Lipshitz matchbox manifolds

A matchbox manifold is a connected, compact foliated space with totally disconnected transversals; or in other notation, a generalized lamination. It is said to be Lipschitz if there exists a metric on its transversals for which the holonomy maps are Lipschitz. Examples of Lipschitz matchbox manifolds include the exceptional minimal sets for $C^1$-foliations of compact manifolds, tiling spaces, the classical solenoids, and the weak solenoids of McCord and Schori, among others. We address the question: When does a Lipschitz matchbox manifold admit an embedding as a minimal set for a smooth dynamical system, or more generally for as an exceptional minimal set for a $C^1$-foliation of a smooth manifold? We gives examples which do embed, and develop criteria for showing when they do not embed, and give examples. We also discuss the classification theory for Lipschitz weak solenoids.Comment: The paper has been significantly revised, with several proofs correcte

Burgers Turbulence

The last decades witnessed a renewal of interest in the Burgers equation. Much activities focused on extensions of the original one-dimensional pressureless model introduced in the thirties by the Dutch scientist J.M. Burgers, and more precisely on the problem of Burgers turbulence, that is the study of the solutions to the one- or multi-dimensional Burgers equation with random initial conditions or random forcing. Such work was frequently motivated by new emerging applications of Burgers model to statistical physics, cosmology, and fluid dynamics. Also Burgers turbulence appeared as one of the simplest instances of a nonlinear system out of equilibrium. The study of random Lagrangian systems, of stochastic partial differential equations and their invariant measures, the theory of dynamical systems, the applications of field theory to the understanding of dissipative anomalies and of multiscaling in hydrodynamic turbulence have benefited significantly from progress in Burgers turbulence. The aim of this review is to give a unified view of selected work stemming from these rather diverse disciplines.Comment: Review Article, 49 pages, 43 figure

Geodesic flow, left-handedness, and templates

We establish that, for every hyperbolic orbifold of type (2, q, $\infty$) and for every orbifold of type (2, 3, 4g+2), the geodesic flow on the unit tangent bundle is left-handed. This implies that the link formed by every collection of periodic orbits (i) bounds a Birkhoff section for the geodesic flow, and (ii) is a fibered link. We also prove similar results for the torus with any flat metric. Besides, we observe that the natural extension of the conjecture to arbitrary hyperbolic surfaces (with non-trivial homology) is false.Comment: Version accepted for publication (Algebraic & Geometric Topology), 60 page

Rokhlin Dimension for Flows

This research was supported by GIF Grant 1137/2011, SFB 878 Groups, Geometry and Actions and ERC Grant No. 267079. Part of the research was conducted at the Fields institute during the 2014 thematic program on abstract harmonic analysis, Banach and operator algebras, and at the Mittag–Leffler institute during the 2016 program on Classification of Operator Algebras: Complexity, Rigidity, and Dynamics.Peer reviewedPostprin

Shape of matchbox manifolds

In this work, we develop shape expansions of minimal matchbox manifolds without holonomy, in terms of branched manifolds formed from their leaves. Our approach is based on the method of coding the holonomy groups for the foliated spaces, to define leafwise regions which are transversely stable and are adapted to the foliation dynamics. Approximations are obtained by collapsing appropriately chosen neighborhoods onto these regions along a "transverse Cantor foliation". The existence of the "transverse Cantor foliation" allows us to generalize standard techniques known for Euclidean and fibered cases to arbitrary matchbox manifolds with Riemannian leaf geometry and without holonomy. The transverse Cantor foliations used here are constructed by purely intrinsic and topological means, as we do not assume that our matchbox manifolds are embedded into a smooth foliated manifold, or a smooth manifold.Comment: 36 pages. Revision of the earlier version: introduction is rewritten. Accepted to a special issue of Indagationes Mathematica

Pattern formation for the Swift-Hohenberg equation on the hyperbolic plane

We present an overview of pattern formation analysis for an analogue of the Swift-Hohenberg equation posed on the real hyperbolic space of dimension two, which we identify with the Poincar\'e disc D. Different types of patterns are considered: spatially periodic stationary solutions, radial solutions and traveling waves, however there are significant differences in the results with the Euclidean case. We apply equivariant bifurcation theory to the study of spatially periodic solutions on a given lattice of D also called H-planforms in reference with the "planforms" introduced for pattern formation in Euclidean space. We consider in details the case of the regular octagonal lattice and give a complete descriptions of all H-planforms bifurcating in this case. For radial solutions (in geodesic polar coordinates), we present a result of existence for stationary localized radial solutions, which we have adapted from techniques on the Euclidean plane. Finally, we show that unlike the Euclidean case, the Swift-Hohenberg equation in the hyperbolic plane undergoes a Hopf bifurcation to traveling waves which are invariant along horocycles of D and periodic in the "transverse" direction. We highlight our theoretical results with a selection of numerical simulations.Comment: Dedicated to Klaus Kirchg\"assne

Embedding solenoids in foliations

In this paper we find smooth embeddings of solenoids in smooth foliations. We show that if a smooth foliation F of a manifold M contains a compact leaf L with H^1(L;R)= 0 and if the foliation is a product foliation in some saturated open neighbourhood U of L, then there exists a foliation F' on M which is C^1-close to F, and F' has an uncountable set of solenoidal minimal sets contained in U that are pair wise non-homeomorphic. If H^1(L;R) is not 0, then it is known that any sufficiently small perturbation of F contains a saturated product neighbourhood. Thus, our result can be thought of as an instability result complementing the stability results of Reeb, Thurston and Langevin and Rosenberg