Hausdorff dimension of graphs of
functions.
Erdos: Is there a positive constant c such that if E is a
Lebesgue measurable set in the plane with measure greater than c
then E contains the vertices of a triangle of area 1?
Separable, completely metrizable spaces
are ubiquitous in mathematics. These spaces are the natural
setting for descriptive set theory. It is therefore of interest
to find conditions which imply that a separable metrizable space
is completely metrizable.
After a quick review of Hurewicz's
criterion and Sierpinski theorem, I will prove Vainstein's
result. Then I will define the class of resolvable maps, and
prove that these preserve complete metrizability. This answers a
recent question of Ostrovsky and is in some sense the best
possible theorem along these lines.
Continued talk about Bernoulli
trial measures m on the Cantor space of consisting all
of infinite sequences on two symbols.
We will give some details about
when such a measure is uniquely ergodic and the problems
of when there is a measure preserving homeomorphism h
leaving such m ergodic and for which h has exactly k
ergodic measures.
The entropy of k-to-1 maps.
ABSTRACT: We consider the dynamics of polynomial
semigroups (i.e. semigroups generated by polynomial maps) on
the Riemann sphere and random complex dynamics of a family
of complex polynomials. We classify (semi-)hyperbolic
semigroups with bounded postcritical set in the plane, in
terms of random dynamics of polynomials. We will show that
there are three types of such semigroups, and any semigroup
of one of these types satisfies that for almost every
sequence, the Random Julia set is a Jordan curve but not a
quasicircle, and the unbounded component of the complement
of the Random Julia set is a John domain. Moreover, we will
give an example of such a semigroup. This is a phenomenon
which occurs in the random dynamics of a family of
polynomials, but does not occur in the usual dynamics of a
single
April 27, 2007
Professor Pieter Allaart
Distribution of the maxima of random Takagi functions
Takagi's continuous nowhere differentiable function is
defined as a series of iterates of the tent map, scaled to
ensure convergence of the series. In this talk, we will
consider a random version where each term of the series is
multiplied by a random sign. We will investigate the
distribution of the maximum value of this random function,
as well as the size of the set of points where the maximum
is attained. I will show how this problem leads naturally to
self-similar measures and random Cantor sets.
Wednesday, March 14, 2007
Professor Eugen Mihailescu
(Rumanian Academy of Science)
Dynamics of skew products and other types
of endomorphisms
I will present some new developments in the study of
skew products and maps with hyperbolic basic sets having
locally constant degree. These will include in particular
metric properties, dimension estimates and properties of
equilibrium measures.
March 2, 2007
Antti Kaenmaki
Uniqueness of equilibrium measures for
self-affine systems.
ABSTRACT: He will discuss some results from his
dissertation and some open problems concerning these
systems.
February 2, 2007
Dan Mauldin
Homeomorphic Bernoulli trial measures on
the Cantor set.
We consider Bernoulli trials with k possible outcomes
with probabilities p_1,...p_k and the induced probability
measure m(p_1,...,p_k) on the infinite product of the space
{1,...,k}. The problem is to determine when two such
measures m_1 and m_2 are topologically equivalent: When is
there an autohomeomorphism h of this Cantor space such
m_1(E) = m_2(h(E)), for all Borel sets E? I will give some
history of the problem and describe some joint work with
Ethan Akin, Randy Dougherty and Andy Yingst.
November 10, 2006
Sara Munday
The Modular Surface and Continued
Fractions.
The aim of this presentation is to make clear the
connection between geodesics on the modular surface M (the
quotient of the hyperbolic plane by the modular group G =
SL(2, Z)) and continued fractions. This connection was first
described by G. Artin in 1924, this presentation follows a
1985 paper of Caroline Series.
October 6, 2006
Prof. Alex Clark
Rigidity of Affine Maps and a Question
of Walters
ABSTRACT: We will give an introduction to rigidity
of affine maps of compact connected abelian groups. We will
then develop a criterion for the rigidity of affine maps
that allows one to construct examples that settles a
question of Walters.
September 29, 2006
Prof. William Cherry
Effective Landau/Schottky Theorems for
Holomorphic Curves in Projective Space
ABSTRACT: Recall that Picard's theorem says that
an entire function with omits 0 and 1 must be constant. The
analagous result for analytic functions in the unit disc is
Landau's Theorem: if f is an analytic function on the unit
disc and omits 0 and 1, then |f'(0)| is explicitly bounded
in terms of f(0). Analagous to Picard's theoorem is the fact
that a holomorphic curve in n dimensional complex projective
space omitting at least 2n+1 hyperplanes in general position
must be constant. In 1944, Dufresnoy published a paper where
he proves an analogous Landau theorem, but his estimate,
using a normal families argument, is not effective and he
comments that the undetermined constant depends in an
"unknown way" on the set of omitted hyperplanes. I will
discuss recent joint work with Alex Eremenko where we use
the potential theoretic method of Eremenko and Sodin to give
an explicit effective estimate on Dufresnoy's constant, that
although very far from sharp, does give some idea how the
constant depends on the omitted hyperplanes.
