About me
I am currently an assistant professor at the Department of Mathematical Engineering of Universidad the Chile.
Research interests
I'm interested in geometry and dynamical systems and their connection with geometry. Currently, I work on Teichmüller dynamics, flat and Riemann surfaces, and dynamically-defined Cantor sets.
Articles
Accepted
Fractal dimensions of the Markov and Lagrange spectra near \(3\) (2022).
With Harold Erazo, Carlos Gustavo Moreira and Sergio Romaña.
Journal of the European Mathematical Society.
Abstract. Preprint. BibTeX.
The Lagrange spectrum \(\mathcal{L}\) and Markov spectrum \(\mathcal{M}\) are subsets of the real line with complicated fractal properties that appear naturally in the study of Diophantine approximations. It is known that the Hausdorff dimension of the intersection of these sets with any half-line coincide, that is, \(\mathrm{dim}_{\mathrm{H}}(\mathcal{L} \cap (-\infty, t)) = \mathrm{dim}_{\mathrm{H}}(\mathcal{M} \cap (-\infty, t)) =: d(t)\) for every \(t \geq 0\). It is also known that \(d(3)=0\) and \(d(3+\varepsilon)>0\) for every \(\varepsilon>0\).
We show that, for sufficiently small values of \(\varepsilon > 0\), one has the approximation \(d(3+\varepsilon) = 2\cdot\frac{W(e^{c_0}|\log \varepsilon|)}{|\log \varepsilon|}+\mathrm{O}\left(\frac{\log |\log \varepsilon|}{|\log \varepsilon|^2}\right)\), where \(W\) denotes the Lambert function (the inverse of \(f(x)=xe^x\)) and \(c_0=-\log\log((3+\sqrt{5})/2) \approx 0.0383\). We also show that this result is optimal for the approximation of \(d(3+\varepsilon)\) by "reasonable" functions, in the sense that, if \(F(t)\) is a \(C^2\) function such that \(d(3+\varepsilon) = F(\varepsilon) + \mathrm{o}\left(\frac{\log |\log \varepsilon|}{|\log \varepsilon|^2}\right)\), then its second derivative \(F''(t)\) changes sign infinitely many times as \(t\) approaches \(0\).
@online{erazoDimensionSpectra,
AUTHOR = {Erazo, Harold and Gutiérrez-Romo, Rodolfo and Moreira, Carlos Gustavo and Romaña, Sergio},
TITLE = {Fractal dimensions of the Markov and Lagrange spectra near $3$},
YEAR = {2022},
EPRINTTYPE = {arxiv},
EPRINTCLASS = {math.NT},
EPRINT = {2208.14830}
}
Kontsevich-Zorich monodromy groups of translation covers of some platonic solids (2022).
With Dami Lee and Anthony Sanchez.
Groups, Geometry and Dynamics.
Abstract. Preprint. BibTeX. Mathematica files.
We compute the Zariski closure of the Kontsevich-Zorich monodromy groups arising from certain square tiled surfaces that are geometrically motivated. Specifically we consider three surfaces that emerge as translation covers of platonic solids and quotients of infinite polyhedra, and show that the Zariski closure of the monodromy group arising from each surface is equal to a power of \(\mathrm{SL}(2, \mathbb{R})\).
We prove our results by finding generators for the monodromy groups, using a theorem of Matheus-Yoccoz-Zmiaikou that provides constraints on the Zariski closure of the groups (to obtain an "upper bound"), and analyzing the dimension of the Lie algebra of the Zariski closure of the group (to obtain a "lower bound").
Moreover, combining our analysis with the Eskin-Kontsevich-Zorich formula, we also compute the Lyapunov spectrum of the Kontsevich-Zorich cocycle for said square-tiled surfaces.
