arXiv subjects
Publications and source records attributed to Jon Chaika.
This paper shows that there exists a non-minimal sequence $\bar{x} \in ([0,1]^2)^{\mathbb{N}}$ such that for any continuous function $f:[0,1]^2 \to [0,1]$, the sequence obtained by mapping terms of $\bar{x}$ by $f$ is minimal.
We discuss discrete one-dimensional Schrödinger operators whose potentials are generated by an invertible ergodic transformation of a compact metric space and a continuous real-valued sampling function. We pay particular attention to the case where the transformation is a minimal interval exchange transformation. Results about the spectrum and the spectral type of these operators are established. In particular, we provide the first examples of transformations for which the associated Schrödinger operators have purely singular spectrum for every non-constant continuous sampling function.
I show that there exist minimal interval exchange transformations with an ergodic measure whose Hausdorff dimension is arbitrarily small, even 0. I will also show that in particular cases one can bound the Hausdorff dimension between $\frac 1 {2r+4}$ and $\frac 1 r$ for any r greater than 1.