Thursday, August 24, 2006 at 2.00 PM -
3.30 PM.
Prof. Marc Kesseboehmer (University of
Bremen)
Limiting modular symbols and their
fractal geometry(Joint work with BO Stratmann)
We introduce Manin's modular symbols for a finite index
subgroup $G$ of PSL$_2(\mathbb{Z})$. It was also Manin who
suggested to 'complete' the set of modular symbols by
defining limiting modular symbols $\ell$ on the boundary of
infinity $\partial M_G$ of the Riemann surface $M_G:=\mathbb{H}/G$.
We will show that this gives rise to non-trivial elements of
the first homology $H_1(M_G,\mathbb{R})$ and determine the
Hausdorff dimensions of the level sets of $\ell$.
May 3, 2006
Prof. Janina Kotus (Warsaw University
of Technology
Introduction to Holomorphic Dynam
April 28, 2006
Steven Muir
"Topological Entropy for piecewise
Monotone Maps of an Interval" (continuation)
March 31, 2006
Pieter Allaart
"Dimension estimates regarding Polya's
space-filling curve"
Abstract: This talk is about the resulting
dynamics when the relaxed Newton's method is applied to the
exponential function F(z)=P(z)exp(Q(z)), where P and Q are
complex polynomials, P is not identically zero and Q is
non-constant. In essence, this becomes an analysis of the
dynamic behavior of a family of rational functions. The
finite fixed points of the relaxed Newton's method are the
attracting fixed points which are the roots of the
polynomial P and infinity is a parabolic fixed point. We
will show that the immediate basins of attraction have
finite area for deg(Q) at least 3 and in fact have a finite
upper bound with respect to the planar Lebesgue measure. The
main idea of these results in based on M.E. Haruta's study.
March 3, 2006
Hasina Akter (UNT)
"Newton's method for transcendental
entire functions"
February 24, 2006
Title: Koebe's Distortion Theorem
Tushar Das (UNT)
February 10, 2006
Title: Divergent Square Averages
Zoltan Buczolich-Eotvos Univ. Budapest, Hungary
January 27, 2006
Title: Ionescu Tulcea-Marinescu
Theorem
Sara Munday
Abstract: A useful theorem about asymptotic behaviour of
iterates of a class of bounded opersators will be proved.
This theorem has profound applications in thermodynamic
formalism.
October 21, 2005
Title: Topological pressure via saddle
points Professor Christian Wolf, Affiliation: Wichita
State
Abstract: et $\Lambda$ be a compact
locally maximal invariant set of a
$C^2$-diffeomorphism $f:M\to M$ on a smooth
Riemannian manifold $M$. In this talk we discuss the
topological pressure $P_{ \rm top}(\varphi)$ (with
respect to the dynamical system $f|\Lambda$) for a wide
class of H\"older continuous potentials and analyze its
relation to dynamical, as well as geometrical,
properties of the system. We show that under a mild
nonuniform hyperbolicity assumption the topological
pressure of $\varphi$ is entirely determined by the
values of $\varphi$ on the saddle points of $f$ in
$\Lambda$.
October 14, 2005
Title: "Properties of unstable manifolds for
non-invertible hyperbolic maps"
Professor Eugen Mihailescu, Romanian
Academy of Science ABSTRACT:
We study the case of a smooth hyperbolic non-invertible map
in higher dimension. First, the unstable dimension is proved
to be the zero of the pressure function of the unstable
potential considered on the space of prehistories. Then we
take a closer look at the construction of local unstable
manifolds for a perturbation g of f, and estimate the speed
of convergence of the unstable dimension when g approaches
f. Also we show that there exist Gibbs measures on the
intersections between local unstable manifolds and basic
sets. Unlike in the case of homeomorphisms, the full
Hausdorff dimension of basic sets does not vary continuously
w.r.t the perturbation. In the holomorphic case we give a
better estimate of the Holder exponent of unstable conjugacy.
