Friday, September 29th, 2pm, GAB 461
speaker: Mariusz Urbanski (UNT)
title: Ruelle's Operator and Conformal Measures with Applications in Fractal Geometry and Number Theory
abstract: Probabilistic invariant measures provide a central tool to describe asymptotic behavior of dynamical systems and their ergodic and stochastic properties.
Starting with natural examples, I will present a method of constructing such measures. It consists in finding fixed points of Ruelle’s operator and is applicable to dynamical systems that include smooth expanding maps, rational functions on the Riemann sphere, and holomorphic endomorphisms of complex projective spaces.
I will tell how spectral properties of Ruelle’s operator entail stochastic properties of a given dynamical system, especially the exponential decay of correlations, the central Limit Theorem, and the Law of Iterated Logarithm. I will also show how these spectral properties lead to the asymptotic of the number of circles in Apollonian packings. Asymptotic of lengths of closed geodesics on some hyperbolic spaces also will be discussed.
Next, I will focus on fractal properties of Julia sets of rational functions and transcendental meromorphic functions. Finally, I will talk about geometric properties of continued fractions.
Friday, October 6th, 2pm, GAB 461
speaker: Mariusz Urbanski (UNT)
title: Ruelle's Operator and Conformal Measures with Applications in Fractal Geometry and Number Theory
abstract: Probabilistic invariant measures provide a central tool to describe asymptotic behavior of dynamical systems and their ergodic and stochastic properties.
Starting with natural examples, I will present a method of constructing such measures. It consists in finding fixed points of Ruelle’s operator and is applicable to dynamical systems that include smooth expanding maps, rational functions on the Riemann sphere, and holomorphic endomorphisms of complex projective spaces.
I will tell how spectral properties of Ruelle’s operator entail stochastic properties of a given dynamical system, especially the exponential decay of correlations, the central Limit Theorem, and the Law of Iterated Logarithm. I will also show how these spectral properties lead to the asymptotic of the number of circles in Apollonian packings. Asymptotic of lengths of closed geodesics on some hyperbolic spaces also will be discussed.
Next, I will focus on fractal properties of Julia sets of rational functions and transcendental meromorphic functions. Finally, I will talk about geometric properties of continued fractions.
Friday, October 20th, 2pm, GAB 461
speaker: Jiajie Zheng (UNT)
title: Twisted recurrence in measurable dynamical systems
abstract: In the study of some dynamical systems, the limsup set of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limsup sets. However, the zero-one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this talk, I will talk about twisted recurrence, which can specialize into both shrinking targets and recurrence. I will introduce some known results about twisted recurrent sets, including their zero-one laws and Hausdorff measures.
Friday, October 27th, 2pm, GAB 461
speaker: Johannes Jaerisch (Nagoya University)
title: Multifractal analysis of growth rates for the geodesic flow on hyperbolic surfaces
abstract: We use multifractal analysis to investigate the long-term behavior of growth rates associated with the geodesic flow on surfaces of constant negative curvature. The growth rates we consider are given by the number of windings around cusps, the number of crossings of sides of a fundamental domain, and the distance travelled on the surface. This talk is based on joint work with Hiroki Takahasi (Keio University) and Manuel Stadlbauer (Universidade Federal do Rio de Janeiro).
Friday, November 3rd, 2pm, GAB 461
speaker: Bunyamin Sari (UNT)
title: Coarse embeddings into Banach spaces
abstract: The notion of a coarse embedding of a metric space into another is very weak form of embedding which can be thought of an embedding which `preserves' the geometry in very large scale. This notion became a central tool in several areas of mathematics; Geometric Group Theory, Topology, Metric Geometry, and even Computer Science. In each of these, roughly the program is to coarsely embed the metric space in question into a nice Banach space and solve the problem there. Usage of coarse embeddings of uniformly discrete metric spaces into sufficiently good Banach spaces in topology was initiated by Gromov. A big breakthrough in Gromov's program was achieved by Yu who showed that if a finitely generated group $G$ admits a coarse embed- ding into some uniformly convex Banach space, then $G$ satisfies the Novikov
conjecture and the coarse Baum-Connes conjecture. This and other related developments naturally brought the question of coarse embedding in Banach spaces, especially finding linear obstructions to such an embedding. We will give a brief overview of the state of the art of the problem, and plan to give proofs of two fundamental questions that started it all. 1. Does every separable Banach space coarsely embed into a Hilbert (reflexive) space? (No, Enflo, Kalton) 2. Does (separable) Hilbert space coarsely embed into every other separable Banach space? (No, Baudier, Lancien, Schlumprecht)
Based on joint work with Steve Jackson and Cory Krause.