@online{gutierrez-romoMonodromyPlatonicSolids,
AUTHOR = {Gutiérrez-Romo, Rodolfo and Lee, Dami and Sanchez, Anthony},
TITLE = {Kontsevich-Zorich monodromy groups of translation covers of some platonic solids},
YEAR = {2022},
EPRINTTYPE = {arxiv},
EPRINTCLASS = {math.DS},
EPRINT = {2208.08460}
}
Preprints
Permutation of periodic points of Veech surfaces in \(\mathcal{H}(2)\) (2021).
With Angel Pardo.
Abstract. Preprint. BibTeX.
We study how are permuted Weierstrass points of Veech surfaces in \(\mathcal{H}(2)\), the stratum of Abelian differentials on Riemann surfaces in genus two with a single zero of order two.
These surfaces were classified by McMullen relying on two invariants: discriminant and spin.
More precisely, given a Veech surface in \(\mathcal{H}(2)\) of discriminant \(D\), we show that the permutation group induced by the affine group on the set of Weierstrass points is isomorphic to \(\mathrm{Dih}_4\), if \(D \mathbin{\equiv_4} 0\); to \(\mathrm{Dih}_5\), if \(D \mathbin{\equiv_8} 5\); and to \(\mathrm{Dih}_6\), if \(D \mathbin{\equiv_8} 1\).
Moreover, these same groups arise when considering only Dehn multitwists of the affine group.
@online{gutierrez-RomoPermutations,
AUTHOR = {Gutiérrez-Romo, Rodolfo and Pardo, Angel},
TITLE = {Permutation of periodic points of Veech surfaces in $\mathcal{H}(2)$},
YEAR = {2021},
EPRINTTYPE = {arxiv},
EPRINTCLASS = {math.DS},
EPRINT = {2111.13638}
}
The flow group of rooted abelian or quadratic differentials (2021).
With Mark Bell, Vincent Delecroix, Vaibhav Gadre and Saul Schleimer.
Abstract. Preprint. BibTeX.
We define the flow group of any component of any stratum of rooted abelian or quadratic differentials (those marked with a horizontal separatrix) to be the group generated by almost-flow loops. We prove that the flow group is equal to the fundamental group of the component. As a corollary, we show that the plus and minus modular Rauzy--Veech groups are finite-index subgroups of their ambient modular monodromy groups. This partially answers a question of Yoccoz.
Using this, and recent advances on algebraic hulls and Zariski closures of monodromy groups, we prove that the Rauzy--Veech groups are Zariski dense in their ambient symplectic groups. Density, in turn, implies the simplicity of the plus and minus Lyapunov spectra of any component of any stratum of quadratic differentials. Thus, we establish the Kontsevich--Zorich conjecture.
@online{bellFlowGroup,
AUTHOR = {Bell, Mark and Delecroix, Vincent and Gadre, Vaibhav and Gutiérrez-Romo, Rodolfo and Schleimer, Saul},
TITLE = {The flow group of rooted abelian or quadratic differentials},
YEAR = {2021},
EPRINTTYPE = {arxiv},
EPRINTCLASS = {math.DS},
EPRINT = {2101.12197}
}
Coding the Teichmüller flow using Veering triangulations (2019).
With Mark Bell, Vincent Delecroix, Vaibhav Gadre and Saul Schleimer.
Abstract. Preprint. BibTeX.
We develop the theory of veering triangulations on oriented surfaces adapted to moduli spaces of half-translation surfaces. We use veering triangulations to give a coding of the Teichmüller flow on connected components of strata of quadratic differentials. We prove that this coding, given by a countable shift, has an approximate product structure and a roof function with exponential tails. This makes it conducive to the study of the dynamics of Teichmüller flow.
@online{bellVeeringTriangulations,
AUTHOR = {Bell, Mark and Delecroix, Vincent and Gadre, Vaibhav and Gutiérrez-Romo, Rodolfo and Schleimer, Saul},
TITLE = {Coding the Teichmüller flow using Veering triangulations},
YEAR = {2019},
EPRINTTYPE = {arxiv},
EPRINTCLASS = {math.DS},
EPRINT = {1909.00890}
}
Simplicity of the Lyapunov spectra of certain quadratic differentials (2017).
Abstract. Preprint. BibTeX.