September 28, 2005
Title: "Strong Dunford-Pettis Sets"
Paul Lewis
March 4, 2005
Title: "Iterated function systems associated
with Gauss-like and Renyi-like maps"
Andrei Ghenciu, ABSTRACT: We are going to
present conformal iterated > function systems and parabolic
iterated function > systems associated with Gauss-like
transformations > and > Renyi-like transformations. Also we
are going to > study > the spectrum made of the Hausdorff
dimensions of the > limit sets of subsystems
2002 - 2003
November 21, 2003
Title: "Metric properties of some family of
meromorphic functions with an asymptotic value mapped onto
pole"
Bartlomiej Skorulski, Faculty of Mathematics and
Information Sciences, Warsaw University of Technology
Abstract: We describe the dynamics of
some non-entire transcendental meromorphic functions with a
finite asymptotic value mapped after some iterations onto a
pole. This situation does not appear in the case of rational
or
entire functions. We study the family of functions given
by the formula $$
f(z)=\frac{a\exp(z^p)+b\exp(-z^p)}{c\exp(z^p)+d\exp(-z^p)},
$$ with the property that there exist a finite asymptotic
value $\xi$ and a positive integer $q$ such that $f^q(\xi)=\infty$.
We will exhibit some metric properties of such a map, e.g.
an estimation of the Hausdorff dimension of the Julia set,
the Hausdorff dimension of the set of points with bounded
trajectory, the Lebesgue measure of the the Julia set and
some others.
September 17, 2004
Title: "The dimensions of Julia sets of expanding
rational semigroups"
Professor Hiroki Sumi from Tokyo Institute of Technology"
Abstract: We estimate the upper and Hausdorff
dimensions of the Julia set of an expanding semigroup
generated by finitely many rational functions, using the
thermodynamic formalism in ergodic theory. Furthermore, we
show Bowen's formula, and the existence and uniqueness of a
conformal measure, for a finitely generated expanding
semigroup satisfying the open set condition.
September, 13, 2002
Title: "Entropy structure and symbolic extensions
in compact dynamical systems"
Tomasz Downarowicz, Wroclaw University of Technology
Abstract: By a (dynamical) system we will
mean an action of a homeomorphism T on a compact metric
space X. A system is called symbolic or subshift) if T=S is
the shift map on a (compact, shift invariant) subset Y of
0,1,2,...,l}^Z. Clearly such Y is zero-dimensional, S is
expansive, and (Y,S) has finite topological entropy. The
following is well known: every measure theoretic system (X,m,T)
with finite entropy has a measure theoretic symbolic
representation (Krieger's generator theorem); every
topological system (X,T) has a zero-dimensional topological
extension (X',T'). Consider the following natural questions:
Does every finite entropy system (X,T) have a symbolic
topological extension? If not, what is the condition? Can
the entropy of a symbolic extension be made equal or
arbitrarily close to the entropy of (X,T)?
In a series of lectures we will present a relatively
complete theory allowing to settle down the above questions
and providing tools to measure the entropy distance between
(X,T) and its symbolic extensions (Y,S). The first lecture
will contain an introduction to the problem and easy
inequalities. The next few lectures will be devoted to some
functional analytic tools concerning upper semicontinuous
functions and affine functions. Last few lectures will apply
these tools to handle the symbolic extensions.
September, 20, 2002
Title: "Entropy structure and symbolic extensions
in compact dynamical systems" (Continued)
Tomasz Downarowicz, Wroclaw University of Technology
September, 27, 2002
Title: "Entropy structure and symbolic extensions
in compact dynamical systems" (Continued)
Tomasz Downarowicz, Wroclaw University of Technology
October 4, 2002
Title: "Entropy structure and symbolic extensions
in compact dynamical systems" (Continued)
Tomasz Downarowicz, Wroclaw University of Technology
October 11, 2002
Title: "Entropy structure and symbolic extensions
in compact dynamical systems" (Continued)
Tomasz Downarowicz, Wroclaw University of Technology
October 18, 2002
Title: "The dynamics of one-dimensional tiling
spaces"
Alex Clark, University of North Texas
Abstract: We will examine the results
obtained by L. Sadun and the speaker on the dynamics of
one-dimensional tiling spaces. In particular, we will focus
on the issue of when varying the size of the tiles of a
self-similar tiling affects the dynamics of the tiling. We
will see that in some cases the dynamics are essentially
unchanged, while in other cases the dynamics are
dramatically changed.
November 8, 2002
Title: "Methods and results in higher dimensional
complex dynamics I"
Eugen Mihailescu, University of North Texas and the
Institute of Mathematics of the Romanian Academy
November 15, 2002
Title: "Methods and results in higher dimensional
complex dynamics II"
Eugen Mihailescu, University of North Texas and the
Institute of Mathematics of the Romanian Academy
January 31, 2003
Title: "Twist-wise flow equivalence"
Michael Sullivan, University of North Texas
Abstract: This will be the first of a two
or maybe three part talk. We will "review" some basic facts
from symbolic dynamics. Once everyone understands what flow
equivalence is and how it is determined, I will define
twist-wise flow equivalence (basically we model how flow
orbits twist around each other) and give examples. The
problems will then be reduced to classifying square matrices
with entries from the group ring Z[Z/2Z] into equivalence
classes determined by certain matrix moves. Then I will
define some easy to compute invariants; but they are not
complete. I will report on joint work with Mike Boyle in
which ideas motivated by K-theory are applied to reframe the
problem of twist-wise flow equivalence in terms of ordinary
matrix SL equivalence, again over the group ring Z[Z/2Z].
