Monday, February 29, 2016
- 06:00 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.
Exley Science Center Tower ESC 638
Tuesday, November 10, 2015
- 01:00 pm
DnA Seminar, Naser Talebizadeh (Princeton): Optimal strong approximation for quadratic forms
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
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+
Exley Science Center (Tower)
Tuesday, April 21, 2015
- 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.
Tuesday, March 24, 2015
- 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).
Tuesday, February 24, 2015
- 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.
Tuesday, November 18, 2014
- 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.
Tuesday, November 11, 2014
- 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.
Tuesday, October 28, 2014
- 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.
Tuesday, September 09, 2014
- 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.