Stable Laminations Research Articles

Newhouse phenomenon for automorphisms of low degree in [formula omitted

We show that there exists a polynomial automorphism f of C3 of degree 2 such that for every automorphism g sufficiently close to f, g admits a tangency between the stable and unstable laminations of some hyperbolic set. As a consequence, for each d≥2, there exists an open set of polynomial automorphisms of degree at most d in which the automorphisms having infinitely many sinks are dense. To prove these results, we give a complex analogous to the notion of blender introduced by Bonatti and Díaz.

Topological classification of Morse–Smale diffeomorphisms without heteroclinic curves on 3-manifolds

We show that, up to topological conjugation, the equivalence class of a Morse–Smale diffeomorphism without heteroclinic curves on a $3$-manifold is completely defined by an embedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space.

Shy shadows of infinite-dimensional partially hyperbolic invariant sets

Let${\mathcal{R}}$be a strongly compact$C^{2}$map defined in an open subset of an infinite-dimensional Banach space such that the image of its derivative$D_{F}{\mathcal{R}}$is dense for every $F$. Let$\unicode[STIX]{x1D6FA}$be a compact, forward invariant and partially hyperbolic set of${\mathcal{R}}$such that${\mathcal{R}}:\unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6FA}$is onto. The$\unicode[STIX]{x1D6FF}$-shadow$W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$of$\unicode[STIX]{x1D6FA}$is the union of the sets$$\begin{eqnarray}W_{\unicode[STIX]{x1D6FF}}^{s}(G)=\{F:\operatorname{dist}({\mathcal{R}}^{i}F,{\mathcal{R}}^{i}G)\leq \unicode[STIX]{x1D6FF}\text{for every }i\geq 0\},\end{eqnarray}$$where$G\in \unicode[STIX]{x1D6FA}$. Suppose that$W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$has transversal empty interior, that is, for every$C^{1+\text{Lip}}$$n$-dimensional manifold$M$transversal to the distribution of dominated directions of$\unicode[STIX]{x1D6FA}$and sufficiently close to$W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$we have that$M\cap W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$has empty interior in$M$. Here$n$is the finite dimension of the strong unstable direction. We show that if$\unicode[STIX]{x1D6FF}^{\prime }$is small enough then$$\begin{eqnarray}\mathop{\bigcup }_{i\geq 0}{\mathcal{R}}^{-i}W_{\unicode[STIX]{x1D6FF}^{\prime }}^{s}(\unicode[STIX]{x1D6FA})\end{eqnarray}$$intercepts a$C^{k}$-generic finite-dimensional curve inside the Banach space in a set of parameters with zero Lebesgue measure for every$k\geq 0$. This extends to infinite-dimensional dynamical systems previous studies on the Lebesgue measure of stable laminations of invariants sets.

Triangulating stable laminations

We study the asymptotic behavior of random simply generated noncrossing planar trees in the space of compact subsets of the unit disk, equipped with the Hausdorff distance. Their distributional limits are obtained by triangulating at random the faces of stable laminations, which are random compact subsets of the unit disk made of non-intersecting chords coded by stable L\'evy processes. We also study other ways to "fill-in" the faces of stable laminations, which leads us to introduce the iteration of laminations and of trees.

On the number of large triangles in the Brownian triangulation and fragmentation processes

The Brownian triangulation is a random compact subset of the unit disk introduced by Aldous. For ϵ>0, let N(ϵ) be the number of triangles whose sizes (measured in different ways) are greater than ϵ in the Brownian triangulation. We determine the asymptotic behavior of N(ϵ) as ϵ→0.To obtain this result, a novel concept of “large” dislocations in fragmentations has been proposed. We develop an approach to study the number of large dislocations which is widely applicable to general self-similar fragmentation processes. This technique enables us to study N(ϵ) because of a bijection between the triangles in the Brownian triangulation and the dislocations of a certain self-similar fragmentation process.Our method also provides a new way to obtain the law of the length of the longest chord in the Brownian triangulation. We further extend our results to the more general class of geodesic stable laminations introduced by Kortchemski.