Potential new invariants and a difficult ring theoretic
problem we be discussed. Lecture notes will be provided.
February 7, 2003
Title: "Twist-wise flow equivalence"
(Continued)
Michael Sullivan, University of North Texas
February 14, 2003
Title: "Twist-wise flow equivalence"
(Continued)
Michael Sullivan, University of North Texas
February 21, 2003
Title: "Micro Tangent Sets of Continuous Functions"
Zoltan Buczolich
Abstract: Motivated by the concept of
tangent measures and by H. Furstenberg's definition of
microsets of a compact set we introduce micro tangent sets
and central micro tangent sets of continuous functions. It
turns out that the typical continuous function has a rich
(universal) micro tangent set structure at many points. The
Brownian motion on the other hand with probability one does
not have graph like, or central graph like micro tangent
sets at all. Finally we show that at almost all points
Takagi's function is graph like, and Weierstrass's nowhere
differentiable function is central graph like. (This talk
might be of interest for students taking the Topics in
Analysis class.)
February 28, 2003
Title: "A solution to the gradient problem of C.
E. Weil"
Zoltan Buczolich, University of North Texas and Lorand
Eotvos University in Budapest, Hungary
Abstract: In this talk I give an answer
to the gradient problem of C. E. Weil. While derivatives in
dimension one have the Denjoy-Clarkson property, my
counterexample shows that this propery does not always hold
for gradients of differentiable real functions with higher
dimensional domains. The gradient problem originated from
the 60s. I have learned it in 1987 and, after 15 years of
work, last summer I managed to find this counterexample.
This talk is independent from my talk given on the 21st of
February and probably will be continued on March 7. In this
first part I will discuss the result and the background
history, while on March 7 I plan to talk about the
construction/proof.
March 7, 2003
Title: "A solution to the gradient problem of C.
E. Weil" (continued)
Zoltan Buczolich, University of North Texas and Lorand
Eotvos University in Budapest, Hungary
2001-2002
September 21, 2001
Title: "Conformal Iterated Function Systems and
Their Applications to Elliptic Functions"
Mariusz Urbanski, University of North Texas
September 28, 2001
Title: "On the Convergence of Sums of Translates
of Measureable
Functions"
Daniel Mauldin, University of North Texas
October 5, 2001
Title: "Fibrewise Expansive Systems"
Mario Roy, University of North Texas
October 13, 2001
Title: "The Linking Homomorphism of
One-dimensional Minimal Sets of Flows"
Alex Clark, University of North Texas
Abstract: We will discuss a way the
speaker and M. C. Sullivan have developed for characterizing
the linking of one-dimensional minimal sets. The talk will
begin with a basic introduction to the linking of periodic
orbits. Then we will examine the structure of a special
class of minimal sets that occur in three-dimensional flows
modeled by Lorenz templates. The structure of these minimal
sets, which have the shape of the wedge of two circles,
naturally leads to the linking homomorphism. We will
consider examples and consider some directions for future
research.
January 18, 2002
Title: "Diffusion to Infinity for Periodic Orbits
in Meromorphic Dynamics"
Professor Janina Kotus, Warsaw Institute of Technology
Abstract: For families of rational
functions, periodic orbits of any period cannot suddenly
appear or dissapear. If $F_{\lambda_0}$ has a periodic point
$p_0$ then under small perturbation of $\lambda$there will
be related periodic points nearby. All periodics orbits of
given period for small changes of the parameter are realized
in this way. For entire functions, however, it becomes
conceivable that as the parameter tends to some value $
\lambda_0$inside the parameter space, the orbit nevertheless
migrates toward $\infty$so that at $\lambda_0$ it has no
counterpart. This appears more likely still for the families
of meromorphic functions, since all that is needed for a
point to escape to $\infty$is that the preceeding point goes
to a pole.
The presented results go in two directions. First, we
show that under certain conditions which are naturally
satisfied by most popular families, diffusion of periodic
orbits to $\infty$ may occur only in the case when a finite
asymptotic value of the map hits the pole. This immediately
"explains" how structural stability is possible if the
asymptotic values are under control. Secondly, we prove that
if the a post asymptotic pole is present, the diffusion can
naturally occur.
