and, as its consequence, the Central Limit Theorem and the Law of Iterated Logarithm for H\"older continuous observables. Finally, the Law of Iterated Logarithm for all linear combinations of H\"older continuous observables and the function $\log|f'|$. Geometric consequences of the Law of Iterated Logarithm lead to comparison of equilibrium states with appropriately generalized Hausdorff measures on the Julia set $J(f)$.

Monday, February 25, 2013

4:00 pm in 241 Altgeld Hall,Monday, February 25, 2013

#### Perturbations of geodesic flows producing unbounded growth of energy

###### Marian Gidea (Northern Illinois University & Institute for Advanced Study)

Abstract: We consider a geodesic flow on a manifold endowed with some generic Riemannian metric. We couple the geodesic flow with a time-dependent potential driven by an external dynamical system, which is assumed to satisfy some recurrence condition. We prove that there exist orbits whose energy grows unboundedly at a linear rate with respect to time; this growth rate is optimal. In particular, we obtain unbounded growth of energy in the case when the external dynamical system is quasi-periodic, of rationally independent frequency vector (not necessarily Diophantine). Our result generalizes Mather's acceleration theorem and is related to Arnold's diffusion problem. It also extends some earlier results by Delshams, de la Llave and Seara

Monday, March 4, 2013

4:00 pm in 241 Altgeld Hall,Monday, March 4, 2013

#### Autocorrelations in Quantum Mechanics and Homogenous Dynamics

###### Francesco Cellarosi (UIUC)

Abstract: I will discuss an application of a result by J. Marklof and myself on the limiting distribution of theta sums to the the study of autocorrelations in certain quantum mechanical systems. I will show how, under quantum evolution, particles ''remember'' their initial profiles. Moreover, when studying the simultaneous evolution of two (or more) related particles, there are some ''forbidden regions'' for the joint autocorrelation functions. All the results are based on a result in homogeneous dynamics, namely equidistribution of horocycles under the action of the geodesic flow. The talk will be elementary and no knowledge of quantum mechanics / homogenous dynamics will be assumed.

Monday, March 11, 2013

4:00 pm in 241 Altgeld Hall,Monday, March 11, 2013

#### Pinball dynamics

###### Maxim Arnold (UIUC)

Abstract: Theory of small perturbations of completely integrable Hamiltonian systems has a long history. Summarily, one could say that Kolmogorov-Arnold-Moser theorem states that if perturbation is sufficiently smooth, then positive measure of invariant tori survives and thus in particular for planar area-preserving transformations there are no trajectories escaping to infinity. In some physical settings one deals with non-smooth perturbations where KAM-technique does not apply. However, there are no general instruments to work with such systems. I shall formulate natural family of such systems which covers many known examples and shall show how to construct an escaping trajectory in one representative example.

Wednesday, March 13, 2013

5:00 pm in 145 Altgeld Hall,Wednesday, March 13, 2013

#### Ergodic Theory by Example

###### Kelly Yancey (UIUC Math)

Abstract: Do you ever wonder if there is a mathematical formula to model the way a baker kneads dough and if so, what it's mathematical properties are? We will explore this question and others using ergodic theory. Throughout this talk we will informally discuss some of the basic definitions in ergodic theory and topological dynamics by studying key examples. We will develop our intuition by analyzing Baker's Transformation and rotations of the circle. There will be free pizza!!!

Monday, March 25, 2013

4:00 pm in 241 Altgeld Hall,Monday, March 25, 2013

#### Benford's Law, Values of L-Functions and the 3x+1 Problem

###### Steven J. Miller (Williams College)

Abstract: Many systems exhibit a digit bias. For example, the first digit (base 10) of the Fibonacci numbers or of $2^n$ equal 1 about 30% of the time; the IRS uses this digit bias to detect fraudulent corporate tax returns. This phenomenon, known as Benford's Law, was first noticed by observing which pages of log tables were most worn from age -- it's a good thing there were no calculators 100 years ago! The first digit of values of L-functions near the critical line also exhibit this bias. A similar bias exists (in a certain sense) for the first digit of terms in the 3x+1 problem, provided the base is not a power of two. For L-functions the main tool is the Log-Normal law; for $3x+1$ it is the rate of equidistribution of $n log_B 2 \mod 1$ and understanding the irrationality measure of $log_B 2$. This work is joint with Alex Kontorovich.

