I will present a short survey of Khintchine type results for subspaces of Euclidean space and specifically discuss a Khintchine type for divergence theorem for affine subspaces.
I consider chaotic (hyperbolic) dynamical systems which have a generating Markov partition. Then, open dynamical systems are built by making one element of a Markov partition a hole through which orbits escape. I compare various estimates of the escape rate which correspond to a physical picture of leaking in the entire phase space. Moreover, I uncover a reason why the escape rate is faster than expected, which is the convexity of the function defining escape rate.
We investigate a phenomenon observed by W. Thurston wherein one constructs a pseudo-Anosov homeomorphism on the omega-limit set of a certain lift of a piecewise-expanding interval map. For a special subclass of such maps, we characterize when this construction produces a pseudo-Anosov in terms of their kneading data, and show that most pseudo-Anosovs obtained in this way have dilatation a Salem number.
We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and set up a framework to use tools from contact and symplectic geometry and topology in the study of questions about Anosov dynamics. If time permits, we will discuss some uniqueness results for the underlying (bi)-contact structure for an Anosov flow, and/or a characterization of Anosovity based on Reeb flows.
Diophantine Approximation is filled with questions that can be viewed as shrinking target problems. Often these approaches require mixing or exponential mixing of a flow on a particular space. One such theorem is Khintchine's Theorem.
This talk will be on a version of Khintchine's Theorem where the rationals are required to satisfy a certain extra requirement. This is a time-honored tradition, the most well-known example being the Duffin-Schaeffer Conjecture, but with many others. The extra requirement in this talk is that the rationals must lie in a fixed neighborhood of some completion of Q.
In a sense, the result quantifies how neighborhoods of rationals become dense under different completions of Q.
In this work we construct Bratteli-Vershik (B-V) models for substitution dynamical systems on a countable infinite alphabets known as left determined. We provide two versions of Rokhlin’s Lemma for such substitution systems. Using the Bratteli-Vershik model we find explicit expressions for invariant and ergodic measures. This is an ongoing joint work with Sergii Bezuglyi and Palle Jorgensen.
In this talk I will present an asymptotic Lindeberg-type of CLT for the measure of maximal entropy of the geodesic flow on non-positively curved manifold. The method is enlightened by a result of Denver, Senti and Zhang, which makes use of specification. This result also extends to several other equilibrium measures and the case of dynamical arrays.