COMBINATORIAL METHODS FOR INTERVAL EXCHANGE TRANSFORMATIONS

International audienceThis is a survey on the big questions about interval exchanges (minimality, unique ergodicity, weak mixing, simplicity) with emphasis on how they can be tackled by mainly combina-torial methods. Interval exchange transformations, defined in Definition 1 below, constitute a famous class of dynamical systems; they were introduced by V. Oseledec [25], and have been extensively studied by many famous authors; up to now, the main results in this swifly-evolving field can be found in the two excellent courses [33] and [34]. To study interval exchanges, three kind of methods can be used: by definition, these systems are one-dimensional, and the first results on them naturally used one-dimensional techniques; then the strongest results on interval exchanges have been obtained by lifting the transformation to higher dimensions and using deep geometric methods. However, many of these results have been reproved by using zero-dimensional methods; these use the codings of orbits to replace the original dynamical system by a symbolic dynamical system, as in Definition 4 below. Now, most of the existing texts, including the two courses mentioned above, focus on the geometric methods; the present survey wants to emphasize what can be achieved by the two other kinds of methods, which have both a strong flavour of combinatorics. The one-dimensional methods yield the basic results, some of which the reader will find in Section 2 below, but also the famous Keane counterexamples described in Section 4, and a very nice new result of M. Bosher-nitzan which is the object of our Section 6; Sections 3 and 5 are devoted to the zero-dimensional methods; the necessary definitions of word combinatorics, symbolic and measurable dynamics are given in Section 1. All those sections are also retracing the colourful history of the theory of interval exchanges, made with big conjectures brilliantly solved after long waits; thus we finish the paper by explaining in Section 7 the last big open question in the domain. This paper stems from a course given during the summer school Dynamique en Cornouaille, which took place in Fouesnant in june 2011; the author is very grateful to the organizer, R. Lep-laideur, for having commandeered it

Harmonic measures for distributions with finite support on the mapping class group are singular

Kaimanovich and Masur showed that a random walk on the mapping class group for an initial distribution with finite first moment and whose support generates a non-elementary subgroup, converges almost surely to a point in the space PMF of projective measured foliations on the surface. This defines a harmonic measure on PMF. Here, we show that when the initial distribution has finite support, the corresponding harmonic measure is singular with respect to the natural Lebesgue measure on PMF.Comment: 43 pages, 16 figures. Minor improvements overall, specifically Section 12. Added reference

K-matrix analysis of the (IJ^{PC}=00^{++})-wave in the mass region below 1900 MeV

We present the results of the current analysis of the partial wave IJ^{PC}=00^{++} based on the available data for meson spectra (pi-pi, K\bar K, eta-eta, eta-eta', pi-pi-pi-pi). In the framework of the K-matrix approach, the analytical amplitude has been restored in the mass region 280 MeV< \sqrt s <1900 MeV. The following scalar-isoscalar states are seen: comparatively narrow resonances f_0(980), f_0(1300), f_0(1500), f_0(1750) and the broad state f_0(1200-1600). The positions of the amplitude poles (masses and total widths of the resonances) are determined as well as pole residues (partial widths to meson channels pi-pi, K\bar K, eta-eta, eta-eta', pi-pi-pi-pi). The fitted amplitude gives us the positions of the K-matrix poles (bare states) and the values of bare-state couplings to meson channels thus allowing the quark-antiquark nonet classification of bare states. On the basis of the obtained partial widths to the channels pi-pi, K\bar K, eta-eta, eta-eta', we estimate the quark/gluonium content of f_0(980), f_0(1300), f_0(1500), f_0(1750), f_0(1200-1600). For f_0(980), f_0(1300), f_0(1500) and f_0(1750), their partial widths testify the q\bar q origin of these mesons though being unable to provide precise evaluation of the possible admixture of the gluonium component in these resonances. The ratios of the decay coupling constants for the f_0(1200-1600) support the idea about gluonium nature of this broad state.Comment: 59 pages, 18 figures, epsfig, references are adde

