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Jana Rodriguez Hertz

Publications and source records attributed to Jana Rodriguez Hertz.

At least 19 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 $π$ from $g$ to $f$. He stated that, in general, there is at least one point $x$ such that $π^{-1}(x)$ is disconnected. We show that this is not the case: for every $x$, the set $π^{-1}(x)$ 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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arXiv2025

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 $C^{1+α}$ 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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arXiv2023

A crash course in Pesin theory

This is more or less the content of a 5-hour mini-course that I have given at KTH in the program ``Master class in low-dimensional dynamics'' during May 1-5, 2023. It was not intended to be exhaustive but to give the students a grasp of basic ideas in Pesin Theory in a very short period. It is to be taken as a first approximation and as an invitation to study it in more detail. As such, there was a lot of hand-waving, probably there were mistakes and inaccuracies too. If you have any remark, correction, comment, you can contact me at {\tt janarhertz@gmail.com}.

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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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arXiv2020

New examples of stably ergodic diffeomorphisms in dimension 3

We prove that in the isotopy class of any volume preserving partially hyperbolic diffeomorphism in a $3$-dimensional manifold, there is a non-partially hyperbolic stably ergodic diffeomorphism. In particular, we provide new examples of stably ergodic diffeomorphisms in 3-dimensional manifolds with respect to a smooth volume measure.

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arXiv2020

Stable minimality of expanding foliations

We prove that generically in $\text{Diff}^{1}_{m}(M)$, if an expanding $f$-invariant foliation $W$ of dimension $u$ is minimal and there is a periodic point of unstable index $u$, the foliation is stably minimal. By this we mean there is a $C^{1}$-neighborhood $\mathcal{U}$ of $f$ such that for all $C^{2}$-diffeomorphisms $g\in \mathcal{U}$, the $g$-invariant analytic continuation of $W$ is minimal. In particular, all such $g$ are topologically mixing. Moreover, all such $g$ have a hyperbolic ergodic component of the volume measure $m$ which is essentially dense. This component is, in fact, Bernoulli. We provide new examples of stably minimal diffeomorphisms which are not partially hyperbolic.

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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

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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arXiv2012

Genericity of non-uniform hyperbolicity in dimension 3

For a generic conservative diffeomorphism of a 3-manifold M, the Oseledets splitting is a globally dominated splitting. Moreover, either all Lyapunov exponents vanish almost everywhere, or else the system is non-uniformly hyperbolic and ergodic. This is the 3-dimensional version of a well-known result by Mañé-Bochi, stating that a generic conservative surface diffeomorphism is either Anosov or all Lyapunov exponents vanish almost everywhere. This result inspired and answers in the positive for dimension 3 a conjecture by Avila and Bochi. We also prove that all partially hyperbolic sets with positive measure and center dimension one have a strong homoclinic intersection. This implies that Cr generically for any r, a diffeomorphism contains no proper partially hyperbolic sets with positive measure and center dimension one.

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arXiv2009

New criteria for ergodicity and non-uniform hyperbolicity

In this work we obtain a new criterion to establish ergodicity and non-uniform hyperbolicity of smooth measures of diffeomorphisms. This method allows us to give a more accurate description of certain ergodic components. The use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of non-zero Lyapunov exponents in some contexts. In the partial hyperbolicity context, we obtain that stably ergodic diffeomorphisms are C^1-dense among volume preserving partially hyperbolic diffeomorphisms with two-dimensional center bundle. This is motivated by a well known conjecture of C. Pugh and M. Shub.

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arXiv2005

Hyperbolic sub-dynamics: compact invariant 3-manifolds

In 1970, Hirsch asked what kind of compact invariant sets could be part of a hyperbolic set. Here we obtain that, in case such an invariant set is a 3D manifold, it is a connected sum of tori with handles quotiented by involutions. Moreover, if the manifold is orientable, the involutions are all trivial. In 1975, Ma{ñ}{é} characterized hyperbolic dynamics restricted to manifolds and called them quasi Anosov. We also classify here quasi-Anosov dynamics in 3D-manifolds.

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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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arXiv2004

Continuum-wise expansive homeomorphisms on Peano continua

On a Peano continuum, all local stable and unstable components of a continuum-wise expansive homeomorphism are non trivial. In particular, there is sensitive dependence on initial conditions. This generalizes results in \cite{h,l} about lack of Lyapunov stable points (weak sinks) and existence of non trivial stable and unstable components for expansive homeomorphisms on Peano continua. We also use this fact to generalize a result in \cite{ktt}: a Peano curve $X$ admitting a continuum-wise expansive homeomorphism is nowhere rim-countable. However, it is not resolved the question of whether such a dynamics could be possible in a locally planar Peano curve. Some other questions are posed.

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