Dynamical Systems and Analysis Seminar 2019 - 2020

Interval exchange transformations are piecewise linear isometries of the unit interval. Almost every interval exchange transformation is rigid. In this talk I will introduce interval exchange transformations, rigidity and mild mixing, and describe work showing a full Hausdorff dimension set of interval exchange transformations is mild mixing.

Let phi be the golden mean. We are interested in the distances between 0-1 polynomials in phi. As a sample question we might ask: Given words a_0,..., a_n and b_0,...., b_n with each a_i and b_i picked independently from {0,1} with equal probability, what is the probability that the difference \[ \sum_{i=0}^n a_i\phi^i -\sum_{i=0}^n b_i\phi^i \] is equal to one? What is the probability that it is equal to d for some other distance d? How does this change as n tends to infinity?

We will explore the answer to these questions, as well as how the answer changes when \phi is replaced with another algebraic integer, and why these questions are relevant to some old questions in fractal geometry.

J. Bowden, S. Hensel, and I introduced an infinite-diameter hyperbolic graph on which the homeomorphism group of a closed surface of positive genus acts by isometries. First I will explain the importance of this new tool. Then I will explain some correspondences with dynamics. The rotation set of a (isotopically-trivial) homeomorphism of the torus is a generalisation of Poincaré's rotation number for the circle. For the torus we show that properties of the rotation set correspond to properties of the isometry (induced by the homeomorphism) on our hyperbolic graph e.g. the rotation set has non-empty interior if and only if the homeomorphism induces a hyperbolic/loxodromic action, an irrational segment implies a parabolic action, and a rational segment with rational points implies an elliptic action. We will discuss examples. This talk is based on ongoing joint projects with J. Bowden, S. Hensel, K. Mann, and E. Militon.

Given an infinite iterated function system (IFS) $\Phi=\{\phi_i: \mathbb{R}^d \to \mathbb{R}^d\}_{i \in \N}$ of contractions, one can define the dimension spectrum $D(\Phi):=\{\dim F_I : I \subset \mathbb{N}\}$, where $F_I$ denotes the attractor of the subsystem $\{\phi_i\}_{i \in I}$.

Dimension spectra were introduced by Kessebohmer and Zhu (Journal of Number Theory, 2006) alongside their proof of the Texan conjecture, which asserted that $D(\Phi)=[0,1]$ when the associated IFS is given by the inverse branches of the Gauss map. Recently there has been interest in the topological properties of $D(\Phi)$ in the case that the maps in $\Phi$ are conformal, (eg. Chousionis, Leykekhman & Urbanski, Transactions of the AMS, 2019 & Selecta Mathematica, to appear; Das & Simmons, preprint, 2019).

In this talk we'll discuss the challenges that arise when investigating the dimension spectrum of IFS consisting of non-conformal maps and interesting new phenomena that emerge in this setting.

In 2012 Hochman and Shmerkin proved that, given Borel probability measures on [0,1] invariant under multiplication by 2 and 3 respectively, the Hausdorff dimension of the orthogonal projection of the product of these measures is equal to the maximum possible value in every direction except the horizontal and vertical directions. Their result holds beyond multiplication by 2,3 to natural numbers m,n which are multiplicatively independent. We discuss a generalisation of this theorem to include random cascade measures on subsets of [0,1] invariant under multiplication by multiplicatively independent m,n. We will define random cascade measures in a heuristic way, as a natural randomisation of invariant measures on symbolic space. The theorem of Hochman and Shmerkin fully resolved a conjecture of Furstenberg originating in the late 1960s concerning sumsets of these invariant sets. We apply our main result to present a random version of this conjecture which holds for products of percolations on $\times m, \times n$-invariant sets.

Saddle connections on flat surfaces are those straight line trajectories connecting zeroes. In this talk I will explain what that means and discuss joint work with Jon Chaika proving that, for almost every flat surface, the sequence of lengths of its saddle connections is uniformly distributed mod 1.

The equidistribution theorem of Weyl gives a sufficient and necessary condition for when a sequence equidistributes on the torus. What if we replace the torus with a higher-degree nilmanifold, such as Heisenberg nilmanifold? I will show how the equidistribution theory on nilmanifolds developed by Leibman, Green and Tao can be applied to the combinatorial problem of counting certain polynomial configurations in subsets of finite fields.