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Rodrigo Treviño

Publications and source records attributed to Rodrigo Treviño.

At least 19 recordsLinked to original sources
arXiv2026

Non-local Dirichlet forms, Gibbs measures, and a cohomological Dirichlet principle for Cantor sets

In this paper I study properties of the generators $\triangle_γ$ of non-local Dirichlet forms $\mathcal{E}^μ_γ$ on ultrametric spaces which are the path space of simple stationary Bratteli diagrams. The measures used to define the Dirichlet forms are taken to be the Gibbs measures $μ_ψ$ associated to Hölder continuous potentials $ψ$ for one-sided shifts. I also define a cohomology $H_{lc}(X_B)$ for $X_B$ which can be seen as dual to the homology of Bowen and Franks. Besides studying spectral properties of $\triangle_γ$, I show that for $γ$ large enough (with sharp bounds depending on the diagram and the measure theoretic entropy $h_{μ_ψ}$ of $μ_ψ$) there is a unique $\mathcal{E}^μ_γ$-minimizing representative of any class $c\in H_{lc}(X_B)$.

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arXiv2025

New results on tilings via cup products and Chern characters on tiling spaces

We study the cohomology rings of tiling spaces $Ω$ given by cubical substitutions. While there have been many calculations before of cohomology groups of such tiling spaces, the innovation here is that we use computer-assisted methods to compute the cup-product structure. This leads to examples of substitution tilings with isomorphic cohomology groups but different cohomology rings. Part of the interest in studying the cup product comes from Bellissard's gap-labeling conjecture, which is known to hold in dimensions $\le 3$, but where a proof is known in dimensions $\ge 4$ only when the Chern character from $K^0(Ω)$ to $H^*(Ω,\mathbb{Q})$ lands in $H^*(Ω,\mathbb{Z})$. Computation of the cup product on cohomology often makes it possible to compute the Chern character. We introduce a natural generalization of the gap-labeling conjecture, called the equivariant gap-labeling conjecture, which applies to tilings with a finite symmetry group. Again this holds in dimensions $\le 3$, but we are able to show that it fails in general in dimensions $\ge 4$. This, plus some of our cup product calculations, makes it plausible that the gap-labeling conjecture might fail in high dimensions.

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arXiv2025

Bratteli diagrams, translation flows and their $C^*$-algebras

In [LT16], Kathryn Lindsey and the second author constructed a translation surface from a bi-infinite Bratteli diagram. We continue an investigation into these surfaces. The construction given in [LT16] was essentially combinatorial. Here, we provide explicit links between the path space of the Bratteli diagram and the surface, including various intermediate topological spaces. This allows us to relate the $C^{*}$-algebras associated with tail equivalence on the Bratteli diagram and the foliation of the surface, under some mild hypotheses. This also allows us to relate the K-theory of the $C^{*}$-algebras involved. We also treat the case of finite genus surfaces in some detail, where the process of Rauzy-Veech induction (and its inverse) provide an explicit construction of the Bratteli diagrams involved.

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arXiv2024

The cohomological equation and cyclic cocycles for renormalizable minimal Cantor systems

For typical properly ordered and minimal Bratteli diagrams $(B,\leq_r)$, it is shown that there are finitely many invariant distributions $\mathcal{D}_i$ which are the only obstructions to solving the cohomological equation $f = u-u\circ ϕ$ for the corresponding adic transformation $ϕ:X_B\rightarrow X_B$ and for $α$-Hölder $f$ with $α$ large enough. These invariant distributions are then used to define cyclic cocycles, a.k.a. traces $τ:K_0(\mathcal{A}_ϕ)\rightarrow \mathbb{R}$ for the crossed product algebra $\mathcal{A}_ϕ$.

