# Seminars and Colloquia

## DnA Seminar

Nov 28

#### DnA Seminar

12:00 pm

Nov 21

#### DnA Seminar

12:00 pm

Nov 14

#### DnA Seminar

12:00 pm

Speaker: Tamara Kucherenko (City College of New York) Title: Measures of maximal entropy for suspension flows over the full shift Abstract: We consider suspension flows with continuous roof function over the full shift on a finite alphabet. For any positive entropy subshift of finite type Y, we show there exists a roof function such that the measure(s) of maximal entropy for the suspension flow over the full shiftare exactly the lifts of the measure(s) of maximal entropy for Y. In the case when Yis transitive, this gives a unique measure of maximal entropy for the flow which is not fully supported. If Yhas more than one transitive component, all with the same entropy, this gives explicit examples of suspension flows over the full shift with multiple measures of maximal entropy. This contrasts with the case of a H\"older continuous roof function where it is well known the measure of maximal entropy is unique and fully supported. This is joint work with Dan Thompson.

Nov 7

#### DnA Seminar

12:00 pm

Oct 31

#### DnA Seminar

12:00 pm

Speaker: Shahriar Mirzadeh, Brandeis University Title: Dimension estimates for the set of points with non-dense orbit in homogeneous spaces. Abstract: In this talk we study the set of points in a homogeneous space whose orbit escapes the complement of a fixed compact subset. We find an upper bound for the Hausdorff dimension of this set. This extends the work of Kadyrov, where he found an upper bound for the Hausdorff dimension of the set of points whose orbit misses a fixed ball of sufficiently small radius in a compact homogeneous space. We can also use our main result to produce new applications to Diophantine approximation. This is joint work with Dmitry Kleinbock.

Oct 17

#### DnA Seminar

12:00 pm

Speaker: Thang Nguyen, NYU Title: Hyperbolic rank rigidity Abstract: Motivated by the question about (Euclidean) rank rigidity, whether a closed nonpositively curved manifold with every geodesic locally contained in a flat is locally symmetric, we consider the question where flat is replaced by hyperbolic plane. The question about Euclidean rank rigidity was answered positive by Ballmann-Brin-Eberline and Burns-Spatzier in 80s'. For the later one, it has not been completely solved yet. It was achieved in many cases by Hamenstadt and Constantine. We give a positive answer for the case quarter-pinched manifolds, which we use different technique with the ones of Hamenstadt or Constantine. The main tool is from dynamics of Lyapunov distributions and a lemma by Foulon. I will talk about difficulty and ideas of proofs. This is a joint work with C. Connell and R. Spatzier.

Oct 10

#### DnA Seminar

12:00 pm

Oct 3

#### DnA Seminar

12:00 pm

Sep 26

#### DnA Seminar

12:00 pm

Sep 19

#### DnA Seminar

12:00 pm

Speaker: Tamara Kucherenko (City College of New York) Title: Measures of maximal entropy for suspension flows over the full shift Abstract: We consider suspension flows with continuous roof function over the full shift on a finite alphabet. For any positive entropy subshift of finite type Y, we show there exists a roof function such that the measure(s) of maximal entropy for the suspension flow over the full shiftare exactly the lifts of the measure(s) of maximal entropy for Y. In the case when Yis transitive, this gives a unique measure of maximal entropy for the flow which is not fully supported. If Yhas more than one transitive component, all with the same entropy, this gives explicit examples of suspension flows over the full shift with multiple measures of maximal entropy. This contrasts with the case of a H\"older continuous roof function where it is well known the measure of maximal entropy is unique and fully supported. This is joint work with Dan Thompson.

Feb 29

#### DnA Seminar

04:30 pm

Dynamics and Analysis Seminar Speaker: Dan Thompson (Ohio State) Title: Generalized beta-transformations and the entropy of unimodal maps Abstract: Generalized beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformationx beta x(mod1),wherebeta> 1, and replacing some of the branches with branches of constant negative slope. We would like to describe the set of beta for which these maps can admit a Markov partition. We know that beta (which is the exponential of the entropy of the map) must be an algebraic number. Our main result is that the Galois conjugates of such beta have modulus less than 2, and the modulus is bounded away from 2 apart from the exceptional case of conjugates lying on the real line. This extends an analysis of Solomyak for the case of beta-transformations, who obtained a sharp bound of the golden mean in that setting. I will also describe a connection with some of the results of Thurston's fascinating final paper, where the Galois conjugates of entropies of post-critically finite unimodal maps are shown to describe an intriguing fractal set. These numbers are included in the setting that we analyze.

Nov 10

#### DnA Seminar, Naser Talebizadeh (Princeton): Optimal strong approximation for quadratic forms

12:00 pm

Abstract : For a non-degenerate integral quadratic form F ( x 1 ,,x d ) in 5 (or more) variables, we prove an optimal strong approximation theorem. Fix any compact subspace d of the affine quadric F ( x 1 ,,x d ) = 1. Suppose that we are given a small ball B of radius 0 < r < 1 inside , and an integer m . Further assume that N is a given integer which satisfies N ( r -1 m ) 4+ for any > 0. Finally assume that we are given an integral vector ( 1 ,, d ) mod m . Then we show that there exists an integral solution x = ( x 1 ,,x d ) of F ( x ) = N such that x i i mod m and B , provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a non-degenerate integral quadratic form F ( x 1 ,, x 4 ) in 4 variables we prove the same result if N ( r -1 m ) 6+

Apr 21

#### DnA Seminar, Daniel Cuzzocreo (Smith): 'Parameter space structures for rational maps'

12:00 pm

Abstract: Fn,d,λ = zn + λ/zd give 1-parameter, n+d degree families of rational maps of the Riemann sphere, which arise as singular perturbations of the polynomial zn. Despite the high degree, symmetries cause these maps to have just a single free critical orbit and thus to form a natural 1-dimensional slice. Due to some similarities with polynomial maps, these families give some of the best-understood examples of non-polynomial rational dynamics in arbitrarily high degree. In this talk we give a survey of some recent results about these maps, with a focus on characterizing some of the fractal structure in the parameter space.

Mar 24

#### DnA Seminar, David Ralston (SUNY Old Westbury): 'Skew Products over Irrational Rotations: Limits of and on Sums'

12:00 pm

Abstract: We will consider two problems involving ergodic sums of bounded-variation functions on the unit circle, where the underlying transformation is irrational rotation. First, while the ergodic sums must return to a small range of values at prescribed times (the Denjoy-Koksma inequality), we may investigate the nondecreasing function which tracks the largest sum (in absolute value) achieved through a given time. We will provide a generic asymptotic upper bound on this function. If we restrict our attention to a specific bounded-variation function (a system known as the infinite staircase), we may also place an almost-sure lower bound on the growth of this function. Second, F. Huveneers established for every rotation the existence of a sequence of times for which the ergodic sums in the infinite staircase obey a central limit theorem, although his technique was only somewhat explicit in determining exactly which times could create a central limit theorem. We will discuss how to make his results more specific and stronger, while also extending them to other piecewise-constant functions (albeit restricted to almost-every instead of every rotation).