Archives: 2010-2011 Logic and Dynamics Seminars | Department of Mathematics

# Archives: 2010-2011 Logic and Dynamics Seminars

The RTG Logic and Dynamics seminars focus on research that is of interest to both logicians and dynamists and speakers usually discuss their own research work, giving detailed proofs of their results.

This activity is sponsored by the RTG in Logic and Dynamics in the Department of Mathematics at UNT. Contact the UMRC series organizer Professor Lior Fishman at Lior.Fishman@unt.edu for more information.

Link to archive for Logic and Dynamics Seminar events occurring after Fall 2011.

# Fall 2011

Professor Mariusz Urbanski, Coordinator

December 2, 2011
Speaker: William Cherry (University of North Texas)
Title: An effective Schottky-Landau theorem for holomorphic curves in projective space
Abstract: In 1944, Dufresnoy published a generalization of Landau's theorem: if a holomorphic map from the unit disc to complex projective n-space omits 2n+1 hyperplanes in general position, then the Fubini-Study derivative of the map at the origin is bounded above by a constant. Dufresnoy's argument, making use of a normal families argument, does not effectively estimate the constant, and Dufresnoy commented that from his argument, the constant depends in an "unkown way" on the omitted hyperplanes. I will discuss joint work with Alex Eremenko about how the potential theoretic method of Eremenko and Sodin can be used to give an effective estimate for Dufresnoy's constant which, although non-sharp, gives a good sense of how Dufresnoy's constant depends on the geometry of the configuration of the omitted hyperplanes and has almost the best possible asymptotic behavior as the hyperplanes degenerate away from general position.

November 18, 2011
Speaker: Aaron Hill (University of North Texas)
Title: Centralizers of concrete rank-1 systems
Abstract: Every totally ergodic, rank-1 measure-preserving transformation is isomorphic to a concrete rank-1 system (X, µ, s), where X is a closed, shift-invariant subset of {0,1}Z, µ is a shift invariant measure supported on X, and s is the shift. We'll prove that for any concrete rank-1 system, the only homeomorphisms of X that commute with s are the integral powers of s. This is something of a topological analogue of King's weak closure theorem.

November 4 and 11, 2011
Speaker: Tushar Das (University of North Texas)
Title: On a theorem of Bishop and Jones
Abstract: Bishop and Jones, in a remarkable paper from Acta '84, proved that for any Kleinian group acting on a finite-dimensional hyperbolic space the Poincare exponent is equal to the Hausdorff dimension of the radial/conical limit set. In joint work with Bernd Stratmann and Mariusz Urbanski we generalize this result to strongly discrete groups acting on infinite-dimensional hyperbolic space. Although the original proof of Bishop and Jones crucially uses the the compactness of the sphere at infinity as well as the fact that finite-dimensional spaces are "doubling", i.e. there is a uniformly bounded number of disjoint balls of a fixed radius inside a ball of twice the radius, our proof avoids such dependence. We first prove a rather general mass-redistribution result that works for complete metric spaces and then use the group action to carefully construct a tree in hyperbolic space to which we apply the former result.

October 28, 2011
Speaker: Dustin Mayeda (University of California Davis)
Title: Diophantine approximation on the limit set of a Kleinian group
Abstract: I will recall three major theorems (Dirichlet, Khinchine, and Jarnik) in the theory of diophantine approximation on the real line. That is the quantification of how well or badly a real number can be approximated by rational ones. Then I will explain how diophantine approximation can be extended to the setting of the limit set of a Kleinian group along with the analogs of the above theorems. The set of badly approximable points on the limit set of a geometrically finite Kleinian group will be analyzed by playing a game on it. In particular it will be shown to be absolutely winning in the sense of McMullen and hence dense in the limit set.

October 14, 2011
Speaker: Kiko Kawamura (University of North Texas)
Title: A classification of self-similar sets determined by two contractions on the plane
Abstract: In the history of mathematics, we have seen some discoveries of strange functions, which gave us a strong impact; for example, the Takagi function, constructed as a simple example of anowhere differentiable but continuous function; the Von Koch curve, a continuous Jordan curve,which admits no tangent line anywhere; and the Levy's dragon curve, which is acontinuous curve but with positive area, and so on. Each of these curves was discovered independently and initially, and no relationships between them were known for a long time. In this talk, we classify binary self-similar sets, which are compact sets determined by two contractions on the plane, into four classes from the viewpoint of functional equations. Binary self-similar sets are the simplest case of self-similar sets; however, they include many interesting special cases; for instance, the Levy curve, the Von Koch curve and Polya's space filling curve. In this classification, we can not only show close relationship among fractal functions (including Cantor's devil stair function, Lebesgue's singular function and the Takagi function) but also give solutions to a few open problems in other fields. The talk will be presented with a lot of pictures: accessible even for undergraduate students. A few open problems will be also introduced.