The Rigidity Conjecture

A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps with a break point. More recent developments show that under similar topological conditions, rigidity does not hold for slightly more general systems. In this paper we state a conjecture which describes how topological classes are organized into rigidity classes.Comment: 6 page

A random tunnel number one 3-manifold does not fiber over the circle

We address the question: how common is it for a 3-manifold to fiber over the circle? One motivation for considering this is to give insight into the fairly inscrutable Virtual Fibration Conjecture. For the special class of 3-manifolds with tunnel number one, we provide compelling theoretical and experimental evidence that fibering is a very rare property. Indeed, in various precise senses it happens with probability 0. Our main theorem is that this is true for a measured lamination model of random tunnel number one 3-manifolds. The first ingredient is an algorithm of K Brown which can decide if a given tunnel number one 3-manifold fibers over the circle. Following the lead of Agol, Hass and W Thurston, we implement Brown's algorithm very efficiently by working in the context of train tracks/interval exchanges. To analyze the resulting algorithm, we generalize work of Kerckhoff to understand the dynamics of splitting sequences of complete genus 2 interval exchanges. Combining all of this with a "magic splitting sequence" and work of Mirzakhani proves the main theorem. The 3-manifold situation contrasts markedly with random 2-generator 1-relator groups; in particular, we show that such groups "fiber" with probability strictly between 0 and 1.Comment: This is the version published by Geometry & Topology on 15 December 200

Hyperbolic Dehn filling in dimension four

We introduce and study some deformations of complete finite-volume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston's hyperbolic Dehn filling. We construct in particular an analytic path of complete, finite-volume cone four-manifolds $M_t$ that interpolates between two hyperbolic four-manifolds $M_0$ and $M_1$ with the same volume $\frac {8}3\pi^2$. The deformation looks like the familiar hyperbolic Dehn filling paths that occur in dimension three, where the cone angle of a core simple closed geodesic varies monotonically from $0$ to $2\pi$. Here, the singularity of $M_t$ is an immersed geodesic surface whose cone angles also vary monotonically from $0$ to $2\pi$. When a cone angle tends to $0$ a small core surface (a torus or Klein bottle) is drilled producing a new cusp. We show that various instances of hyperbolic Dehn fillings may arise, including one case where a degeneration occurs when the cone angles tend to $2\pi$, like in the famous figure-eight knot complement example. The construction makes an essential use of a family of four-dimensional deforming hyperbolic polytopes recently discovered by Kerckhoff and Storm.Comment: 60 pages, 23 figures. Final versio

On canonical triangulations of once-punctured torus bundles and two-bridge link complements

We prove the hyperbolization theorem for punctured torus bundles and two-bridge link complements by decomposing them into ideal tetrahedra which are then given hyperbolic structures, following Rivin's volume maximization principle.Comment: This is the version published by Geometry & Topology on 16 September 2006. Appendix by David Fute

Bifix codes and interval exchanges

We investigate the relation between bifix codes and interval exchange transformations. We prove that the class of natural codings of regular interval echange transformations is closed under maximal bifix decoding.Comment: arXiv admin note: substantial text overlap with arXiv:1305.0127, arXiv:1308.539

Combinatorics of embeddings

We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to every cell there corresponds a unique cell with the complementary set of vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere; there exist precisely two 3-dimensional dichotomial cell complexes, and their 1-skeleta are K_5 and K_{3,3}; and precisely six 4-dimensional ones, and their 1-skeleta all but one graphs of the Petersen family. In higher dimensions n>2, we observe that in order to characterize those compact n-polyhedra that embed in S^{2n} in terms of finitely many "prohibited minors", it suffices to establish finiteness of the list of all (n-1)-connected n-dimensional finite cell complexes that do not embed in S^{2n} yet all their proper subcomplexes and proper cell-like combinatorial quotients embed there. Our main result is that this list contains the n-skeleta of (2n+1)-dimensional dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell complexes include (apart from the three joins of the i-skeleta of (2i+2)-simplices) at least ten non-simplicial complexes.Comment: 49 pages, 1 figure. Minor improvements in v2 (subsection 4.C on transforms of dichotomial spheres reworked to include more details; subsection 2.D "Algorithmic issues" added, etc