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Raul Ures

Publications and source records attributed to Raul Ures.

11 recordsLinked to original sources
arXiv2026

Entropy and semiconjugacy on surfaces

Let $g$ be a $C^\infty$ diffeomorphism in the isotopy class of a pseudo-Anosov homeomorphism $f$ such that $g$ and $f$ have the same topological entropy. In 1988, Handel proved that this implies the existence of a semiconjugacy $\pi$ from $g$ to $f$. He stated that, in general, there is at least one point $x$ such that $\pi^{-1}(x)$ is disconnected. We show that this is not the case: for every $x$, the set $\pi^{-1}(x)$ is the intersection of a nested sequence of closed topological disks, and hence is connected. We also prove that there is a unique $g$-invariant probability measure projecting to the measure of maximal entropy of $f$. This measure is entropy-maximizing, hyperbolic, and Bernoulli, and the semiconjugacy induces a metric isomorphism between the corresponding measure-preserving systems.

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arXiv2026

Partially Hyperbolic Dynamics on $\mathbb T^4$: Existence of Compact Center-Unstable Leaves

We show that for $n\ge2$, if a partially hyperbolic diffeomorphism $f:\mathbb T^{n+1}\to \mathbb T^{n+1}$ with $\dim E^s=\dim E^c=1$ has an invariant center-unstable foliation with a compact incompressible leaf, then this foliation has a transverse closed curve in the universal cover. Also, if $f$ is leaf conjugate to its linear part, it has no compact incompressible center-unstable submanifold. In particular, by the incompressibility result we obtained on Anosov tori, the incompressibility assumptions can be removed when $f$ is defined on $\mathbb T^4$.

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arXiv2024

Measures of maximal entropy that are SRB

A smooth conservative DA-diffeomorphism is smoothly conjugated to its Anosov linear part if and only if all Lyapunov exponents coincide almost everywhere with those of its linear part. A more general result for entropy maximizing measures of $C^{1+\alpha}$ partially hyperbolic diffeomorphisms isotopic to Anosov (DA-diffeomorphisms) on $T^3$ is that they are SRB measures if and only if the sum of its positive Lyapunov exponents coincides with that of the linear Anosov map on all periodic orbits of the support of the measure. In that case, the measure is also the unique physical measure. This rigidity result is not as strong as in the A. Katok rigidity conjecture. Examples are provided.

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arXiv2022

Maximal transverse measures of expanding foliations

For an expanding (unstable) foliation of a diffeomorphism, we use a natural dynamical averaging to construct transverse measures, which we call \emph{maximal}, describing the statistics of how the iterates of a given leaf intersect the cross-sections to the foliation. For a suitable class of diffeomorphisms, we prove that this averaging converges, even exponentially fast, and the limit measures have finite ergodic decompositions. These results are obtained through relating the maximal transverse measures to the maximal $u$-entropy measures of the diffeomorphism.

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arXiv2016

Partially hyperbolic dynamics in dimension 3

Partial hyperbolicity appeared in the sixties as a natural generaliza- tion of hyperbolicity. In the last 20 years in this area there has been great activity. Here we survey the state of the art in some topics, focusing especially in partial hyperbolicity in dimension 3. The reason for this is not only that it is the smallest dimension in which non-degenerate partial hyperbolicity can occur, but also that the topology of 3-manifolds influences this dynamics in revealing ways.

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arXiv2014

A non-dynamically coherent example on ${\mathbb T}^3$

In this paper we give the first example of a non-dynamically coherent partially hyperbolic diffeomorphism with one-dimensional center bundle. The existence of such an example had been an open question since 1975.

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arXiv2004

Dynamics in the isotopy class of a pseudo-Anosov map

Despite its homotopical stability, new relevant dynamics appear in the isotopy class of a pseudo-Anosov homeomorphism. We study these new dynamics by identifying homotopically equivalent orbits, obtaining a more complete description of the topology of the corresponding quotient spaces, and their stable and unstable sets. In particular, we get some insight on how new periodic points appear, among other corollaries. A list of further questions and problems is added at the end of the paper.

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arXiv2001

On manifolds supporting quasi Anosov diffeomorphisms

Let $M$ be an $n$-dimensional manifold supporting a quasi Anosov diffeomorphism. If $n=3$ then either $M={\mathbb T}^3$, in which case the diffeomorphisms is Anosov, or else its fundamental group contains a copy of ${\mathbb Z} ^6$. If $n=4$ then $Π_1(M)$ contains a copy of ${\mathbb Z} ^4$, provided that the diffeomorphism is not Anosov.

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