Monday, April 1, 2013

4:00 pm in 241 Altgeld Hall,Monday, April 1, 2013

#### Mixing properties in tiling dynamical systems

###### Younghwan Son (Ohio State)

Abstract: 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

Tuesday, April 2, 2013

2:00 pm in Altgeld Hall 347,Tuesday, April 2, 2013

#### Invariance Principle for theta sums

###### Francesco Cellarosi (UIUC Math)

Abstract: Theta sums are very classical objects in Number Theory and Physics and can be seen as sums of strongly dependent random variables. I will present several results concerning these sums, such as a non-standard CLT (exhibiting a form of anomalous diffusion), and a weak invariance principle. The standard probabilistic methods for sums of weakly dependent random variables fail in this situation. I will present an approach that uses ergodic theory and homogeneous dynamics. Joint work with Jens Marklof (Bristol, U.K.)

Monday, April 8, 2013

4:00 pm in 241 Altgeld Hall,Monday, April 8, 2013

#### Bounds on coycles

###### Joseph Rosenblatt (UIUC)

Abstract: A cocycle $S_n^\tau f = \sum\limits_{k=0}^{n-1} f\circ \tau^k$ is a coboundary if and only if the norms $\|S_n^\tau f\|_2$ are uniformly bounded. We consider what can be said when the cocycle is bounded only along some subsequence of $\|S_n^\tau f\|_2$ for specific transformations and specific functions.

Thursday, April 11, 2013

1:00 pm in 137 Henry Administration Building,Thursday, April 11, 2013

#### Homeomorphic measures on Cantor sets and dimension groups

###### Sergey Bezuglyi (Institute for Low Temperature Physics, Kharkov, Ukraine)

Abstract: Two measures, m and m' on a topological space X are called homeomorphic if there is a self-homeomorphism f of X such that m(f(A)) = m'(A) for any Borel set A. The question when two Borel probability non-atomic measures are homeomorphic has a long history beginning with the work of Oxtoby and Ulam who gave a criterion when a probability Borel measure on the cube [0, 1]^n is homeomorphic to the Lebesgue measure. The situation is more interesting for measures on a Cantor set. There is no complete characterization of homeomorphic measures so far. But, for the class of the so called good measures (introduced by E. Akin), the answer is simple: two good measures are homeomorphic if and only if the sets of their values on clopen sets are the same. I will focus in the talk on the study of probability measures invariant with respect to a minimal (or aperiodic) homeomorphism. These measures are in one-to-one correspondence with traces on a corresponding dimension group. The technique of dimension groups allows us to apply new methods for studying good traces. A good trace is characterized by its kernel having dense image in the annihilating set of affine functions on the trace space. A number of examples with seemingly paradoxical properties is considered. The talk will be based on a joint paper with D. Handelman.

Monday, April 15, 2013

4:00 pm in 241 Altgeld Hall,Monday, April 15, 2013

#### The distribution of directions in an affine lattice: two-point correlations and mixed moments

###### Ilya Vinogradov (University of Bristol, U.K.)

Abstract: We consider an affine Euclidean lattice and record the directions of all lattice vectors of length at most T. Marklof and Strombergsson proved that the distribution of gaps between the lattice directions has a limit as T tends to infinity. For a typical affine lattice, the limiting gap distribution is universal and has a heavy tail; it differs markedly from the gap distribution observed in a Poisson process which is exponential. We show that the limiting two-point correlation function of the projected lattice points exists and is Poissonian, and answer a recent question of Boca, Popa, and Zaharescu [arXiv:1302.5067]. The existence of the limit is subject to a certain Diophantine condition. We also establish the convergence of more general mixed moments. Joint work with D. El-Baz and J. Marklof.

