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Raúl Ures

Publications and source records attributed to Raúl Ures.

10 recordsLinked to original sources
arXiv2026

Anosov Diffeomorphisms of Finite-Type Surfaces

We prove that a complete surface of finite topological type satisfying Condition A is homeomorphic to the two-torus. More generally, we show that an isolated planar end of a complete open surface satisfying Condition A cannot be periodic under the induced action on the space of ends. Condition A includes dense periodic points, a uniformly hyperbolic splitting into one-dimensional subbundles for a complete metric, and invariant line fields that uniquely integrate to transverse stable and unstable foliations. These results give a partial answer to the question of whether a complete surface satisfying Condition A must be compact.

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arXiv2025

Accessibility and central integrability in the absence of periodic points

We consider a partially hyperbolic diffeomorphism $f: M \to M$ without periodic points on a closed manifold $M$. We prove that $f$ is accessible when $M$ is a 3-manifold with non-virtually-solvable fundamental group $π_1(M)$. In the case where $\dim E^c = 1$, we demonstrate that the center bundle $E^c$ is uniquely integrable if $f$ lacks accessibility. Furthermore, we provide a complete characterization of accessibility classes for such systems with one-dimensional center bundles.

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arXiv2025

Partially hyperbolic diffeomorphisms homotopic to the identity in dimension three

We show that any conservative partially hyperbolic diffeomorphism homotopic to the identity is accessible unless the fundamental group of its ambient 3-manifold is virtually solvable. As a consequence, such diffeomorphisms are ergodic, giving an affirmative answer to the Hertz-Hertz-Ures Ergodicity Conjecture in the homotopy class of identity.

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arXiv2025

Accessibility and Ergodicity of Partially Hyperbolic Diffeomorphisms without Periodic Points

We prove that every $C^2$ conservative partially hyperbolic diffeomorphism of a closed 3-manifold without periodic points is ergodic, which gives an affirmative answer to the Ergodicity Conjecture by Hertz-Hertz-Ures in the absence of periodic points. We also show that a partially hyperbolic diffeomorphism of a closed 3-manifold $M$ with no periodic points is accessible if the non-wandering set is all of $M$ and the fundamental group $π_1(M)$ is not virtually solvable.

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arXiv2025

On the Ergodicity of Rotation Extensions of Hyperbolic Endomorphisms

We study the ergodicity of partially hyperbolic endomorphisms, focusing on skew products where the base dynamics are governed by Anosov endomorphisms. For this family, we establish ergodicity and prove that accessibility holds for an open and dense subset. By analyzing the topological implications of accessibility, we demonstrate that conservative accessible partially hyperbolic endomorphisms are topologically transitive. Leveraging accessibility, we further show ergodicity for skew products with $\mathbb{S}^1$-fibers. Finally, although out the context of rotation extensions, we prove ergodic stability results for partially hyperbolic endomorphisms with $\dim(E^c) = 1.$

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arXiv2021

Robust minimality of strong foliations for DA diffeomorphisms: $cu$-volume expansion and new examples

Let $f$ be a $C^2$ partially hyperbolic diffeomorphisms of ${\mathbb T}^3$ (not necessarily volume preserving or transitive) isotopic to a linear Anosov diffeomorphism $A$ with eigenvalues $$λ_{s}<1<λ_{c}<λ_{u}.$$ Under the assumption that the set $$\{x: \,\mid\log \det(Tf\mid_{E^{cu}(x)})\mid \leq \log λ_{u} \}$$ has zero volume inside any unstable leaf of $f$ where $E^{cu} = E^c\oplus E^u$ is the center unstable bundle, we prove that the stable foliation of $f$ is $C^1$ robustly minimal, i.e., the stable foliation of any diffeomorphism $C^1$ sufficiently close to $f$ is minimal. In particular, $f$ is robustly transitive.\par We build, with this criterion, a new example of a $C^1$ open set of partially hyperbolic diffeomorphisms, for which the strong stable foliation and the strong unstable foliation are both minimal.

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arXiv2019

Maximal entropy measures of diffeomorphisms of circle fiber bundles

We characterize the maximal entropy measures of partially hyperbolic C^2 diffeomorphisms whose center foliations form circle bundles, by means of suitable finite sets of saddle points, that we call skeletons. In the special case of 3-dimensional nilmanifolds other than the torus, this entails the following dichotomy: either the diffeomorphism is a rotation extension of an Anosov diffeomorphism -- in which case there is a unique maximal measure, with full support and zero center Lyapunov exponents -- or there exist exactly two ergodic maximal measures, both hyperbolic and whose center Lyapunov exponents have opposite signs. Moreover, the set of maximal measures varies continuously with the diffeomorphism.

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arXiv2018

On the three-legged accessibility property

We show that certain types of the three-legged accessibility property of a partially hyperbolic diffeomorphism imply the existence of a unique minimal set for one strong foliation and the transitivity of the other one. In case the center dimension is one, we also give a criteria to obtain three-legged accessibility in a robust way. We show some applications of our results to the time-one map of Anosov flows, skew products and certain Anosov diffeomorphisms with partially hyperbolic splitting.

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arXiv2016

On the non-robustness of intermingled basins

It is well-known that it is possible to construct a partially hyperbolic diffeomorphism on the 3-torus in a similar way than in Kan's example. It has two hyperbolic physical measures with intermingled basins supported on two embedded tori with Anosov dynamics. A natural question is how robust is the intermingled basins phenomenon for diffeomorphisms defined on boundaryless manifolds? In this work we will show that on the 3-torus the only partially hyperbolic examples having hyperbolic physical measures with intermingled basins are not robust.

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