Research
Interests: I continue to be interested in several
different areas and their
interaction. This includes topics from analysis, topology,
set theory, dynamics and
ergodic theory.
Recently, I have been interested in
problems concerning geometric measure theory (particlarly
properties of Hausdorff and packing measures); nowhere
differentible functions (particularly the dimension
of the Weierstrass-Hardy type functions), and representation
of functions by series of translates or by superpositions
a la Kolmogorov.
In topology I have been interested in
topological characterizations of types of continua and
characterization of homeomorphic measures (with connections
to number theory, dynamics and ergodic theory).
In set theory I have been interested
in problems originating from my solution with S. Jackson
to Steinhaus' lattice problem and descriptive set theoretic
problems concerning representations of families of sets.
In dynamics I have been interested in
nonconventional ergodic averages and the convergence
of various associated linear and nonlinear operators.
This topic involves a good deal of interaction with
number theory.
In dynamical systems, I continue to
work with M. Urbanski on infinite iterated function
systems including the graph directed systems developed
in our recent book. There are many applications of these
systems to various areas of dynamics, number theory
and geometry. I have also been working with an interdisciplinary
team on applications of dynamics to biocomplex systems.
unt.edu