January 25, 2002
Title: "A geometric characterization of uniqueness
polynomials for entire functions"
Professor William Cherry, University of North Texas
Abstract: A polynomial P is called a
strong uniqueness polynomial for the family of non-constant
entire functions on the complex plane if one cannot find two
distinct non-constant entire functions f and g and a
non-zero constant c such that P(f)=cP(g). In joint work with
Julie Wang, we give necessary and sufficient geometric
conditions on zeros of $P$ that $P$ be a strong uniqueness
polynomial for the family of non-constant entire functions
on the
complex plane.
February 15, 2002
Title: "Finer Julia set for a class of entire
functions"
Professor Anna Zdunik, University of Warsaw
Abstract:We consider the maps $f_\lambda(z)=\lambda\exp(z)$
such that $f_\lambda$ has an attracting periodic orbit. Let
$J(f_\lambda)$ be the Julia set of $f_\lambda$. It is known
that the Hausdorff dimension of $J(f_\lambda)$ is equal to
$2$, while the $2$-dimensional Hausdorff measure of $J(f_\lambda)$
is zero.
We introduce the "finer Julia set" $J_r(f_\lambda)=\{z\in
J(f_\lambda):f^n_\lambda(z)$ does not tend to $\infty\}$ and
prove that its Hausdorff dimension $h_\lambda$ is less than
$2$. We also prove that $h_\lambda$-dimensional Hausdorff
measure of $J_r(f_\lambda)$ is non-zero and finite on each
horizontal strip, while the $h_\lambda$-dimensional packing
measure is locally infinite at each point of $J_r(f_\lambda)$.
Next, we introduce an appropriate topological pressure,
Perron-Frobenius operators, generalized geometric and
invariant conformal measures. Using these tools we prove
that (in a neighbourhood of a parameter $\lambda_0$
satisfying our assumptions) $\HD(J_r(f_\lambda))$ is a
real-analytic function of $\lambda$.
These results were obtained jontly with Mariusz
Urba'nski.
February 15, 2002
Title: "Finer Julia set for a class of entire
functions"
Anna Zdunik, University of Warsaw
March 1, 2002
Title: "Lattice Problems of Steinhaus"
Andrew Yingst, University of North Texas
April 19, 2002
Title: "Polynomial and rational approximation in
complex analysis"
Alex Izzo, Bowling Green State University and Texas A&M
Abstract: When can the continuous
functions on a compact set be approximated uniformly by
complex polynomials? This question is completely understood
for sets in the plane. In higher dimensions, the situation
is much more complicated, and despite many partial results,
the question is still wide open. In this talk we will
discuss some of the know results (both in the plane and in
higher dimensions) and present some open problems. The talk
will be aimed at a general mathematical audience and will be
accessible to anyone with a basic knowledge of complex
anaysis in ONE variable.
May 16, 2002
Title: "Geometry of conformal dynamical systems"
Bernd Stratmann, University of St. Andrews
Abstract: Limit sets of Kleinian groups
and Julia sets of rational maps represent highly inspiring
examples of fractal sets. For the Kleinian case we sketch
the construction of how to derive these conformal attractors
from within the associated hyperbolic 3-manifolds, and then
give a brief survey of some of the most important structural
theorems (Ahlfors, Thurston, Sullivan, Bishop and Jones). On
the basis of this discussion, we then highlight for the
geometrically finite scenario the strong analogies between
these two twin stars in complex dynamics (radial limit sets,
conformal measures, weak multifractal and thermodynamic
phase transitions). Finally, there will be a brief
discussion of the geometrically infinite cases, introducing
concepts such as Joergensen points, horospherical points and
the NCP-condition for Kleinian groups.
2000-2001
September 8, 2000
Title: "Quantization Dimension for conformal
Iterated Function Systems"
Larry Lindsay, UNT Graduate Student
September 15, 2000
Title: "Billiards and Veech Groups"
Pascal Hubert, Institut de Mathematique de Luminy, France
Abstract: The study of some polygonal
billiards is closely related to the study of linear flows on
Riemann surfaces. These Riemann surfaces are endowed with a
translation structure (or an holomorphic form). Very
interesting examples were found ten years ago, by W. Veech.
He defined groups (now called Veech groups) which play an
important role for the understanding of these dynamical
systems.I will try to explain applications of these groups
from different points of view (dynamical systems and
algebraic geometry).
September 29, 2000
Title: "Outer measure constructions, Fubini type
theorems and descriptive
set theory"
R. Daniel Mauldin, University of North Texas
Abstract: We indicate the interaction
between descriptive set theory and some problems from
geometric measure theory and Fubini type theorems - some
solved some unsolved. Time permitting we give applications
to generalized Hausdorff measures and packing dimensions.
