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Rafael Potrie

Publications and source records attributed to Rafael Potrie.

At least 19 recordsLinked to original sources
arXiv2026

Robust transitivity versus trapping regions for partially hyperbolic diffeomorphisms

We show that among partially hyperbolic diffeomorphisms with one dimensional center there is a $C^1$-open and dense subset for which either there is a proper quasi-attractor or both strong foliations are minimal. The same result holds among volume-preserving partially hyperbolic diffeomorphisms and has consequences about robust transitivity beyond the conservative setting. The proofs involve a careful study of minimal $\mathcal{W}^u$-saturated sets and interact with recent results in the subject.

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arXiv2026

Exotic codimension one Anosov flows

We construct Anosov flows in certain circle bundles over closed hyperbolic 3-manifolds, producing counterexamples to a conjecture of Verjovsky. Some of these 4-manifolds admit infinitely many distinct Anosov flows up to orbit equivalence. The construction is made by using Cannon-Thurston maps associated to pseudo-Anosov quasigeodesic flows in hyperbolic $3$-manifolds.

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arXiv2026

Propagation of regularity along unstable manifolds

Let $\varphi_t : M \to M$ be a flow on a smooth closed connected manifold $M$ that preserves and expands a foliation $F$. We establish a theorem of propagation of regularity along the leaves of $F$ for sections of vector bundles satisfying a transport equation involving the generator of a cocycle over $\varphi_t$. As a consequence, we prove a regularity result for Pollicott-Ruelle resonant states: if such state is smooth in restriction to a piece of an unstable leaf, then it is in fact smooth over the entire manifold. We also announce further applications related to joint integrability of extreme bundles of partially hyperbolic diffeomorphisms. The proofs rely on a leafwise semiclassical pseudodifferential calculus adapted to a foliated space, which may be of independent interest.

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arXiv2025

A simultaneous Abels-Margulis-Soifer lemma

The Abels-Margulis-Soifer lemma states that if a semigroup $\Gamma$ acts strongly irreducibly by linear transformations on a finite-dimensional real vector space, then any element of $\Gamma$ can be multiplied by an element of some fixed finite subset of $\Gamma$ so that it becomes proximal (i.e. it acts on the corresponding projective space with an attracting fixed point and a repelling projective hyperplane) and even uniformly proximal (i.e. the distance between the attracting fixed point and the repelling projective hyperplane is uniformly bounded from below and the contraction towards the attracting fixed point is uniformly strong). We prove a version of this lemma simultaneously for linear representations of a semigroup $\Gamma$, acting on the corresponding projective spaces, and for representations of $\Gamma$ to isometry groups of (not necessarily proper) Gromov hyperbolic metric spaces, acting on the corresponding Gromov boundaries.

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arXiv2025

Invariant sets for homeomorphisms of hyperbolic 3-manifolds

We prove that under some assumptions on how points escape to infinity in the universal cover, homeomorphisms of hyperbolic 3-manifolds are forced to have several invariant sets (in particular, they cannot be minimal). For this, we use some shadowing techniques which, when the homeomorphism has positive speed with respect to a uniform foliation, allow us to obtain strong consequences on the structure of the invariant sets. We discuss also homological rotation sets and end the paper with some extensions to other manifolds as well as posing some general problems for the understanding of minimal homeomorphisms of 3-manifolds.

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arXiv2025

Horizontality of partially hyperbolic foliations

We show exactly which Seifert manifolds support partially hyperbolic dynamical systems. In particular, a circle bundle over a higher-genus surface supports a partially hyperbolic system if and only if it supports an Anosov flow. We also show for these systems that the center-stable and center-unstable foliations can be isotoped so that their leaves are transverse to the circle fibering. As a consequence, every partially hyperbolic system defined on the unit tangent bundle of a higher-genus surface is a collapsed Anosov flow.

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arXiv2025

On transverse $R$-covered minimal foliations

We study minimal transverse foliations which are $R$-covered. If in addition the dimension of the ambient manifold is $3$, and the foliations are Anosov foliations we give necessary and sufficient conditions for the intersected foliation to be the orbit foliation of an Anosov flow.

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arXiv2024

Quasi-isometric center action in dimension 3

We study transitive partially hyperbolic diffeomorphisms in dimension 3 preserving a center foliation on which they act quasi-isometrically. We show that the diffeomorphism is up to finite lift and iterate, either a skew-product or a discretised Anosov flow.

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arXiv2024

Quasi-isometric free group representations into SL_3(R)

We study quasi-isometric representations of finitely generated non-abelian free groups into some higher rank semi-simple Lie groups which are not Anosov, nor approximated by Anosov. We show in some cases that these can be perturbed to be non-quasi-isometric, or to have some instability properties with respect to their action on the flag space.

