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Sergio Romaña

Publications and source records attributed to Sergio Romaña.

At least 19 recordsLinked to original sources
arXiv2026

Partially hyperbolic geodesic flow via conformal deformation

This paper presents a new construction of non-Anosov Partially Hyperbolic Geodesic flows. Our construction is closely related to the construction made by Carneiro and Pujals, the novelty is the use of conformal deformations to produce the examples. Some of the necessary conditions appear more naturally and are easier to check. Besides that, we could enumerate the conditions required to produce partially hyperbolic geodesic flow examples. We show how to produce examples with metrics that are non-positively curved and with a finner analysis we can prove ergodicity for the Liouville measure and uniqueness of the measure of maximal entropy. These examples lie on the boundary of Anosov metrics, allowing us to also produce metrics with partially hyperbolic geodesic flows and conjugate points.

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arXiv2026

Isolated Flat Points and $C^2$-Robustness of the No Focal Points Property

We prove a local stability criterion for the no focal points property on compact Riemannian surfaces. More precisely, if a smooth metric $g$ has non-positive Gaussian curvature and its zero-curvature set is finite, then $g$ belongs to the $C^2$-interior of the set of metrics without focal points. Thus, every sufficiently small $C^2$-perturbation of $g$ still has no focal points, although arbitrarily small perturbations may create regions of positive Gaussian curvature. By Ruggiero's characterization of the $C^2$-interior of the set of metrics without conjugate points, all metrics in the resulting neighborhood are Anosov. We also show that, starting from any hyperbolic metric on a compact surface, one can prescribe an arbitrary finite set as the zero set of the Gaussian curvature of a smooth conformal non-positively curved metric. These metrics can be approximated smoothly by negatively curved metrics, so they lie on the boundary of the negatively curved regime while remaining interior points of the no-focal-points regime. The proof of the stability theorem combines uniform local convexity, Gulliver's bound on the length of geodesic segments contained in small balls, and an inductive argument on the Riccati equation that controls successive passages through the regions where positive curvature may appear.

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arXiv2026

Ruelle's inequality and the Pesin formula for geodesic flows on manifolds without conjugate points

In this paper, we prove the existence of Lyapunov exponents for the geodesic flow on complete Riemannian manifolds without conjugate points whose sectional curvature is bounded below. Under suitable additional curvature assumptions, we establish Ruelle's inequality. Furthermore, assuming the same geometric conditions, we obtain the Pesin entropy formula for $C^1$-Hölder geodesic flows on finite-volume manifolds without conjugate points.

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arXiv2026

Singular Suspension Flows and Infinite Topological Entropy

We construct a dense set of homeomorphisms with infinite topological entropy whose associated pseudo-singular suspension flows have finite entropy -- and indeed, arbitrarily small positive values can be achieved -- showing that infinite entropy is not preserved under singular time changes. Complementing this, we prove that for a residual set of homeomorphisms, all pseudo-singular suspensions retain positive entropy. We also prove that for any $n \geq 2$, there exists a compact n-dimensional manifold admitting a minimal homeomorphism with infinite topological entropy; for such minimal homeomorphisms, a suitably chosen pseudo-singular suspension with a single singularity has zero entropy. Our results reveal that while positivity of entropy is generically stable, its infinitude is fragile under singular reparametrizations.

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arXiv2026

Robust transitivity of geodesic flows from metrics with conjugate points

In this article, we prove a general criterion for obtaining robust transitivity of partially hyperbolic geodesic flows. As a consequence, we exhibit the first example of a $C^2$ open set of Riemannian metrics with conjugate points and transitive geodesic flow. In particular, such a set does not intersect the set of Riemannian metrics with Anosov geodesic flows.

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arXiv2026

Rotation Sets and Topological Entropy for Random Circle Endomorphisms

We study the topological dynamics of random circle endomorphisms of degree one over an ergodic measure-preserving dynamical system. Under an integrability assumption, we prove that the random rotation set is almost surely the compact interval whose endpoints are the mean random rotation numbers of the associated lower and upper random maps. We also show that the natural orbitwise versions of the random rotation set agree almost surely, and on the same full-measure set, every value in this interval is realised as the asymptotic average displacement along an individual orbit. In addition, every closed subinterval of the random rotation set is realised, on a full-measure set, as the set of accumulation values of the displacement averages along a single orbit. Finally, we prove that a positive length of the random rotation set implies a positive random topological entropy, in contrast to random monotone maps, which have zero random topological entropy. We illustrate the theory by computing the random rotation set and random topological entropy for a piecewise linear example.

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arXiv2026

On the Birkhoff Spectrum for Hyperbolic Dynamics

In this paper, we study the structure of Birkhoff spectra for hyperbolic dynamical systems. Given a Hölder observable \(f\) on a basic set \(Λ\), we obtain the following results: First, we characterize when the Birkhoff spectrum of \(f\) is dense in the positive (or negative) real line. Second, we prove that a bounded Birkhoff spectrum forces \(f\) to be cohomologous to zero, which constitutes an extension of Livšic's theorem. Moreover, we show that if the spectrum exhibits an ``arithmetically sparse'' structure, then \(f\) is cohomologous to a constant. \\ \indent We then extend these results to continuous time. For Anosov flows -- including geodesic flows on Anosov manifolds -- we establish analogous density results for Birkhoff integrals over closed orbits. In particular, we generalize a theorem of Dairbekov--Sharafutdinov \cite{Dairbekov} by proving that a bounded (resp.~arithmetically sparse) spectrum forces a smooth function to vanish (resp.~be constant).

