We resolve a long-standing open question on the relationship between measure-theoretic
dynamical complexity and symbolic complexity by establishing the exact word complexity at …

We prove that for a suitably nice class of random substitutions, their corresponding subshifts
have automorphism groups that contain an infinite simple subgroup and a copy of the …

We show that there is a distortion element in a finitely generated subgroup G of the
automorphism group of the full shift, namely an element of infinite order whose word norm …

We introduce a class of rank-one transformations, which we call extremely elevated
staircase transformations. We prove that they are measure-theoretically mixing and, for any …

We exhibit, for arbitrary $\epsilon> 0$, subshifts admitting weakly mixing (probability)
measures with word complexity p satisfying $\limsup p (q)/q< 1.5+\epsilon $. For arbitrary $ f …

The overarching topic of this thesis is the study of topics at the intersection of topological
dynamics and descriptive theory. I provide additional motivation for each problem at the …

1In Lemma 2.4, we denote NC1 for “Nick's Class” of complexity of level 1, ie the class of
languages L⊆ Σ∗ such that L is decidable by Boolean circuits with a polynomial number of …