Abstracts

Abstract: In joint work with S. Lalley, J. Sapir, and M. Wroten, we show that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting Poisson line process.

Abstract: There is an extensive literature on the entropy behavior of polynomial interval maps as they vary in families. In particular, the monotonicity problem (due to Milnor) which asks about the connectedness of the level sets of the entropy function (the isentropes) has been of immense interest and is very well studied in the context of polynomial interval maps. In contrast, the entropy behavior of real rational maps is a much less studied albeit much more general setting. After introducing a real entropy function on a moduli space of real rational maps, we focus on the case of real quadratic rational maps where, supported by experimental evidence, we show that the isentropes are connected in certain dynamically defined regions of the moduli space while in general, they become disconnected due to a non-polynomial behavior.

Abstract: We introduce a Riemannian metric on every hyperboblic component of the Mandelbrot set which is conformal equivalent to the standard pressure metric. As an application, we discuss the Hausdorff dimension function on hyperbolic components. Along the way, we introduce multiplier functions for invariant probability measures on Julia sets, which is a key ingredient in the construction of our metric. This is joint work with Hongming Nie.

Abstract: In 1984, Mahler asked how well typical points on Cantor’s set can be approximated by rational numbers. His question fits within a program, set out by himself in the 1930s, attempting to determine conditions under which subsets of R^n inherit the Diophantine properties of the ambient space. Since approximability of typical points in Euclidean space by rational points is governed by Khinchine’s classical theorem, the ultimate form of Mahler’s question asks whether an analogous zero-one law holds for fractal measures. Progress has been achieved in recent years, albeit, almost all known results have been of ``convergence type”.

In this talk, we present the first instances where a complete analogue of Khinchine’s theorem for fractal measures is obtained. The main new ingredient is an effective equidistribution theorem for certain fractal measures on the space of unimodular lattices. The latter is established via a new technique involving the construction of S-arithmetic Markov operators possessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal. Joint work with Manuel Luethi

Abstract: The standard continued fraction algorithm come from the Euclidean algorithm. We can also describe this algorithm using a dynamical system of [0,1), where the transformation that takes x to the fractional part of 1/x is said to generate the continued fraction expansion of x. From there, we ask two questions: What happens to the continued fraction expansion when we change the domain to something other than [0,1)? What happens to the dynamical system when we impose restrictions on the continued fraction expansion, such as finding the nearest odd integer instead of the floor? This talk will focus on the case where we first restrict to odd integers, then start shifting the domain [\alpha-2,\alpha).

This talk is based on joint work with Florin Boca and animations done by Xavier Ding, Gustav Jennetten, and Joel Rozhon as part of an Illinois Geometry Lab project.

Abstract: There are a number of useful quantities in dynamical systems which give information about a given dynamical systems (e.g., Entropy, Lyapunov exponents, dimension of limit sets etc.). Even for very simple one-dimensional dynamical systems these can be very difficult to estimate efficiently. We will describe how (thermodynamic) pressure can be a useful tool. This circle of ideas includes application to other areas of mathematics, including Zaremba Theory and Lagrange-Markov spectra in number theory.

Abstract: In their recent paper, S. Kiriki, Y. Nakano, and T. Soma introduced a concept of pointwise emergence to measure the complexity of irregular orbits. They constructed a residual subset of the full shift with high pointwise emergence. In our joint work with Yushi Nakano we consider the set of points with high pointwise emergence for topologically mixing subshifts of finite type. We show that this set has full topological entropy, full Hausdorff dimension, and full topological pressure for any H\"older continuous potential. Furthermore, we show that this set belongs to a certain class of sets with large intersection property.