arXiv subjects
Publications and source records attributed to Carlos Ospina.
We study the real Rel orbits of certain translation surfaces in the stratum $\mathcal{H}(1,1)$, specifically surfaces that are tremors of the locus of branched double covers of tori. We give necessary and sufficient conditions on tremors of a surface so that the real Rel orbit is recurrent. As a consequence, we are able to provide explicit examples of trajectories of real Rel that are not recurrent.
In this note, we investigate coboundaries of interval exchange transformations of three intervals, or 3-IETs. More precisely, we study differentiable functions whose derivative is absolutely continuous, and whose integral is zero and whose derivative also has integral zero. We show that these functions are coboundaries for a typical 3-IET if and only if their values at the endpoints of the domain are zero. We also show the existence of exceptional counterexamples for both possible endpoint behaviors. Our results are obtained by studying the properties of associated skew products.
Let $k\geq 2$ be a positive integer. It is known that the set of visible lattice points from the origin in $\mathbb{Z}^k$ has a translation bounded pure point diffraction spectrum. We investigate these properties for sets of points simultaneously visible from a finite set of lattice points $ \{\mathbf{x}_1,\dots,\mathbf{x}_n\} \subseteq \mathbb{Z}^k$. We provide explicit formulas for the coefficients of the diffraction spectrum. Additionally, we generalize our procedure to show that the set of visible points from the origin in certain classes of cut-and-project sets has a translation bounded pure point diffraction spectrum.
We define variational properties for dynamical systems with subexponential complexity, and study these properties in certain specific examples. By computing the value of slow entropy directly, we show that some subshifts are not variational, while a class of interval exchange transformations are variational.
The catalytic sites of MoS$_{2}$ monolayers towards hydrogen evolution are well known to be vacancies and edge-like defects. However, it is still very challenging to control the position, size, and defective areas on the basal plane of Mo$S_{2}$ monolayers by most of defect-engineering routes. In this work, the fabrication of etched arrays on ultra-large supported and free-standing MoS$_{2}$ monolayers using focused ion beam (FIB) is reported for the first time. By tuning the Ga+ ion dose, it is possible to confine defects near the etched edges or spread them over ultra-large areas on the basal plane. The electrocatalytic activity of the arrays toward hydrogen evolution reaction (HER) was measured by fabricating microelectrodes using a new method that preserves the catalytic sites. We demonstrate that the overpotential can be decreased up to 290 mV by assessing electrochemical activity only at the basal plane. High-resolution transmission electron microscopy images obtained on FIB patterned freestanding MoS$_{2}$ monolayers reveal the presence of amorphous regions and X-ray photoelectron spectroscopy indicates sulfur excess in these regions. Density-functional theory calculations provide identification of catalytic defect sites. Our results demonstrate a new rational control of amorphous-crystalline surface boundaries and future insight for defect optimization in MoS$_{2}$ monolayers.
In his paper, Thurston shows that a positive real number $h$ is the topological entropy for an ergodic traintrack representative of an outer automorphism of a free group if and only if its expansion constant $λ= e^h$ is a weak Perron number. This is a powerful result, answering a question analogous to one regarding surfaces and stretch factors of pseudo-Anosov homeomorphisms. However, much of the machinery used to prove this seminal theorem on traintrack maps is contained in the part of Thurston's paper on the entropy of postcritically finite interval maps and the proof difficult to parse. In this expository paper, we modernize Thurston's approach, fill in gaps in the original paper, and distill Thurston's methods to give a cohesive proof of the traintrack theorem. Of particular note is the addition of a proof of ergodicity of the traintrack representatives, which was missing in Thurston's paper.