Monday, April 22, 2013

4:00 pm in 241 Altgeld Hall,Monday, April 22, 2013

#### Continued fractions on the Heisenberg group

###### Joseph Vandehey (UIUC)

Abstract: We will discuss some recent joint work with Anton Lukyanenko developing a theory of continued fractions on the Heisenberg group. These appear to be a very natural higher-dimensional analog of continued fractions on the real line; there are many formulas which have analogs on the Heisenberg group but which are missing from other higher-dimensional continued fraction algorithms (such as Jacobi-Perron). We will discuss the basic development of these continued fractions as well as difficulties associated with proving ergodicity of the associated shift map.

Monday, April 29, 2013

4:00 pm in 241 Altgeld Hall,Monday, April 29, 2013

#### Entropy of Transformations Preserving an Infinite Measure

###### Rachel Bayless (University of North Carolina, Chapel Hill)

Abstract: The entropy of a system measures the amount of information gained with each application of an experiment or transformation, and higher entropy corresponds to more disorder and less predictable systems. Classical measure-theoretic entropy is only well-defined for finite-measure-preserving transformations, and there is no universal analogue for transformations preserving an infinite measure. Three possible definitions have been given independently by Krengel (1967), Parry (1969), and Roy (2009). Although two of these definitions have been around for over 40 years, there exist very few examples where any of these entropies have been computed explicitly. In this talk we provide a method of computing the Krengel entropy for all conservative rational functions which preserve Lebesgue measure on the real line.

Monday, September 9, 2013

4:00 pm in 241 Altgeld Hall,Monday, September 9, 2013

#### Coboundaries and ergodic sums.

###### Joe Rosenblatt (Illinois)

Abstract: The behavior of the norms of ergodic sums can be used to characterize coboundaries. But the behavior of the norms of ergodic sums can be fairly chaotic. Moreover, for a given function, which classes of transformations have that function as a coboundary is a complex issue. These types of things will be considered in some detail for general ergodic transformations of a probability space.

Monday, September 16, 2013

4:00 pm in 241 Altgeld Hall,Monday, September 16, 2013

#### Anomalous modulus of continuity for the theta process and logarithm laws for geodesics.

###### Francesco Cellarosi (Illinois)

Abstract: Theta sums are particular exponential sums with deep number-theoretical and physical connections. The planar curves obtained by linearly interpolating their partial sums are sometimes called 'curlicues' because of their rich geometric structure of spirals arranged in an approximate multi-fractal structure. Analogously to the construction of the Brownian Motion starting from simple symmetric random walks, I will briefly explain how to obtain a random process (the Theta Process) using equidistribution of horocycles under the action of the geodesic flow on a suitable hyperbolic manifold. Among the properties of this process, I will discuss the anomalous modulus of continuity of typical realizations of this process (different from that of a typical Brownian path), and derive this property using a logarithm law for geodesics due to Kleinbock and Margulis. This implies, in particular H\"older continuity of typical realizations for any exponent less than 1/2. Joint work with Jens Marklof (Bristol)

Monday, September 30, 2013

4:00 pm in 241 Altgeld Hall,Monday, September 30, 2013

#### Finite generating partitions for countable group actions

###### Anush Tserunyan (Illinois)

Abstract: Consider a continuous action of a countable group G on a Polish space space X. A countable Borel partition P of X is called a generator if GP={gA: g in G, A in P} generates the Borel sigma-algebra of X. For G=Z, the Kolmogorov-Sinai theorem gives a measure-theoretic obstruction to the existence of finite generators: they don't exist in the presence of an invariant probability measure with infinite entropy. It was asked by Weiss in the late 80s whether the nonexistence of any invariant probability measure would guarantee the existence of a finite generator. We show that the answer is positive for an arbitrary countable group G and sigma-compact X (in particular, for locally compact X).

