Abstract: Substitution schemes provide a classical method for constructing tilings
of Euclidean space. Allowing multiple scales in the scheme, we introduce
a rich family of sequences of tile partitions generated by the substitution
rule, which include the sequence of partitions of the unit interval
considered by Kakutani as a special case. In this talk we will use new path counting
results for directed weighted graphs to show that such sequences
of partitions are uniformly distributed, thus extending Kakutani's
original result. Furthermore, we will describe certain limiting frequencies
The geodesic flow on a normal cover of a compact hyperbolic surface admits a "random walk" on the group of decks transformations $G$. In this talk, I'll provide some recent results which connect this walk to the geometric properties of the cover and $G$.
Abstract: I will discuss two stochastic target problem in the Brownian framework .
The first problem has a nice solution which I will present. The second problem
is much more complicated and for now remains open. I will discuss the challenges and connection with other fields in probability theory.
Abstract: In this talk I will show that for finite entropy countable Markov shifts the entropy map is upper semi-continuous when restricted to the set of ergodic measures. This is joint work with Mike Todd and Anibal Velozo.
Probabilistic limit theorems for (distance expanding and hyperbolic) dynamical systems is a well studied
topic. In this talk I will present conditions guaranteeing that a local central limit theorem holds true for certain families of distance
expanding random dynamical systems. If time permits, I will also discuss a version of the Berry-Esseen theorem.
Joint work with Yuri Kifer.
Abstract: This talk describes two classes of symbolic topological systems, the odometer based and the circular systems. The odometer based systems are ubiquitous--when equipped with invariant measures they form an upwards closed cone in the space of ergodic transformations (in the pre-ordering induced by factor maps). The circular systems are a small class, but represent the diffeomorphisms of the 2-torus built using the Anosov-Katok technique of approximation by conjugacy.
For fixed t>1 and L>3 we establish sharp asymptotic formula for
the log-probability that the number of cycles of length L in the Erdos - Renyi
random graph G(N,p) exceeds its expectation by a factor t, assuming only that
p >> log N/sqrt(N). In a narrower range of p=p(N) we obtain such sharp upper tail
for general subgraph counts and for the Schatten norms of the corresponding adjacency
matrices.
In this talk, based on a joint work with Nick Cook, I will explain our approach,
based on convex-covering and a new quantitative refinement of
Abstract: In Diophantine approximation we are often interested in the Lebesgue and Hausdorff measures of certain $\limsup$ sets. In 2006, Beresnevich and Velani proved a remarkable result --- the Mass Transference Principle --- which allows for the transference of Lebesgue measure theoretic statements to Hausdorff measure theoretic statements for $\limsup$ sets arising from sequences of balls in $\mathbb{R}^k$.
Although each cocycle for a action of the integers is
specified by the sequence of Birkhoff sums of a function,
it is relatively difficult to specify cocycles for the actions of
multidimensional groups such as $\Bbb Z^2$.
We'll see that if $(X,T)$ is a transitive action of the finitely
generated (countable) group $\Gamma$ by homeomorphism of the polish space $X$,
and $\Bbb B$ is a separable Banach space, there is a cocycle
$F:\Gamma\times X \to\Bbb B$
with each $x\mapsto F(g,x)$ bounded and continuous
so that the skew product action $(X x \Bbb B,S)$ is transitive where
Abstract: An inhomogeneous Markov chain X_n is a Markov chain whose state spaces and transition kernels change in time. A “local limit theorem” is an asymptotic formula for probabilities of th form
Prob[S_N-z_N\in (a,b)] , S_N=f_1(X_1,X_2)+....+f_N(X_N,X_{N+1})
in the limit N—>infinity. Here z_N is a “suitable” sequence of numbers.
I will describe general sufficient conditions for such results.
If time allows, I will explain why such results are needed for the study of certain problems related to irrational rotations.
This is joint work with Dmitry Dolgopyat.
Krieger’s generator theorem shows that any free invertible ergodic measure preserving action (Y,\mu, S) can be modelled by A^Z (equipped with the shift action) provided the natural entropy constraint is satisfied; we call such systems (here it is A^Z) universal. Along with Tom Meyerovitch, we establish general specification like conditions under which Z^d-dynamical systems are universal. These conditions are general enough to prove that
1) A self-homeomorphism with almost weak specification on a compact metric space (answering a question by Quas and Soo)
Suppose that a countable group $G$ acts on a compact metric space $X$ and that $S \subset G$ is a semigroup not containing the identity of $G$. If every continuous function $f$ on $X$ is contained in the closed algebra generated by $\{sf : s \in S\}$, the action is said to be $S$-predictable.
In this talk, we consider the following question due to Hochman:
When $G$ is amenable, does $S$-predictability imply zero topological entropy?
To provide an affirmative answer, we introduce the notion of a random invariant order.
A longstanding open question concerning the horocycle flow on moduli space of translation surfaces, is whether one can classify the invariant measures and orbit-closures for this action. Related far-reaching results of Eskin, Mirzakhani and Mohammadi indicated that the answer might be positive. However, in recent work with Jon Chaika and John Smillie, we find unexpected examples of orbit-closures; e.g. orbit closures which are not generic for any measure, and others which have fractional Hausdorff dimension. Such examples exist even in genus 2.