Friday, April 19th, 2pm, GAB 461
speaker: Nathan Dalaklis (UNT)
title: Extremal $F$-Exponents of Finitely Irreducible CGDMS's
abstract: Given a H\"older family of functions $F$ and a finitely irreducible CGDMS $\Phi$ encoded by a symbolic representation $E_A^{\infty}$, one may associate to each coding $\omega\in E_A^{\infty}$ a Birkhoff average $\xi(\omega)$ called the $F$-exponent of $\omega$, should it exist. The ergodic optimization of these exponents by way of zero-temperature style limits is important to the characterization of Birkhoff spectra for CGDMSs. In this talk, we will introduce the objects at play in this optimization problem and the results which address this problem for a large collection of possible families $F$.
Friday, May 10th, 2pm, GAB 461
speaker: Anna Zdunik (University of Warsaw)
title: Hausdorff and packing measure for limit sets of conformal repellers and iterated function systems.
abstract: In this talk, I will present some recent results concerning the (numerical value of) Hausdorff and packing measures (H_h and P_h) on limit sets of a class of conformal iterated function systems or conformal repellers
For such systems, the way of finding exact value of Hausdorff/packing dimension was established long time ago (versions of Bowen’s formula). Also, a lot is known about dependence of the dimension on parameter in naturally parametrized families of such systems.
Studying the numerical value of Hausdorff/packing measure of the limit set (evaluated in its Hausdorff/packing dimension) is a challenging problem. There is no visible way of determining the value of Hausdorff (or: packing) measure in terms of thermodynamic formalism.
In our joint paper with Mariusz Urbański, we considered the sequence of iterated function systems approximating the Gauss map. The limit set J_n of such n-th systems is exactly the set of irrational numbers in [0,1] whose continued fraction expansion has entries bounded by n.
Denote by h_n the Hausdorff dimension of J_n. The asymptotic of h_n is known (Hensley), namely:
(1-h_n)⋅n→6/π^2 as n→∞.
In the above mentioned work with Mariusz, we proved continuity of Hausdorff measure: H_(h_n ) (J_n)→1 as n→∞.
In the present joint work with Mariusz Urbański and Rafał Tryniecki, we study much more delicate question, namely -asymptotic of the value 1-H_(h_n ) (J_n). We do it for the Gauss map and its piecewise linear analogue.
I will also mention some recent results of R Tryniecki concerning analogous questions of continuity of Hausdorff and packing measure for other classes of infinite iterated function systems.
Friday, September 30th, 2pm, GAB 461
speaker: Harrison Gaebler (UNT)
title: Growth of orbits for operator semigroups on Banach spaces with Schauder bases
abstract: Let $X$ be a Banach space with a normalized (Schauder) basis $(b_k)_k$ and let $\{T(t)\}_{t\geq 0\}$ be a $C_{0}$-semigroup on $X$ with generator $A:D(A)\subset X\to X$. I will in this talk introduce a new estimate for the semigroup orbits of initial data in $D(A^{2})$ that have with respect to $(b_{k})_{k}$ ``absolutely summable graph norm." This result is noteworthy because (1) there is no requirement that $z\mapsto R(z,A)x$ has a bounded analytic extension to the right half-plane and (2) the only structural condition on $X$ is the existence of $(b_{k})_{k}$.
Friday, October 7th, 2pm, STaRS seminar
Friday, October 14th, 2pm, STaRS seminar
Friday, October 21st, 2pm, Zoom
speaker: Yusheng Luo (Stony Brook)
title: Quasiconformal nonequivalence of Julia set and limit set
abstract: The Julia set of a rational map and the limit set of a Kleinian group share many common features. In many cases, the two fractal sets can be homeomorphic. However, these homeomorphisms are usually not quasiconformal. In fact, we do not know any non-trivial examples where a Julia set of a rational map is quasiconformally homeomorphic to a limit set of a Kleinian group.
In this talk, I will discuss some non-equivalent phenomenon for some classes of Julia set and limit set. In particular, I will explain why a Julia set can never be quasiconformally homeomorphic to the Apollonian gasket.
The talk is based on a joint work with Yongquan Zhang.
Friday, November 11, 2pm, GAB 461
speaker: Kiko Kawamura (UNT)
title: Relationship between Okamoto’s functions and Terdragons
abstract: Okamoto's functions were introduced in 2005 as a one-parameter family of self-affine functions, which are expressed by ternary expansion of x on the interval [0,1]. By changing the parameter, one can produce interesting examples: Perkin’s nowhere differentiable function, Bourbaki-Katsuura function and Cantor’s Devil’s staircase function. Recently, Dalaklis-Kawamura-Mathis-Paizanis studied the partial derivative of Okamoto’s function with respect to the parameter.
