Tuesday, April 21, 2015

12:00 pm
- 01:00 pm

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

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.

ESC 638

Tuesday, March 24, 2015

12:00 pm
- 01:00 pm

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

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).

ESC 638

Tuesday, February 24, 2015

12:00 pm
- 01:00 pm

DnA Seminar, Scott Kaschner (University of Arizona): 'Dynamical Degrees, Invariant Foliations, and Measure of Maximal Entropy'

Abstract: We present a simple rational map of the complex projective plane whose first and second dynamical degrees coincide, but which does not have any invariant foliation. This answers a question posed by Guedj in 2006 and is joint work with Rodrigo Perez and Roland Roeder. Background in dynamical degrees of rational maps of two dimensional complex projective space and singular holomorphic foliations will also be presented. Using a family of maps with equal dynamical degrees and no invariant foliation, we construct a measure of maximal entropy. This is joint work with Rodrigo Perez and Roland Roeder.

ESC 638

Tuesday, November 18, 2014

12:00 pm
- 01:00 pm

Dynamics and Analysis Seminar, Daniel Wang (Trinity): 'Variable Hardy Spaces'

Abstract: Classically, Hardy space are natural generalization of Lebesgue spaces L^p when p is between 0 and 1. A more modern generalization of Lebesgue spaces is the variable L^p space, where the exponent p is allowed to vary. In this talk, we introduce the variable Hardy space, combining classical Hardy space theory and weighted theory. This is based on the work done with David Cruz-Uribe of Trinity College.

ESC 638

Tuesday, November 11, 2014

12:00 pm
- 01:00 pm

Dynamics and Analysis Seminar, David Constantine (Wes): 'Volume near standard anti-de Sitter 3-manifolds'

Abstract: Anti-de Sitter manifolds are the Lorentzian analogues of hyperbolic manifolds. In dimension 3, there is an interesting moduli space of compact AdS 3-manifolds, which is still not well understood. In this talk I'll introduce anti-de Sitter manifolds, and describe what's known about the compact examples. Then I'll present a first result on a very basic question: What are the volumes of these manifolds? Almost nothing is known about this question. I will show volume is not constant and characterize its behavior near the 'standard' AdS 3-manifolds. If time permits, I'll speak briefly on some further questions in this direction.

ESC 638

Tuesday, October 28, 2014

12:00 pm
- 01:00 pm

Dynamics and Analysis Seminar, Dale Winter (Yale): 'Uniform exponential mixing for finitely generated subgroups of SL_2(Z) and applications'

Abstract: Mixing properties of geodesic flows on hyperbolic surfaces are known to have consequences in many different areas of mathematics, including representation theory, spectral theory and affine sieve. Often we require very precise mixing results, such as exponential error terms which are valid uniformly for congruence covers. I'll outline some of these applications, and then describe an approach to proving uniform exponential mixing by combining the theory of expander graphs with the techniques from hyperbolic dynamics. This is joint work with Hee Oh.

ESC 638

Tuesday, September 09, 2014

12:00 pm
- 01:00 pm

Dynamics and Analysis Seminar: Joanna Furno, Wesleyan 'Dynamics of singular and nonsingular p-adic transformations'

Abstract: Orbit equivalence is weaker than isomorphism as a notion of equivalence between transformations that are nonsingular and ergodic. In this talk, we will give examples of p-adic transformations in different orbit equivalence classes and show that iterates of a transformation are not always in the same orbit equivalence class. Some of the examples use independent and identically distributed (i.i.d.) Bernoulli measures beyond Haar measure. These non-Haar i.i.d. Bernoulli measures also give examples of singular systems when the transformation is translation by a rational number.