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arXiv2025

Mean dimension explosion of induced homeomorphisms

Given $X$ a compact metric space and $T: X \to X$ a continuous map, the induced hyperspace map $T_\mathcal{K}$ acts on the hyperspace $\mathcal{K}(X)$ of closed and nonempty subsets of $X$, and on the continuum hyperspace $\mathcal{C}(X) \subset \mathcal{K}(X)$ of connected sets. This work studies the mean dimension explosion phenomenon: when the base system $T$ has zero topological entropy, but the mean dimension of the induced map $T_\mathcal{K}$ is infinite. In particular, this phenomenon occurs for Morse-Smale diffeomorphisms. Furthermore, for a circle homeomorphism $H$, the mean dimension explosion does not occur if and only if $H$ is conjugate to a rotation. For the metric mean dimension, a different result is obtained: we establish sufficient conditions for the induced hyperspace map to have zero or infinite metric mean dimension.

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arXiv2025

Curvature Rigidity Through Level Sets of Lyapunov Exponents in Geodesic Flows

In this paper, we establish new geometric rigidity results through the study of Lyapunov exponent level sets via invariant measures. First, we prove that for a manifold $M$ without focal points, if the zero Lyapunov exponent level set has full measure with respect to a fully supported invariant measure, then $M$ must be flat. This result recovers and extends a result of Freire and Mañé (cf. [9]). Second, we prove that if the level set of the Lyapunov exponents has full measure with respect to some fully supported measure, then the sectional curvature must be constant. This advances the resolution of Conjecture 1 in [17]. Furthermore, we establish curvature relationships between manifolds with $1$-equivalent geodesic flows, yielding a new criterion to obstruct smooth conjugacy for flows on manifolds without conjugate points. Our techniques provide unified proofs for all rigidity results in [17] as corollaries, and additionally yield rigidity theorems for totally geodesic submanifolds in settings without conjugate points. Notably, several key results hold without compactness assumptions.

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arXiv2025

Entropy of Singular Suspensions

In this work, we investigate diffeomorphisms whose positiveness of topological entropy is destroyed by singular suspensions. We show that this phenomenon is rare in the set of $C^1$-diffeomorphisms. Precisely, we prove that for an open and dense set of $C^1$-diffeomorphism positive topological entropy is preserved by singular suspensions, even for suspensions with infinitely many singularities. We prove a similar result to the conservative diffeomorphisms. We apply our techniques to show that every expansive singular suspension $C^{1+ε}$-flow over a three-dimensional manifold has positive topological entropy. Finally, we explore this phenomenon for Anosov dynamics, showing that to nullify the topological entropy for Anosov suspension flow, the set of singularities must capture the non-wandering set.

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arXiv2024

Riemannian manifolds with Anosov geodesic flow do not have conjugate points

This paper establishes a significant result concerning the absence of conjugate points in certain complete Riemannian manifolds. Specifically, we demonstrate that any complete non-compact manifold with curvature bounded below and an Anosov geodesic flow does not possess conjugate points. This resolves an open problem left by R. Mañé in [9] and subsequently highlighted by [7].

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arXiv2024

Rigidity of Lyapunov Exponents for Geodesic Flows

In this paper, we study rigidity problems between Lyapunov exponents along periodic orbits and geometric structures. More specifically, we prove that for a surface M without focal points, if the value of the Lyapunov exponents is constant over all periodic orbits, then $M$ is the flat 2-torus or a surface of constant negative curvature. We obtain the same result for the case of Anosov geodesic flow for surface, which generalizes Butler's result in dimension two. Using completely different techniques, we also prove an extension of Butler's result to the finite volume case, where the value of the Lyapunov exponents along all periodic orbits is constant, being the maximum or minimum possible.

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arXiv2024

Some Rigidity Theorem for Anosov Geodesic Flows

In this paper, we prove that if the geodesic flow of a complete manifold without conjugate points with sectional curvatures bounded below by $-c^2$ is of Anosov type, then the constant of contraction of the flow is $\geq e^{-c}$. Moreover, if $M$ has finite volume, the equality holds if and only if the sectional curvature is constant. We also apply this result to get a certain rigidity bi-Lipschitz conjugation, and consequently, for $C^1$-conjugacy between two geodesic flows.

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arXiv2023

Typical conservative homeomorphisms have total metric mean dimension

Given a compact smooth boundaryless manifold with dimension greater than one endowed with a locally positive non-atomic measure $μ$, we prove that typical $μ$-preserving homeomorphisms have upper metric mean dimension, with respect to the Riemannian distance, equal to the dimension of the manifold. Moreover, we prove that $μ$ is a measure of maximal metric mean dimension, with respect to the variational principle established in [VV17].

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arXiv2023

Fractal dimensions of the Markov and Lagrange spectra near $3$

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 $0$.

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arXiv2023

Geometric conditions to obtain Anosov geodesic flow in non-compact manifolds

Let $(M, g)$ be a complete Riemannian manifold without focal points and curvature bounded below. We prove that when the average of the sectional curvature in tangent planes along geodesics is negative and uniformly away from zero, then the geodesic flow is of Anosov type. We use this result to construct a non-compact manifold of non-positive curvature with the geodesic flow of Anosov type.

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