Terdragon is a famous tiling fractal constructed by infinitely many paper-folds. In this talk, I will discuss a relationship between Okamoto’s functions and Terdragons.
This is an on-going project. The talk is very accessible for undergraduate students and includes many computer graphics.
Friday, November 18, 2pm, GAB 461
speaker: Pieter Allaart (UNT)
title: The beta-transformation with a hole at 0
abstract: An open dynamical system is a dynamical system with an open subset (the “hole”) of the domain that must be avoided. One studies in particular the survivor set of points whose orbit never enters the hole. In this talk I will focus on an open dynamical system arising from the beta transformation on [0,1) with a hole (0,t). It was shown recently by Kalle, Kong, Langeveld and Li that the Hausdorff dimension of the survivor set, as a function of t, is a descending devil’s staircase. Of particular interest is the critical value of t where this function reaches zero. I will discuss this question, as well as our current work toward proving an intriguing conjecture from the paper of Kalle et al. This is joint work with Derong Kong.
Friday, December 2, 2pm, (zoom)
speaker: Fabrizio Bianchi (Universite de Lille)
title: A Spectral Gap for the Transfer Operator on Complex Projective Spaces.
abstract: We study the transfer (Perron-Frobenius) operator on Pk(C) induced by a generic holomorphic endomorphism f and a suitable continuous weight. We prove the existence of a unique equilibrium state and we introduce various new invariant functional spaces, including a dynamical Sobolev space, on which the action of f admits a spectral gap. This is one of the most desired properties in dynamics. It allows us to obtain a list of statistical properties for the equilibrium states. Most of our results are new even in dimension 1 and in the case of constant weight function, i.e., for the operator f_*. Our construction of the invariant functional spaces uses ideas from pluripotential theory and interpolation between Banach spaces. This is a joint work with Tien-Cuong Dinh
Friday, December 9, 2pm, GAB 473
speaker: James Waterman (Stony Brook University)
title: Eremenko's Conjecture and Wandering Lakes of Wada
abstract: In 1989, Eremenko investigated the set of points that escape to infinity under iteration of a transcendental entire function, the so-called escaping set. He proved that every component of the closure of the escaping set is unbounded and conjectured that all the components of the escaping set are unbounded. Much of the recent work on the iteration of transcendental entire functions is involved in investigating properties of the escaping set, motivated by Eremenko's conjecture. We will begin by introducing many of the basic dynamical properties of iterates of an analytic function, and finally discuss constructing a transcendental entire function with a point connected component of the escaping set, providing a counterexample to Eremenko's conjecture. This is joint work with David Martí-Pete and Lasse Rempe.
Friday, March 31st, 2pm, GAB 461
speaker: Vladimir Dragovic (University of Texas at Dallas)
title: Chebyshev dynamics, isoharmonic deformations, and constrained Schlesinger systems
abstract: The talk emphases interrelations between integrable billiards, extremal polynomials, Riemann surfaces, potential theory, and isomonodromic deformations. We introduce and study dynamics of Chebyshev polynomials on several intervals. We introduce a notion of isoharmonic deformations and study their isomonodromic properties. We formulate a new class of constrained Schlesinger systems and provide explicit solutions to these systems. One line of motivation for this research goes back to works of Hitchin in mid 90’s. The talk is based on joint results with Vasilisa Shramchenko, including work in progress.
Friday, April 21st, 2pm, zoom
speaker: Alan Haynes (University of Houston)
title: The distribution of reduced rationals in the unit interval
abstract: The main part of this talk will aim to answer the question, what is the expected value of the smallest denominator of a rational number in a randomly chosen interval of fixed radius? Along the way, we will provide a cautionary tale to explain classical results that demonstrate how similar "easy" questions about the distribution of reduced rationals end up being extremely difficult to solve. For the adventurous, we will give an "easy" reformulation of the Riemann hypothesis. For the sober-minded, we will also present several other open problems.
Friday, April 28th, 2pm, zoom
speaker: Andrew Torok (University of Houston)
title: Stable laws for random dynamical systems
abstract: For a random system consisting of beta-transformations, or more general
uniformly expanding maps, we consider the convergence to a stable law (the
analogue of the Central Limit Theorem for certain observations that have
infinite second moments). We obtain quenched convergence (that is, for
almost each choice of the sequence of maps) in the Skorokhod 𝐽1
topology, by extending results of Marta Tyran-Kaminska. We obtain some of
these results also for random systems of intermittent maps.
This is joint work with Matthew Nicol and Romain Aimino.