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Rafael Potrie

Publications and source records attributed to Rafael Potrie.

At least 37 records · Page 2Linked to original sources
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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arXiv2022

Ergodicity of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds

We study conservative partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We show that they are always accessible and deduce as a result that every conservative $C^{1+}$ partially hyperbolic in a hyperbolic 3-manifold must be ergodic, giving an afirmative answer to a conjecture of Hertz-Hertz-Ures in the context of hyperbolic 3-manifolds. Some of the intermediary steps are also done for general partially hyperbolic diffeomorphisms homotopic to the identity.

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arXiv2022

Topology of leaves for minimal laminations by hyperbolic surfaces

We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness which may be of independent interest. The main step in establishing this relative version of residual finiteness is to obtain finite covers with control on the \emph{second systole} of the surface, which is done in the appendix. In a companion paper, the case of other generic leaves is treated.

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arXiv2021

Minimality of the action on the universal circle of uniform foliations

Given a uniform foliation by Gromov hyperbolic leaves on a $3$-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are $\mathbb{R}$-covered and we give a new description of the universal circle of $\mathbb{R}$-covered foliations with Gromov hyperbolic leaves in terms of the JSJ decomposition of $M$.

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arXiv2021

Accessibility and ergodicity for collapsed Anosov flows

We consider a class of partially hyperbolic diffeomorphisms introduced in [BFP] which is open and closed and contains all known examples. If in addition the diffeomorphism is non-wandering, then we show it is accessible unless it contains a su-torus. This implies that these systems are ergodic when they preserve volume, confirming a conjecture by Hertz-Hertz-Ures for this class of systems.

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arXiv2021

A remark on uniform expansion

For every $\mathcal{U} \subset \mathrm{Diff}^\infty_{vol}(\mathbb{T}^2)$ there is a measure of finite support contained in $\mathcal{U}$ which is uniformly expanding.

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arXiv2021

Partially hyperbolic dynamics and 3-manifold topology

This is an expository note intended to illustrate current research in topological study of partially hyperbolic diffeomorphisms in dimension 3 with a beautiful result due to Margulis and Plante-Thurston on topological obstructions for a manifold to admit an Anosov flow.

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arXiv2020

Anosov representations and dominated splittings

We provide a link between Anosov representations introduced by Labourie and dominated splitting of linear cocycles. This allows us to obtain equivalent characterizations for Anosov representations and to recover recent results due to Guéritaud-Guichard-Kassel-Wienhard and Kapovich-Leeb-Porti by different methods. We also give characterizations in terms of multicones and cone-types inspired in the work of Avila-Bochi-Yoccoz and Bochi-Gourmelon. Finally we provide a new proof of the higher rank Morse Lemma of Kapovich-Leeb-Porti.

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arXiv2020

Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends work of Bonatti, Gogolev, Hammerlindl and Potrie to the whole isotopy class. We relate the techniques with the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in the previous paper by the authors. The appendix reviews some consequences of the Nielsen-Thurston classification of surface homeomorphisms to the dynamics of lifts of such maps to the universal cover.

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arXiv2019

Finiteness of partially hyperbolic attractors with one-dimensional center

We prove that the set of diffeomorphisms having at most finitely many attractors contains a dense and open subset of the space of $C^1$ partially hyperbolic diffeomorphisms with one-dimensional center. This is obtained thanks to a robust geometric property of partially hyperbolic laminations that we show to hold after perturbations of the dynamics. This technique also allows to prove that $C^1$-generic diffeomorphisms far from homoclinic tangencies in dimension $3$ either have at most finitely many attractors, or satisfy Newhouse phenomenon.

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arXiv2019

Anomalous partially hyperbolic diffeomorphisms III: abundance and incoherence

Let $M$ be a closed 3-manifold which admits an Anosov flow. In this paper we develop a technique for constructing partially hyperbolic representatives in many mapping classes of $M$. We apply this technique both in the setting of geodesic flows on closed hyperbolic surfaces and for Anosov flows which admit transverse tori. We emphasize the similarity of both constructions through the concept of $h$-transversality, a tool which allows us to compose different mapping classes while retaining partial hyperbolicity. In the case of the geodesic flow of a closed hyperbolic surface $S$ we build stably ergodic, partially hyperbolic diffeomorphisms whose mapping classes form a subgroup of the mapping class group $\mathcal{M}(T^1S)$ which is isomorphic to $\mathcal{M}(S)$. At the same time we show that the totality of mapping classes which can be realized by partially hyperbolic diffeomorphisms does not form a subgroup of $\mathcal{M}(T^1S)$. Finally, some of the examples on $T^1S$ are absolutely partially hyperbolic, stably ergodic and robustly non-dynamically coherent, disproving a conjecture by F. Rodriguez Hertz, J. Rodriguez Hertz and R. Ures.

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arXiv2018

Seifert manifolds admitting partially hyperbolic diffeomorphisms

We characterize which 3-dimensional Seifert manifolds admit transitive partially hyperbolic diffeomorphisms. In particular, a circle bundle over a higher-genus surface admits a transitive partially hyperbolic diffeomorphism if and only if it admits an Anosov flow.

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arXiv2018

Robust dynamics, invariant structures and topological classification

This text is about geometric structures imposed by robust dynamical behaviour. We explain recent results towards the classification of partially hyperbolic systems in dimension 3 using the theory of foliations and its interaction with topology. We also present recent examples which introduce a challenge in the classification program and we propose some steps to continue this classification. Finally, we give some suggestions on what to do after classification is achieved.

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arXiv2017

Rotation intervals and entropy on attracting annular continua

We show that if $f$ is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of $f$ is positive. Further, the entropy is shown to be associated to a $C^0$-robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors so that the rotation interval is uniformly large but the entropy approaches zero as much as desired. The developed techniques allow us to obtain similar results in the context of Birkhoff attractors.

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arXiv2017

Eigenvalues and Entropy of a Hitchin representation

We show that the critical exponent of a representation in the Hitchin component of $PSL(d,\mathbb{R})$ is bounded above, the least upper bound being attained only in the Fuchsian locus. This provides a rigid inequality for the area of a minimal surface on $ρ\backslash X,$ where $X$ is the symmetric space of $PSL(d,\mathbb{R}).$ The proof relies in a construction useful to prove a regularity statement: if the Frenet equivariant curve of $ρ$ is smooth, then $ρ$ is Fuchsian.

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arXiv2017

Many intermingled basins in dimension 3

We construct a diffeomorphism of $\mathbb{T}^3$ admitting any finite or countable number of physical measures with intermingled basins. The examples are partially hyperbolic with splitting $T\mathbb{T}^3 = E^{cs} \oplus E^u$ and can be made volume hyperbolic and topologically mixing.

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