Random stable laminations of the disk

We study large random dissections of polygons. We consider random dissections of a regular polygon with $n$ sides, which are chosen according to Boltzmann weights in the domain of attraction of a stable law of index $\theta\in(1,2]$. As $n$ goes to infinity, we prove that these random dissections converge in distribution toward a random compact set, called the random stable lamination. If $\theta=2$, we recover Aldous’ Brownian triangulation. However, if $\theta\in(1,2)$, large faces remain in the limit and a different random compact set appears. We show that the random stable lamination can be coded by the continuous-time height function associated to the normalized excursion of a strictly stable spectrally positive Lévy process of index $\theta$. Using this coding, we establish that the Hausdorff dimension of the stable random lamination is almost surely $2-1/\theta$.

Dissecting the circle, at random

Random laminations of the disk are the continuous limits of random non-crossing congurations of regular polygons. We provide an expository account on this subject. Initiated by the work of Aldous on the Brownian triangulation, this eld now possesses many characters such as the random recursive triangulation, the stable laminations and the Markovian hyperbolic triangulation of the disk. We will review the properties and constructions of these objects as well as the close relationships they enjoy with the theory of continuous random trees. Some open questions are scattered along the text.

Non-integrability of open billiard flows and Dolgopyat-type estimates

AbstractWe consider open billiard flows in ℝn and show that the standard symplectic form dα in ℝn satisfies a specific non-integrability condition over their non-wandering sets Λ. This allows one to use the main result in Stoyanov [Spectra of Ruelle transfer operators for Axiom A flows. Preprint, 2010, arXiv:0810.1126v4 [math.DS]] and obtain Dolgopyat-type estimates for the spectra of Ruelle transfer operators under simpler conditions. We also describe a class of open billiard flows in ℝn(n≥3) satisfying a certain pinching condition, which in turn implies that the (un)stable laminations over the non-wandering set are C1.

Arc exchange systems and renormalization

We exhibit the construction of stable arc exchange systems from the stable laminations of hyperbolic diffeomorphisms. We prove a one-to-one correspondence between (i) Lipshitz conjugacy classes of C 1+H stable arc exchange systems that are C 1+H fixed points of renormalization and (ii) Lipshitz conjugacy classes of C 1+H diffeomorphisms f with hyperbolic basic sets Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. Let HD s (Λ) and HD u (Λ) be, respectively, the Hausdorff dimension of the stable and unstable leaves intersected with the hyperbolic basic set Λ. If HD u (Λ) = 1, then the Lipschitz conjugacy is, in fact, a C 1+H conjugacy in (i) and (ii). We prove that if the stable arc exchange system is a fixed point of renormalization with bounded geometry, then the stable arc exchange system is smooth conjugate to an affine stable arc exchange system.

Distances of Heegaard splittings

J Hempel [Topology, 2001] showed that the set of distances of the Heegaard splittings (S,V, h^n(V)) is unbounded, as long as the stable and unstable laminations of h avoid the closure of V in PML(S). Here h is a pseudo-Anosov homeomorphism of a surface S while V is the set of isotopy classes of simple closed curves in S bounding essential disks in a fixed handlebody. With the same hypothesis we show the distance of the splitting (S,V, h^n(V)) grows linearly with n, answering a question of A Casson. In addition we prove the converse of Hempel's theorem. Our method is to study the action of h on the curve complex associated to S. We rely heavily on the result, due to H Masur and Y Minsky [Invent. Math. 1999], that the curve complex is Gromov hyperbolic.

On mixing properties of compact group extensions of hyperbolic systems

We study compact group extensions of hyperbolic diffeomorphisms. We relate mixing properties of such extensions with accessibility properties of their stable and unstable laminations. We show that generically the correlations decay faster than any power of time. In particular, this is always the case for ergodic semisimple extensions as well as for stably ergodic extensions of Anosov diffeomorphisms of infranilmanifolds.

Markov neighborhoods for zero-dimensional basic sets

We extend the local stable and unstable laminations for a zero-dimensional basic set to semi-invariant laminations of a neighborhood, and use these extensions to construct the appropriate analog of a Markov partition, which we call a Markov neighborhood. The main applications we give are in the perturbation theory for stable and unstable manifolds; in particular, we prove a transversality theorem. For these applications we require not only that the basic sets be zero dimensional but that they satisfy certain tameness assumptions. This leads to global results on improving stability properties via small isotopies.