We prove that the “plus” Rauzy–Veech groups of all connected components of all strata of meromorphic quadratic differentials defined on Riemann surfaces of genus at least one having at most simple poles and at least three singularities (zeros or poles), not all of even order, are finite-index subgroups of their ambient symplectic groups. This shows that the “plus” Lyapunov spectrum of such strata is simple. Moreover, we show that the index of the “minus” Rauzy–Veech group is also finite for connected components of strata satisfying the same conditions and having exactly two singularities of odd order. This shows that the “minus” Lyapunov spectrum of such strata is simple.
@online{gutierrez-romoSimplicityLyapunovSpectra2017,
AUTHOR = {Gutiérrez-Romo, Rodolfo},
TITLE = {Simplicity of the Lyapunov spectra of certain quadratic differentials},
YEAR = {2017},
EPRINTTYPE = {arxiv},
EPRINTCLASS = {math.DS},
EPRINT = {1711.02006}
}
Published
Lower bounds on the dimension of the Rauzy gasket (2020).
With Carlos Matheus.
Bulletin de la Société Mathématique de France 148.2, pp. 321–327.
Abstract. Journal. Postprint. BibTeX.
The Rauzy gasket \(R\) is the maximal invariant set of a certain renormalization procedure for special systems of isometries naturally appearing in the context of Novikov's problem in conductivity theory for monocrystals.
It was conjectured by Novikov and Maltsev in 2003 that the Hausdorff dimension \(\dim_{\mathrm{H}}(R)\) of Rauzy gasket is strictly comprised between \(1\) and \(2\).
In 2016, Avila, Hubert and Skripchenko confirmed that \(\dim_{\mathrm{H}}(R)<2\). In this note, we use some results by Cao–Pesin–Zhao in order to show that \(\dim_{\mathrm{H}}(R)>1.19\).
@online{gutierrez-romoLowerBoundsDimensionRauzyGasket,
AUTHOR = {Gutiérrez-Romo, Rodolfo and Matheus, Carlos},
TITLE = {Lower bounds on the dimension of the Rauzy gasket},
YEAR = {2020},
JOURNAL = {Bull. Soc. Math. Fr},
VOLUME = {148},
NUMBER = {2},
PAGES = {321--327},
DOI = {10.24033/bsmf.2807}
}
Counting saddle connections in a homology class modulo \(q\) (with an appendix by Rodolfo Gutiérrez-Romo) (2020).
By Michael Magee and Rene Rühr.
Journal of Modern Dynamics 15, pp. 237–262.
Abstract. Journal. Preprint. BibTeX.
We give effective estimates for the number of saddle connections on a translation surface that have length \(\leq L\) and are in a prescribed homology class modulo \(q\). Our estimates apply to almost all translation surfaces in a stratum of the moduli space of translation surfaces, with respect to the Masur-Veech measure on the stratum.
@online{mageeCountingSaddleConnections2019,
AUTHOR = {Magee, Michael and Rühr, Rene},
TITLE = {Counting saddle connections in a homology class modulo $q$ (with an appendix by Rodolfo Gutiérrez-Romo)},
YEAR = {2019},
JOURNAL = {J. Mod. Dyn.},
VOLUME = {15},
PAGES = {237--262},
DOI = {10.3934/jmd.2019020}
}
A family of quaternionic monodromy groups of the Kontsevich–Zorich cocycle (2019).
Journal of Modern Dynamics 14, pp. 227–242.
Abstract. Journal. Postprint. Preprint. BibTeX.
For all \(d\) belonging to a subset of density \(1/8\) of the natural numbers, we give an example of a square-tiled surface conjecturally realizing the group \(\mathrm{SO}^*(2d)\) in its standard representation as the Zariski-closure of a factor of its monodromy. We prove that this conjecture holds for the first elements of this subset, showing that the group \(\mathrm{SO}^*(2d)\) is realizable for every \(11 \leq d \leq 299\) such that \(d = 3 \bmod 8\), except possibly for \(d = 35\) and \(d = 203\).