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arXiv2024

Anosov flows in dimension 3: an outside look

These notes were intended as support material for a minicourse on Anosov flows in the conference "Symplectic geometry and Anosov flows'' which took place in Heidelberg in July 2024 organized by Peter Albers, Jonathan Bowden and Agust\'in Moreno. I took the invitation to present the subject as asking from an outsider view of the subject, given the fact that my research uses both ideas and results from the theory of Anosov flows. The point of view of the course is to provide an overview of the main results and questions in the subject, with emphasis on the interaction with topology, geometry, specially symplectic geometry and contact aspects of the theory. Some detail is given in the presentation of the Barbot-Fenley theory of leaf spaces. Hopefully the notes will contribute in gaining a working knowledge of the theory and its many beautiful connections.

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arXiv2023

Intersection of transverse foliations in 3-manifolds: Hausdorff leafspace implies leafwise quasi-geodesic

Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be transverse two dimensional foliations with Gromov hyperbolic leaves in a closed 3-manifold $M$ whose fundamental group is not solvable, and let $\mathcal{G}$ be the one dimensional foliation obtained by intersection. We show that $\mathcal{G}$ is \emph{leafwise quasigeodesic} in $\mathcal{F}_1$ and $\mathcal{F}_2$ if and only if the foliation $\mathcal{G}_L$ induced by $\mathcal{G}$ in the universal cover $L$ of any leaf of $\mathcal{F}_1$ or $\mathcal{F}_2$ has Hausdorff leaf space. We end up with a discussion on the hypothesis of Gromov hyperbolicity of the leaves.

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arXiv2023

Transverse minimal foliations on unit tangent bundles and applications

We show that if $\mathcal{F}_1$ and $\mathcal{F}_2$ are two transverse minimal foliations on $M = T^1S$ then either they intersect in an Anosov foliation or there exists a Reeb-surface in the intersection foliation. The existence of a Reeb surface is incompatible with partially hyperbolic foliations so we deduce from this that certain partially hyperbolic diffeomorphisms in unit tangent bundles are collapsed Anosov flows. We also conclude that every volume preserving partially hyperbolic diffeomorphism of a unit tangent bundle is ergodic.

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arXiv2023

Geometric properties of partially hyperbolic measures and applications to measure rigidity

We give a geometric characterization of the quantitative non-integrability, introduced by Katz, of strong stable and unstable bundles of partially hyperbolic measures and sets in dimension 3. This is done via the use of higher order templates for the invariant bundles. Using the recent work of Katz, we derive some consequences, including the measure rigidity of $uu$-states and the existence of physical measures.

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arXiv2022

Collapsed Anosov flows and self orbit equivalences

We propose a generalization of the concept of discretized Anosov flows that covers a wide class of partially hyperbolic diffeomorphisms in 3-manifolds, and that we call collapsed Anosov flows. They are related with Anosov flows via a self orbit equivalence of the flow. We show that all the examples constructed in a paper by Bonatti, Gogolev, Hammerlindl and the third author belong to this class, and that it is an open and closed class among partially hyperbolic diffeomorphisms. We provide some equivalent definitions which may be more amenable to analysis and are useful in different situations. Conversely we describe the class of partially hyperbolic diffeomorphisms that are collapsed Anosov flows associated with certain types of Anosov flows.

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arXiv2022

Eigenvalue gaps for hyperbolic groups and semigroups

Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the i-th and (i+1)-th Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index i. We establish a similar result for sofic subshifts coming from word hyperbolic groups, in relation with Anosov representations of such groups. We discuss the case of finitely generated semigroups, and propose a notion of Anosov representation in this setting.

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arXiv2022

Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part I: The dynamically coherent case

We study 3-dimensional dynamically coherent partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the transverse geometry and topology of the center stable and center unstable foliations, and the dynamics within their leaves. We find a structural dichotomy for these foliations, which we use to show that every such diffeomorphism on a hyperbolic or Seifert fibered 3-manifold is leaf conjugate to the time one map of a (topological) Anosov flow. This proves a classification conjecture of Hertz-Hertz-Ures in hyperbolic 3-manifolds and in the homotopy class of the identity of Seifert manifolds.

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arXiv2022

Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part II: Branching foliations

We study $3$-dimensional partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the geometry and dynamics of Burago and Ivanov's center stable and center unstable \emph{branching} foliations. This extends our study of the true foliations that appear in the dynamically coherent case (see \emph{Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part I: The dynamically coherent case}, arxiv:1908.06227v3). We complete the classification of such diffeomorphisms in Seifert fibered manifolds. In hyperbolic manifolds, we show that any such diffeomorphism is either dynamically coherent and has a power that is a discretized Anosov flow, or is of a new potential class called a \emph{double translation}.

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