Thursday, October 3, 2013

1:00 pm in Altgeld Hall 347,Thursday, October 3, 2013

#### Factors of IID on Trees

###### Russel Lyons (Indiana University)

Abstract: Classical ergodic theory for integer group actions uses entropy as a complete invariant for isomorphism of IID (independent, identically distributed) processes (a.k.a. product measures). This theory holds for amenable groups as well. Despite recent spectacular progress of Bowen, the situation for non-amenable groups, including free groups, is still largely mysterious. We discuss a few known results and open questions on free groups, which are particularly interesting in combinatorics, statistical physics, and probability. No background will be assumed.

Monday, October 14, 2013

4:00 pm in 241 Altgeld Hall,Monday, October 14, 2013

#### Distribution of digits of rational power sequences

###### Yiannis Konstantoulas (Illinois)

Abstract: In this talk, we will describe recent history of and the author's ongoing work on the distribution of digits of the power sequence $r_n=\left(\frac{3}{2}\right)^n$ and other rational power sequences. Using algebraic and combinatorial tools, we will show that for the $n$-fold dyadic partition of the interval $[0,1]$, one can find among the $2^n$ subintervals at least $cn$ distinct intervals containing a limit point of the fractional parts of $r_n$ for an absolute constant $c>0$. Finally, we will introduce a topological space whose large scale structure reflects the metric theory of the map $\{(r_n)\xi\}$ and link the problem of distribution of digits of $r_n$ to properties of random walks on that topological space. The talk will contain many questions, a few answers and hopefully an interesting new perspective on rational power sequence maps.

Thursday, October 17, 2013

4:00 pm in 245 Altgeld Hall,Thursday, October 17, 2013

#### Foliations - Thurston Zebras to Cantor Geometries

###### Steve Hurder (University of Illinois at Chicago)

Abstract: It was forty years ago that William Thurston electrified the theory of foliations with his celebrated constructions of foliations, and raised the hope of classifying these geometric structures on manifolds. This was followed by an intense period of development over the next ten years of the theory of characteristic classes for foliations. The study of foliations since then has taken many twists and turns, finding applications in dynamical systems and ergodic theory, and incorporating many new ideas in the quest for a viable classification scheme.

In this talk, I will survey three aspects of these developments: the role of algebra in the works of Kamber and Tondeur in the 1970's on secondary class theory; the rise of dynamical and ergodic theory methods in the 1980's and 1990's; and the current development of topological methods and the techniques of Cantor actions to gain a new understanding of the structure of foliations.

Monday, October 28, 2013

4:00 pm in 241 Altgeld Hall,Monday, October 28, 2013

#### Hodge theory and rigidity in Teichmuller dynamics

###### Simion Filip (University of Chicago)

Abstract: I will start by introducing the basic concepts of Teichmuller dynamics. While many questions are of intrinsic interest, they also have direct applications to more classical systems such as polygonal billiards and interval exchanges. I will then discuss some connections to algebraic geometry and how techniques from Hodge theory can give rigidity results for dynamics.

Thursday, October 31, 2013

2:00 pm in 243 Altgeld Hall,Thursday, October 31, 2013

#### Current results and future directions regarding fractal billiard tables.

###### Robert Niemeyer (University of New Mexico)

Abstract: In this talk, we will discuss recent results on three different fractal billiard tables: the Koch snowflake, a self-similar Sierpinski carpet and (for lack of a better name and none ever given in the literature) the T-fractal. Initially, an investigation of the flow on a fractal billiard table was made on the Koch snowflake. Results on the Koch snowflake motivated us to investigate the other two billiard tables. Using a result of J. Tyson and E. Durand-Cartagena, we show that there are periodic orbits of a self-similar Sierpinski carpet billiard table. J. P. Chen and I have begun investigating whether or not it makes sense to discuss the existence of a dense orbit of a self-similar Sierpinski carpet billiard table. Substantial progress has been made in determining periodic orbits of the T-fractal billiard table. We detail some of the recent results concerning periodic orbits, determined in collaboration with M. L. Lapidus and R. L. Miller. Less has been done to determine what may constitute a dense orbit of the T-fractal billiard. We provide substantial experimental and theoretical evidence in support of the existence of an orbit that is dense in the T-fractal billiard table but is not a space-filling curve. We briefly touch on a long-term goal of determining a topological dichotomy for the flow on a fractal billiard table, namely that, in a fixed direction, the flow is either closed or minimal. Parts of this talk will be suitable for an advanced undergraduate/beginning graduate student audience.

