A similarity is a map from Rd to itself that
uniformly scales distances. When one repeatedly applies randomly
chosen contracting similarities, the resulting Markov chain
converges to a limiting stationary distribution known as a
self-similar measure...
The generalized Markoff equation gives rise to a dynamical
system via the Markoff group acting on the solution set. This talk
considers the resulting dynamics over finite fields, which has
strong connections with combinatorial group theory and...
Rank 1 transformations are a long studied, flexible class of
measure preserving transformations. King proved a variety of deep
results about rank 1 rank transformations including: A) A rank 1
transformation’s centralizer is the weak closure of its...
We present solutions to two problems on indefinite integral
ternary quadratic forms. The first, highlighted by Margulis in
1990, concerns the distribution of the ternary Markoff spectrum
associated with minima of forms. The second, initiated by...
The classical Rokhlin Lemma asserts that for an aperiodic
measure-preserving transformation $T$ of a probability space, one
can find a "tower" of sets on which $T$ acts by translation and
which covers almost all of the space. This result is a basic...
A translation surface is a closed surface that is obtained by
gluing edges of a polygon in parallel. The group GL2(R) acts on the
collection translation surfaces of a fixed genus g. For a fixed
translation surface S and t greater than 0, we obtain a...
Zimmer proposed the study of higher-rank semisimple group
actions in the 1980's after showing certain rigidity results,
especially about associated cocycles in the measurable category. In
this talk, I will describe recent progress towards...
The geodesic flow (for the hyperbolic metric) of an infinite
Riemann surface is ergodic if and only if Brownian motion is
recurrent, which is also equivalent to the divergence of the
Poincaré series. Surfaces with ergodic geodesic flows are
most...
In this talk I will describe new results establishing the
existence of a strong spectral gap for geometrically finite,
n dimensional, hyperbolic manifold with critical exponent greater
than (n−1)/2. This settles a conjecture of Mohammadi and Oh...