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arXiv2024

On transversal Hölder regularity for flat Wieler solenoids

This paper studies various aspects of inverse limits of locally expanding affine linear maps on flat branched manifolds, which I call flat Wieler solenoids. Among the aspects studied are different types of cohomologies, the rates of mixing given by the Ruelle spectrum of the hyperbolic map acting on this space, and to solutions of the cohomological equation for primitive substitution subshifts for Hölder functions. The overarching theme is that considerations of $α$-Hölder regularity on Cantor sets go a long way.

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arXiv2023

Quantitative weak mixing for random substitution tilings

For $N$ compatible substitution rules on $M$ prototiles $t_1,\dots,t_M$, consider tilings and tiling spaces constructed by applying the different substitution rules at random. These give (globally) random substitution tilings. In this paper I obtain bounds for the growth on twisted ergodic integrals for the $\mathbb{R}^d$ action on the tiling space which give lower bounds on the lower local dimension of spectral measures. For functions with some extra regularity, uniform bounds on the lower local dimension are obtained. The results here extends results of Bufetov-Solomyak to tilings of higher dimensions.

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arXiv2022

Spectral cocycle for substitution tilings

The construction of spectral cocycle from the case of 1-dimensional substitution flows by Bufetov-Solomyak [arXiv:1802.04783] is extended to the setting of pseudo-self-similar tilings in ${\mathbb R}^d$, allowing expanding similarities with rotations. The pointwise upper Lyapunov exponent of this cocycle is used to bound the local dimension of spectral measures of deformed tilings. The deformations are considered, following Treviño [arXiv:2006.16980], in the simpler, non-random setting. We review some of the results on quantitative weak mixing from [arXiv:2006.16980] in this special case and illustrate them on concrete examples.

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arXiv2021

Statistical properties of Lorentz gases on aperiodic tilings, Part 1

We consider the Lorentz gas model of category A (that is, with no corners and of finite horizon) on aperiodic repetitive tilings of $\mathbb{R}^2$ of finite local complexity. We show that the compact factor of the collision map has the K property, from which we derive mixing for pattern-equivariant functions as well as the planar ergodicity of the Lorentz gas flow.

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arXiv2021

Tilings and traces

This paper deals with (globally) random substitutions on a finite set of prototiles. Using renormalization tools applied to objects from operator algebras we establish upper and lower bounds on the rate of deviations of ergodic averages for the uniquely ergodic $\mathbb{R}^d$ action on the tiling spaces obtained from such tilings. We apply the results to obtain statements about the convergence rates for integrated density of states for random Schrödinger operators obtained from aperiodic tilings in the construction.

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arXiv2019

Equilibrium configurations for generalized Frenkel-Kontorova models on quasicrystals

I study classes of generalized Frenkel-Kontorova models whose potentials are given by almost-periodic functions which are closely related to aperiodic Delone sets of finite local complexity. Since such Delone sets serve as good models for quasicrytals, this setup presents Frenkel-Kontorova models for the type of aperiodic crystals which have been discovered since Shechtman's discovery of quasicrystals. Here I consider models with configurations $u:\mathbb{Z}^r \rightarrow \mathbb{R}^d$, where $d$ is the dimension of the quasicrystal, for any $r$ and $d$. The almost-periodic functions used for potentials are called pattern-equivariant and I show that if the interactions of the model satisfies a mild $C^2$ requirement, and if the potential satisfies a mild non-degeneracy assumption, then there exist equilibrium configurations of any prescribed rotation rotation number/vector/plane. The assumptions are general enough to satisfy the classical Frenkel-Kontorova models and its multidimensional analogues. The proof uses the idea of the anti-integrable limit.

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arXiv2019

Random Substitution Tilings and Deviation Phenomena

Suppose a set of prototiles allows $N$ different substitution rules. In this paper we study tilings of $\mathbb{R}^d$ constructed from random application of the substitution rules. The space of all possible tilings obtained from all possible combinations of these substitutions is the union of all possible tilings spaces coming from these substitutions and has the structure of a Cantor set. The renormalization cocycle on the cohomology bundle over this space determines the statistical properties of the tilings through its Lyapunov spectrum by controlling the deviation of ergodic averages of the $\mathbb{R}^d$ action on the tiling spaces.