October 7, 2011
Speaker: Mrinal Kanti Roychowdhury (The University of Texas-Pan American)
Title: Quantization Dimension for Infinite Self-Similar Probabilities
Abstract: The term quantization' in the title originates in the theory of signal processing and denotes a process of discretising signals. As a mathematical theory quantization concerns the best approximation of probabilities by discrete probabilities with a given number of points in their support. Quantization dimension' for a probability measure gives the rate at which some specified measure of the error goes to zero as n goes to infinity. Recently, I determined the quantization dimension for an infinite iterated function system consisting of self-similar mappings, and established its functional relationship with the temperature function of the thermodynamic formalism arising in mulitfractal analysis. In my talk, I will present it.

September 23 and 30, 2011
Speaker: David Simmons (University of North Texas)
Title: Existence and uniqueness of equilibrium states for random iteration of rational functions
Abstract: Thermodynamic formalism is the study of dynamical systems from an information-theoretic point of view. A key notion is the concept of an equilibrium measure, a measure which maximixes the entropy of the system plus the integral of a potential function. For any dynamical system T:X-> X an important question is whether equilibrium states exist, and if so, whether they are unique. If X is the Riemann sphere and if T is a rational function then uniqueness of equilibrium states was proven by Denker and Urbanski ('91) [existence was proven earlier by Yomdin]. The question becomes more complicated if instead of considering a single rational function T, you choose a sequence of rational functions according to some probability measure. Under certain assumptions, I proved the existence and uniqueness of equilibrium states in this case. In this talk I describe the general setup and give a sketch of my proof.

September 2 and 9, 2011
Speaker: Lior Fishman (University of North Texas)
Title: Badly approximable numbers, Schmidt's game and fractals, Parts I, II
Abstract: The set of badly approximable numbers is a classical and a well studied set within Diophantine approximation theory. It is well known that this set is "small" measureand category-wise, i.e., Lebesgue null and meager but nevertheless, "large" Hausdorff dimension-wise, possessing full dimension. In these talks we will first define the set of badly approximable numbers and other important Diophantine sets and discuss their properties. We will also survey some famous results of Dirichlet, Liouville, Borel, Jarnik and others, and introduce the Hausdorff measure and dimension. Schmidt's game and results will follow, allowing us to describe recent advances made with respect to Diophantine sets and fractals, a notion which will be defined and discussed. I strongly emphasize that these talks are accessible to everyone interested, and in particular undergrads. I will define everything needed for proving/understanding the theorems discussed!

# Summer 2011

Professor Su Gao, Coordinator

August 2, 2011
Speaker: Longyun Ding (Nankai University, China)
Title: Good Scales and Surjectively Universal Polish Groups
Abstract: In my last talk I constructed a surjectively universal Polish group by defining a very simple scale on the free group over the Baire space. The rest of the construction follows a standard procedure defined in an earlier paper of Ding and Gao on Graev-type metrics. One problem with the group constructed is that the metric is very hard to compute, and as a consequence it is very hard to explore further properties of the group. In this talk I give a different scale, a good one in the sense that it is possible to compute the metric on the resulting group easily, and still show that it gives rise to a surjectively universal Polish group.

July 26, 2011
Speaker: Longyun Ding (Nankai University, China)
Title: Surjectively Universal Polish Groups
Abstract: A problem in descriptive set theory and topological group theory that has been open for more than a decade is whether there exists a Polish group so that every other Polish group is a continuous homomorphic image of it. Such groups are called surjectively universal. In this talk I will present a self-contained proof of the existence of surjectively universal groups. The proof uses some techniques developed by Ding and Gao about Graev-type metrics on free groups.

# Spring 2011

April 22, 2011
Speaker: Aaron Hill (University of Illinois at Urbana-Champaign)
Title: Topological similarity in the group of invertible measure-preserving transformations
Abstract: Two elements g and h of a Polish group G are topologically similar if for every sequence <in> of integers, <gin> converges to the identity iff <hin> converges to the identity. We will explore this notion in the group of invertible measure-preserving transformations, showing connections to mixing properties, centralizers, and conjugacy.

April 15, 2011
Speaker: Mariusz Urbanski (University of North Texas)
Title: Regularity properties of Hausdorff dimension
Abstract: There will be an overview of results on the behavior of Hausdorff dimension in natural families of invariant sets appearing in conformal dynamical systems. The primary examples will comprise the quadratic family, the exponential family, more general classes of transcendental meromorphic functions, semi-hyperbolic generalized polynomial-like mappings, parabolic cubic polynomials, random conformal expanding systems, and non-autonomous systems of hyperbolic endomorphisms of the Riemann sphere. In all these families the Hausdorff dimension of the underlying Julia sets or the limit sets depends in a real-analytic manner on the parameter. In fact in all but the last one, where the real analyticity breaks down and all one can get is Hölder continuity. The main idea of the proofs of real analyticity will be sketched. It is based on Bowen's equation and Kato-Rellich Perturbation Theorem. Lastly, the continuity of Hausdorff measures will be also briefly discussed.