Monday, November 4, 2013

4:00 pm in 241 Altgeld Hall,Monday, November 4, 2013

#### The Ricci Flow for Homogeneous Spaces

###### Tracy Payne (Idaho State)

Abstract: The Ricci flow evolves a metric g_0 on a manifold M according to the PDE g_t = -2 ric(g_t). If M is a simply connected homogeneous space, the Ricci flow preserves the homogeneity of M. Let X be a class of simply connected homogeneous spaces that remains invariant under the Ricci flow. It is possible to encode the Ricci flow on X as a system of ODEs. We will describe the general set-up of the system of ODEs and describe methods for qualitative analysis of such systems, giving examples of the analysis for various specific subclasses.

Monday, November 11, 2013

4:00 pm in 241 Altgeld Hall,Monday, November 11, 2013

#### Schmidt’s game, nondense orbits, and the set of badly approximable systems of linear forms

###### Ryan Broderick (Northwestern)

Abstract: We will discuss an infinite game which produces a class of full-dimension sets that is closed under countable intersection and under taking images by C^1 diffeomorphisms. In particular, the countable intersection of C^1 diffeomorphic images of such a set meets any sufficiently regular fractal in a set of full Hausdorff dimension. We present joint work with L. Fishman and D. Simmons, where we have shown that, for any surjective endomorphism of a torus, the set of nondense orbits belongs to the above-mentioned class. We also show that the set of badly approximable systems of linear forms has this property. We will then discuss recent work with D. Kleinbock which uses the game, along with techniques from homogeneous dynamics, to produce complements to the above results concerning `uniform’ versions of the sets.

Monday, November 18, 2013

4:00 pm in 241 Altgeld Hall,Monday, November 18, 2013

#### To Be Announced

###### Anton Lukyanenko (Illinois)

Thursday, November 21, 2013

2:00 pm in 345 Altgeld Hall,Thursday, November 21, 2013

#### Rigid and Weakly Mixing Cutting and Stacking Constructions

###### Kelly Yancey (Maryland)

Abstract: In the setting of infinite ergodic theory, measure-preserving transformations that are rigid and spectrally weakly mixing are generic in the sense of Baire category. During this talk we will discuss rigid verses various types of weakly mixing in infinite ergodic theory. We will also construct examples of transformations that have these desired properties. Our examples will be via the method of cutting and stacking. This is joint work with Rachel Bayless. (Note special time and date.)

Monday, December 2, 2013

4:00 pm in 241 Altgeld Hall,Monday, December 2, 2013

#### Thin groups: arithmetic and beyond

###### Elena Fuchs (California-Berkeley)

Abstract: In 1643, Rene Descartes discovered a formula relating curvatures of circles in Apollonian circle packings, constructed by Apollonius of Perga in 200 BC. This formula has recently led to a connection between the construction of Apollonius and orbits of a certain so-called thin subgroup G of GL_4(Z). This connection is key in recent results on the arithmetic of Apollonian packings, which I will describe in this talk. A crucial ingredient in the proofs is the spectral gap coming from families of expander graphs associated to G -- this gap is far less understood in the case of thin groups than that of non-thin groups. Motivated by this problem, I will then discuss the ubiquity of thin groups and present results on thinness of monodromy groups of hypergeometric equations in the case where these groups act on hyperbolic space.