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arXiv2018

Canonical diffusions on the pattern spaces of aperiodic Delone sets

We consider pattern spaces of aperiodic and repetitive Delone sets of finite local complexity. These spaces are compact metric spaces and constitute a special class of foliated spaces. We define new Sobolev spaces with respect to the unique invariant measure and prove the existence of the unitary Schrödinger semigroup, which in physics terms describe the evolution of phasons. We define and study natural leafwise diffusion processes on these pattern spaces. These processes have Feller, but lack strong Feller and hypercontractivity properties, and heat kernels do not exist. The associated Dirichlet forms are regular, strongly local, irreducible and recurrent, but not strictly local. For harmonic functions we prove new Liouville and Helmholtz-Hodge type theorems.

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arXiv2018

Traces of random operators associated with self-affine Delone sets and Shubin's formula

We study operators defined on a Hilbert space defined by a self-affine Delone set $Λ$ and show that the usual trace of a restriction of the operator to finite-dimensional subspaces satisfies a certain $\limsup$ law controlled by traces on a certain subalgebra. The asymptotic traces are defined through asymptotic cycles, or $\mathbb{R}^d$-invariant distributions of a dynamical system defined by $Λ$. We use this to refine Shubin's trace formula for self-adjoint operators and show that the errors of convergence in Shubin's formula are given by these traces.

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arXiv2018

Diophantine approximations for translation surfaces and planar resonant sets

We consider Teichmüller geodesics in strata of translation surfaces. We prove lower and upper bounds for the Hausdorff dimension of the set of parameters generating a geodesic bounded in some compact part of the stratum. Then we compute the dimension of those parameters generating geodesics that make excursions to infinity at a prescribed rate. Finally we compute the dimension of the set of directions in a rational billiard having fast recurrence, which corresponds to a dynamical version of a classical result of Jarník and Besicovich. Our main tool are planar resonant sets arising from a given translation surface, that is the countable set of directions of its saddle connections or of its closed geodesics, filtered according to length. In an abstract setting, and assuming specific metric properties on a general planar resonant set, we prove a dichotomy for the Hausdorff measure of the set of directions which are well approximable by directions in the resonant set, and we give an estimate on the dimension of the set of badly approximable directions. Then we prove that the resonant sets arising from a translation surface satisfy the required metric properties.

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arXiv2017

Indiscriminate covers of infinite translation surfaces are innocent, not devious

We consider the interaction between passing to finite covers and ergodic properties of the straight-line flow on finite area translation surfaces with infinite topological type. Infinite type provides for a rich family of degree $d$ covers for any integer $d>1$. We give examples which demonstrate that passing to a finite cover can destroy ergodicity, but we also provide evidence that this phenomenon is rare. We define a natural notion of a random degree $d$ cover and show that, in many cases, ergodicity and unique ergodicity are preserved under passing to random covers. This work provides a new context for exploring the relationship between recurrence of the Teichmüller flow and ergodic properties of the straight-line flow.

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arXiv2017

Self affine Delone sets and deviation phenomena

We study the growth of norms of ergodic integrals for the translation action on spaces coming from expansive, self-affine Delone sets. The linear map giving the self-affinity induces a renormalization map on the pattern space and we show that the rate of growth of ergodic integrals is controlled by the induced action of the renormalizing map on the cohomology of the pattern space up to boundary errors. We explore the consequences for the diffraction of such Delone sets, and explore in detail what the picture is for substitution tilings as well as for cut and project sets which are self-affine. We also explicitly compute some examples.

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arXiv2017

Logarithmic laws and unique ergodicity

We show that Masur's logarithmic law of geodesics in the moduli space of translation surfaces does not imply unique ergodicity of the translation flow, but that a similar law involving the flat systole of a Teichmüller geodesic does imply unique ergodicity. It shows that the flat geometry has a better control on ergodic properties of translation flow than hyperbolic geometry.

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