April 8, 2011
Speaker: David Simmons (University of North Texas)
Title: On fixed points of isometries of hyperbolic space, Part II
Abstract: We define what it means for a map to be conformal, explain the relation of such maps to isometries of hyperbolic space, tell you about a generalization of a classical theorem proved first by Liouville in 1850 and then proceed to give an independent (of the previous similarly titled RTG seminar) proof of the existence of fixed points of an isometry of infinite-dimensional hyperbolic space using the "Spectral Theorem".

April 1, 2011
Speaker: Tushar Das (University of North Texas)
Title: On fixed points of isometries of hyperbolic space, Part I
Abstract: One may (and often does) use the Brouwer fixed point theorem to prove the existence of fixed points of isometries of hyperbolic space of any finite dimension. However this strategy does not generalize to infinite-dimensional models of hyperbolic space. We introduce the concept of what a hyperbolic space is, remind you what it means to be an isometry in such a setting and prove the existence of fixed points of such a map via "Gromov products".

March 25, 2011
Speaker: Scott Crass (California State University Long Beach)
Title: Dynamics with 168-fold symmetry
Abstract: The collineation group of seven-point projective space consists of 168 elements that exhibit rich combinatorial behavior. Felix Klein discovered an elegant representation of this group on the complex plane. Developing some of the action's rich algebraic and geometric properties leads to a special holomorphic map on CP2 that is symmetric (or equivariant) under the group. The map is distinguished by its critically-finite behavior from which follow several global dynamical results. The map's dynamics provides the main tool in an algorithm that solves certain "heptic" equations. The discussion will treat the interplay between the map's geometric and dynamical properties.

March 4 and 11, 2011
Speaker: John Clemens (University of North Texas)
Title: Complemented sets of integers and weakly wandering sequences for transformations
Abstract: A set A of integers is said to be complemented if there is a second set B such that each integer n can be uniquely written as the sum n = a + b with a in A and b in B. Complemented sets are closely related to both difference sets and weakly wandering sequences for measure-preserving transformations, where a set A is a difference set if there is some set B such that A = {n - m : n,m in B}, and a set W is weakly wandering for a transformation T on X if there is a measurable subset Z of X of positive measure such that for all distinct m and n in W we have that the translated Tm[Z] and Tn[Z] are disjoint.

I will explain the connections between these notions, and then use descriptive set-theoretic techniques to show that it is difficult to determine which sequences have the above properties; specifically, the following are all complete analytic sets: the set of complemented sequences, the set of sequences containing an infinite difference set, the set of sequences which are weakly wandering sequences for some transformation, and several variants of these. I will then use the same techniques to produce weakly wandering sequences with special properties, such as a sequence which is exhaustive weakly wandering for some transformation but which is not weakly wandering for any ergodic transformation.

February 11, 18, and 25, 2011
Speaker: Su Gao (University of North Texas)
Title: The construction of a 2-coloring
Abstract: It was promised in Steve Jackson's talks last semester that 2-colorings exist on any countably infinite group. Here I will give the promised construction.

# Fall 2010

November 5, 12, 19, and December 3, 2010
Speaker: Steve Jackson (University of North Texas)
Title: Bernoulli subflows and countable group colorings
Abstract: I will consider some basic properties about the Bernoulli shift action of a general countable group and show how they can be formulated in various equivalent ways. This talk serves as an introduction to many problems and results that will be discussed in this seminar in subsequent talks. I will prove some elementary results about the concept of 2-colorings on a countable group. A recurring scheme is that notions significant from the point of view of dynamical systems admit equivalent combinatorial characterizations.

October 29, 2010
Speaker: John Clemens (University of North Texas)
Title: Generic trees, pointed trees, and injective selectors
Abstract: I will continue discussing the notion of weakly pointed trees (ones
with branches able to effectively compute the original tree), and show that suitably generic trees can not be weakly pointed. I will then show how to convert this into a topological argument showing that there there are Borel plane sets with uncountable sections (on a comeager set) which do not admit injective selectors with non-meager domain.

October 22, 2010
Speaker: John Clemens (University of North Texas)
Title: Injective selectors, pointed trees, and effective randomness
Abstract: I will discuss a theorem guaranteeing the existence of injective partial selectors and consider effective versions related to algorithmic randomness. In particular, I will show that any suitably random uniformly branching tree has branches which can effectively compute the structure of the entire tree.

October 8 and 15, 2010
Speaker: David Simmons (University of North Texas)
Title: Rokhlin's Disintergration Theorem
Abstract: In this talk I will discuss the concept of conditional measures and Rokhlin's disintegration theorem, which states that conditional measures exist in non-pathological cases. A proof will be provided based on a generalized version of the Lebesgue differentiation theorem. If there is time I will talk about generalizations, and about how to calculate conditional measures if the maps are differentiable.

September 17, 24, and October 1, 2010
Speaker: Alex McLinden and Dan Mauldin (University of North Texas)
Title: Translating the Cantor set by a random
Abstract: Alex McLinden will be discussing the basics of constructive dimension and Kolmogorov complexity and the relationship between constructive dimension and classical Hausdorff dimension. Then Dan Mauldin will be discussing translations of the Cantor set by a random real number, and discuss how the Cantor set "cancels randomness" to some degree.