ESC 638

Tuesday, October 15, 2013

12:00 pm
- 01:00 pm

DnA Seminar, Han Li (Yale): 'Effective Discreteness of the 3 Dimensional Markov Spectrum'

Abstract: Let O denote the set of non-degenerate, indefinite, real quadratic forms in 3-variables. We define for every such quadratic form Q, the Markov infimum m(Q)=inf{|Q(v)|^3/|det(Q)|: v is a nonzero integral vector in R^3}. This normalization makes the infimum invariant after rescaling the quadratic form. The set M={m(Q): Q in O} is called the 3-dimensional Markov spectrum. An early result of Cassels-Swinnerton-Dyer combined with Margulis' proof of the Oppenheim conjecture asserts that M consists of rational numbers, and for every a>0 there are only finitely many numbers in M which are greater than a. In this lecture we will discuss an effective improvement of this result. This is an ongoing joint work with Prof. Margulis. The key ingredient is to study the compact orbits of the SO(2,1) action on SL(3, R)/SL(3, Z), and our method involves techniques from the geometry of numbers, dynamics on homogeneous spaces and automorphic representations.

ESC 638

Tuesday, November 27, 2012

12:00 pm
- 01:00 pm

DnA Seminar, Ron Blei - UCONN: 'Measurements of randomness and interdependence'

Abstract: Interdependence and randomness can be calibrated by indices based separately on combinatorial measurements, p-variations, and tail-probability estimates. These notions had naturally originated in a context of harmonic analysis, and appeared later in stochastic settings. I intend in this talk to survey and explain these ideas, and (hopefully) also shed some new light on them. No formal proofs will be given. I will speak heuristically, but will try to be precise.

ESC 638

Tuesday, November 06, 2012

12:00 pm
- 01:00 pm

DnA Seminar, Jon Chaika - University of Chicago: 'Diophantine approximation for interval exchange transformations'

Abstract: Interval exchange transformations (IETs) are invertible, piecewise order preserving isometries of the unit interval with finitely many discontinuities. They generalize rotations of the circle. Motivated by this connection several diophantine properties will be presented. In particular, this talk will answer the questions of typically how quickly does the orbit of a point under an IET approach another point or itself. This is joint work with M. Boshernitzan and D. Constantine.

ESC 638

Tuesday, October 30, 2012

12:00 pm
- 01:00 pm

DnA Seminar, Giulio Tiozzo - Harvard: 'Renormalization and alpha-continued fractions'

Abstract: Alpha-continued fraction transformations are a one-parameter family of maps which arise from generalized continued fraction algorithms. The average speed of convergence of these algorithms (which corresponds to the entropy of the maps) varies wildly with the parameter, and is known to be locally monotone outside a closed fractal set E. Surprisingly, such a set has the same structure as the real slice of the Mandelbrot set, making it possible to apply ideas from complex dynamics to continued fractions: in this talk, we will investigate the self-similar structure of E and characterize the plateaux occurring in the graph of entropy.

ESC 638

Tuesday, October 23, 2012

12:00 pm
- 01:00 pm

DnA Seminar, Stephen Shea-St Anslem: 'Topological Conjugacy on the Complement of the Periodic Points

Abstract: Let Per(X) denote the periodic points of a subshift X. I will say two subshifts X and Y are essentially conjugate if there exists a topological conjugacy from X\Per(X) to Y\Per(Y). In 1990, Susan Williams presented an example of a sofic shift that is not topologically conjugate to a renewal system. I will show that this example is essentially conjugate to a renewal system. I will also present an example of a renewal system that is essentially conjugate to a shift of finite type but not topologically conjugate to a shift of finite type. Lastly, we will provide a sufficient condition for a renewal system to be essentially conjugate to a shift of finite type.

ESC 638

Tuesday, October 09, 2012

12:00 pm
- 01:00 pm

DnA Seminar, David Constantine, Wesleyan: 'Frame flow and rank-rigidity'

Abstract: I will describe the dynamics of the frame flow on a compact manifold of negative curvature, as well as some extensions to certain non-positively curved manifolds. I will then use this to prove that a compact manifold with 'higher hyperbolic rank' is actually hyperbolic, under a few curvature assumptions. This fits into a family of rank-rigidity theorems/open problems.

ESC 638