@article{gutierrez-romoFamilyQuaternionicMonodromyGroups,
AUTHOR = {Gutiérrez-Romo, Rodolfo},
TITLE = {A family of quaternionic monodromy groups of the Kontsevich--Zorich cocycle},
YEAR = {2019},
JOURNAL = {J. Mod. Dyn.},
VOLUME = {14},
PAGES = {227--242},
DOI = {10.3934/jmd.2019008}
}
Classification of Rauzy–Veech groups: proof of the Zorich conjecture (2019).
Inventiones Mathematicae 215.3, pp. 741–778.
Abstract. Journal. Postprint. Preprint. BibTeX.
We classify the Rauzy–Veech groups of all connected components of all strata of the moduli space of translation surfaces in absolute homology, showing, in particular, that they are commensurable to arithmetic lattices of symplectic groups. As a corollary, we prove a conjecture of Zorich about the Zariski-density of such groups.
@article{gutierrez-romoClassificationRauzyVeechGroups2018,
AUTHOR = {Gutiérrez-Romo, Rodolfo},
TITLE = {Classification of Rauzy--Veech groups: proof of the Zorich conjecture},
YEAR = {2019},
JOURNAL = {Invent. Math.},
VOLUME = {215},
NUMBER = {3},
PAGES = {741--778},
DOI = {10.1007/s00222-018-0836-7}
}
Characterization of minimal sequences associated with self-similar interval exchange maps (2018).
With Milton Cobo and Alejandro Maass.
Nonlinearity 31.4, pp. 1121–1154.
Abstract. Journal. Postprint. BibTeX.
The construction of affine interval exchange maps with wandering intervals that are semi-conjugate to a given self-similar interval exchange map is strongly related to the existence of the so called minimal sequences associated with local potentials, which are certain elements of the substitution subshift arising from the given interval exchange map. In this article, under the condition called unique representation property, we characterize such minimal sequences for potentials coming from non-real eigenvalues of the substitution matrix.
We also give conditions on the slopes of the affine extensions of a self-similar interval exchange map that determine whether it exhibits a wandering interval or not.
@article{coboCharacterizationMinimalSequences2018,
AUTHOR = {Cobo, Milton and Gutiérrez-Romo, Rodolfo and Maass, Alejandro},
TITLE = {Characterization of minimal sequences associated with self-similar interval exchange maps},
YEAR = {2018},
JOURNAL = {Nonlinearity},
VOLUME = {31},
NUMBER = {4},
PAGES = {1121--1154},
DOI = {10.1088/1361-6544/aa9a87}
}
Wandering intervals in affine extensions of self-similar interval exchange maps: the cubic Arnoux–Yoccoz map (2018).
With Milton Cobo and Alejandro Maass.
Ergodic Theory and Dynamical Systems 38.7, pp. 2537–2570.
Abstract. Journal. Postprint. Preprint. BibTeX.
In this article we provide sufficient conditions on a self-similar interval exchange map, whose renormalization matrix has complex eigenvalues of modulus greater than one, for the existence of affine interval exchange maps with wandering intervals and semi-conjugate with it. These conditions are based on the algebraic properties of the complex eigenvalues and the complex fractals built from the natural substitution emerging from self-similarity. We show that the cubic Arnoux–Yoccoz interval exchange map satisfies these conditions.
@article{coboWanderingIntervalsAffine2018,
AUTHOR = {Cobo, Milton and Gutiérrez-Romo, Rodolfo and Maass, Alejandro},
TITLE = {Wandering intervals in affine extensions of self-similar interval exchange maps: the cubic Arnoux--Yoccoz map},
YEAR = {2018},
JOURNAL = {Ergodic Theory Dynam. Systems},
VOLUME = {38},
NUMBER = {7},
PAGES = {2537--2570},
DOI = {10.1017/etds.2016.143}
}
More about me
I'm fluent in Spanish and English. Some people say I speak French.
I contribute to Etilmercurio, a math and science popularization website in Spanish.