October 6, 2000
Title: "Rigidity of Connected Limit Sets of
Conformal IFS"
Dr. Mariusz Urbanski; University of North Texas
Abstract: We consider infinite conformal iterated
function systems in the phase space $\R^d$with $d\ge 2$. Let
$J$ be the limit set of such a system. Under a mild
technical assumption which is always satisfied if the system
is finite, we prove that either the Hausdorff dimension of
$J$ exceeds $1$ or else the closure of $J$ is a proper
compact segment of either a geometric circle or a straight
line if $d\ge 3$ or an analytic interval if $d=2$
October 20, 2000
Title: "Stable Hausdorff dimension for Axiom A
endomorphisms"
Prof. Eugene Mihailescu- Texas A&M at College Statio
Abstract: "Given an s-hyperbolic
endomorphism of P^{2}, Fornaess-Sibony have introduced the
set K^{-} which is the analogue of the set of points with
bounded backward iterates from the case of diffeomorphisms.
In this talk we will consider the Hausdorff dimension for
the set K^{-} of a holomorphic endomorphism of P^{2} and
also, in connection with this, the Hausdorff dimension of
the intersection between stable manifolds and basic sets. In
this case there are several important differences from the
situation of Henon maps."
January 26, 2001
Title: "On the wild components of the family
$\lambda \tan(z)$"
Janina Kotus, Warsaw Instutute of Technology
Abstract: It is natural to ask if the
hyperbolic components of the tangent family ${\cal
F}=\{\lambda \tan(z): \lambda \in \C-\{0\}\}$ are the only
open components of the $J$-stable maps. The evidence of the
computer pictures indicates an affirmative answer to this so
called {\it Densiy conjecture}. If the density conjecture is
not true, there are non-trivial components of the $J$-stable
set containing non-hyperbolic maps; we call them {\it wild
components}. We prove that there are no wild components in
the family $\cal F$ for which the orbits of the asymptotic
values are unbounded, what is a partial positive answer to
the density conjecture.
February 2, 2001
Title: "The Rotation Class of a Flow"
Alex Clark, University of North Texas
Abstract: Generalizing a construction of
A. Weil, we introduce the rotation class of a flow: a
topological invariant for flows on compact, connected,
finite dimensional, Abelian, topological groups. We will
start with some simple examples of flows on the torus and
calculate Weil's invariant and then see how this naturally
leads to the more general rotation class. Then we will
calculate the rotation class for some examples and compare
the information it gives with the information yielded by
other invariants.
February 9, 2001
Title: "Geometry and dynamics of transcendental
meromorphic functions"
Mariusz Urbanski, University of North Texas
February 23, 2001
Title: "On a lattice problem of Steinhaus"
Dan Mauldin, University of North Texas
Abstract: In 1957 H. Steinhaus asked: Is
there a set in the plane which meets each isometric copy of
Zz in exactly one point? We discuss the history and status
of the problem including the recent positive solution of
Steve Jackson and myself.
March 2, 2001
Title: "On a lattice problem of Steinhaus"
(continued)
Dan Mauldin, University of North Texas Abstract: In 1957
H. Steinhaus asked: Abstract: Is there a
set in the plane which meets each isometric copy of Zz in
exactly one point? We discuss the history and status of the
problem including the recent positive solution of Steve
Jackson and myself.
April 6, 2001
Title: "Cellular Automata and Graph Directed
Construction"
Larry Lindsay, University of North Texas
Abstract: We describe a class of cellular
automata and the conversion of them to graph directed
construction and how to analyze the limit sets.
April 20, 2001
Title: "Computing Hausdorff dimension of sets
invariant under conformal expanding dynamical systems"
Professor Oliver Jenkinson, Queen Mary, University of
London
Abstract: As is well known, the
Hausdorff dimension of the classical "middle-third" Cantor
set is log2/log3. For more general fractal sets it is
impossible to obtain an explicit expression for its
Hausdorff dimension, so there is interest in efficiently
approximating the dimension. C. McMullen (Amer. J. Math.
1998) gave an algorithm for computing the dimension of
various dynamically defined sets, for example Julia sets of
holomorphic maps, limit sets of various Kleinian groups,
etc. In this talk I will present two algorithms for
computing Hausdorff dimension of such sets.
The first algorithm is joint work with Mark Pollicott,
and is theoretically more efficient than that of McMullen,
and in practice (ie working within the constraints of modern
computers) is more efficient for many interesting examples.
The method relies on understanding the periodic orbits of
the underlying dynamical system. The lecture will contain a
pot pourri of examples: fractal sets arising in number
theory, continued fractions, Diophantine approximation,
Markoff and Lagrange spectra; Julia sets for holomorphic
maps, especially quadratic maps z^2+c, dimension as a
function of the parameter c, the Mandelbrot set; limit sets
for certain Kleinian groups - Schottky groups, Fuchsian
groups, quasi-Fuchsian groups. We will then review intuitive
and formal definitions of dimension, Bowen's "pressure
formula" for Hausdorff dimension, and move onto the
thermodynamic ideas underlying the algorithm - transfer
operators, nuclear operators, Fredholm determinants,
periodic points, trace formulae. We will briefly discuss
computational issues, and give some explicit numerical
results.
The second algorithm is designed to approximate Hausdorff
dimension of infinite branch systems, with the key example
being Complex Continued Fractions (as studied by Gardner \&
Mauldin, and Mauldin & Urbanski). (The above periodic point
method is much less effective in this case, since for any
given $n$ there are infinitely many period-$n$ points). The
algorithm works by projecting the transfer operator onto a
finite dimensional subspace, and analysing the spectral
radius of the resulting finite rank operator.
1999-2000
September 17, 1999
Title: "Maps of Matchbox Manifolds Homotopic to
the Identity and the Non-existence of Expansive
Homeomorphisms"
Alex Clark, UNT Math Faculty
Abstract: The nature of the set of fixed
points of a map of a matchbox manifold homotopic to the
identity is examined. Also, it is shown that all solenoids
from a certain class admit no expansive homeomorphism.
October 1, 1999
Title: "Mixing and the Foundations of Statistical
Mechanics"
Constantino Tsallis, UNT Physics Faculty
Abstract: A great variety of physical
systems badly acomodate within standard, Boltzmann-Gibbs
statistical mechanics. A proposal has been done one decade
ago which consists in using a nonextensive entropy to derive
generalized statistical mechanics. This entropy has been
observed to be of remarkable interest for describing
nonlinear dynamical systems with power-law, instead of
exponential, mixing. This seems to directly lead to a
power-law distribution for equilibrium, instead of the
Boltzmann-Gibbs exponential one.
October 22, 1999
Title: "Porosities and Dimensions of Measures"
Esa Jarvenpaa, University of Jyvaskyla, Finland
[Joint work with Jean-Pierre Eckmann and Maarit Jarvenpaa
Abstract: Can one estimate the size of a
measure using information about the holes it contains? I
answer to this question by defining the concept "porosity of
measure" and by proving that big porosity implies small
dimension.
October 29, 1999
Title" "Random walks on the Sierpinski gasket"
Artemi Berlinkov, UNT Math Teaching Fellow
Abstract: We consider the construction
of a random walk on the Sierpinski gasket, find its
dimension and prove that its packing measure is positive and
finite.
November 5, 1999
Title: "Classification of Anosov families"
Albert Fisher, Faculty, IME-USP, Sao Paulo, Brazil
Abstract: An Anosov family is a sequence
of diffeomorphisms along compact manifolds such that the
tangent bundles split into expanding and contracting
subspaces. A single Anosov diffeomorphism corresponds to a
constant family.
Another type of example comes from renormalization
theory. From a geometrical perspective these families
(multiplicative families) are given by a sequence of
Teichmuller maps of the torus, taken along a geodesic in
Teichmuller space, with the direction determined by a pair
of foliations. The class of all such Anosov families
naturally imbeds in a flow on a torus fiber bundle over the
unit tangent bundle of the Teichmuller space of the torus.
This flow, the Teichmuller mapping flow, extends the
classical modular flow (which is the Teichmuller flow of the
torus).
A natural flow cross-section is a skew product
transformation whose base is the multiplicative continued
fraction (whence the name multiplicative family). This
crossection (and hence the collection of all multiplicative
families) forms a natural completion to the collection of
all Anosov maps, with the Anosov maps corresponding to
periodic flow orbits, and equivalently periodic two-sided
continued fraction expansions and corresponding subshifts of
finite type.
Other examples of Anosov families come from random
dynamics: a random product of invertible integer matrices,
with nonzero Lyapunov exponents, defines an Anosov family on
the d-dimensional torus.
For the case of the two-torus, we classify linear Anosov
families up to the equivalence relation generated by
conjugacy in the modular group plus gathering and dispersal:
the partial composition of maps and its converse, inserting
more maps. The statement is that any Anosov family is
conjugate to a unique multiplicative family.
November 12, 1999
Title: "Gibbs states on the symbolic space over an
infinite alphabet"
Mariusz Urbanski, UNT Math Faculty
November 19, 1999
Title: "Gibb States and Applications to
Multifractals"
R. Daniel Mauldin, UNT Math Faculty
Abstract: We will give some specific
examples of Gibb states arising in the study of continued
fraction systems and Apollonian packing.
Febuary 25, 1999
Title: "Complexity of fractal sets and 'P is not
equal to NP?' problem"
Dr. Kiko Kawamura, post-doctoral visitor, Nara Women's
University, Japan
Abstract: The aim of this study is to
find a mathematical tool other than fractal dimensions and
considered to be an estimation of complexity in fractal
geometry. We investigate self-similar sets from viewpoint of
Pour-El & Richards style computable analysis, and propose
computational complexity as one of the tools to estimates
complexity of self-similar sets.
March 3, 1999
Title: "Some Problems in Measure Theory and
Geometry"
Dan Mauldin, UNT Faculty
Abstract: We will discuss come
long-standing simply stated elementary unsolved problems.
March 10, 1999
Title: "Positivity of disintegration kernels of
random measures generated by cascading exchangeable
processes"
Stanley C. Williams, Utah State University
Abstract: Cascading likelihood ratios
associated with exchangeable processes leads to random
measures which are generalization of martingales of
Mandelbrot, and which fall within the theory of
T-martingales of Kahane. Disintegrations of these measures
in terms of ergodic limits are investigated in terms of
linkage and separation. The effects of X-factors leading to
multiplicative cascades are studied.
March 31, 1999
Title: "On the Visibility of Invisible Sets"
Dr. Marianna Csörnyei, University College London
Abstract: A planar set A is called
invisible, if its orthogonal projection is of measure 0 in
almost every direction; A is visible, if it is not
invisible. We say that A is invisible from a point, if
almost all lines through that point do not hit the set. P.
Mattila raised the poetic question whether the set of points
from which an invisible set is visible is invisible. R. O.
Davies proved in 1952 that an arbitrary measurable planar
set A can be covered by lines in such a way that the set of
all points covered by these lines has the same Lebesgue
measure as A. We prove that the same result holds not only
for Lebesgue measure, but for every sigma-finite Borel
measure on the plane. We show how this result may be used to
answer the question of Mattila, and we characterize the sets
of the plane from which an invisible set is visible.
April 7, 1999
Title: "Hausdorff dimension of harmonic measures
for self-conformal sets"
Mariusz Urbanski, UNT faculty
Abstract: Under some technical
assumptions it will shown that the Hausdorff dimension of
the harmonic measure on the limit set of a conformal
infinite iterated function system is strictly less than the
Hausdorff dimension of the limit set itself if the l limit
set is contained in a real-analytic curve, if the iterated
function system consists of similarities only, or if this
system is irregular. As a consequence of this general result
the same statement will discussed for hyperbolic and
parabolic Julia sets, finite parabolic iterated function
systems and generalized polynomial-like mappings. Also
sufficient conditions will be provided for a limit set to be
uniformly perfect and for the harmonic measure to have the
Hausdorff dimension less than 1.
April 14, 1999
Title: "Homogeneity in Continua"
Alex Clark, UNT faculty
Abstract: First we cover general
background material, defining continua and homogeneity and
giving some of the history of results on the homogeneity of
planar continua. After showing that (up to homeomorphism)
the circle is the only homogeneous planar continuum that
contains an arc, Bing raised the following question: If X is
a homogeneous continuum and if the only proper subcontinua
of X are arcs, must X be homeomorphic to a solenoid?
Hagopian answered this question in the affirmative. Another
way of thinking of this result is that any such homogeneous
continuum supports a compatible abelian topological group
structure. We generalize this result in the following way:
If X is a homogeneous continuum and if X admits a fiber
bundle projection with totally disconnected fibers p:X ->T
onto some torus T of countable dimension, then X supports a
compatible abelian topological group structure. We explain
what a fiber bundle projection is and how such spaces X
occur in natural settings.
April 21, 1999
Title: "Unique Range Sets for Polynomials"
William Cherry, UNT faculty
Abstract: A subset S of the complex
numbers is called a unique range set for non-constant
polynomials if there do not exist two distinct non-constant
polynomials f and g with f^{-1}(S)=g^{-1}(S). The concept of
unique range set first came up in the study of iteration and
factorization of holomorphic functions, and I have been
interested in giving necessary and sufficient conditions for
a set to be unique range for various classes of functions. A
few years ago a nice geometric characterization of finite
unique range sets for degree d polynomials was given, the
case of variable degree remains open. A very recent preprint
by Tien-Cuong Dinh uses methods traditionally used to study
complex dynamics to give a necessary and sufficient
condition for a set of positive logarithmic capacity to be
unique range. I will give a brief history of the concept of
unique range sets, and I will attempt to explain the Dinh
paper in some detail and make some connections with Julia
sets of polynomials.
