André de Carvalho

Pruning fronts and the formation of horseshoes

Let f : π → π be a homeomorphism of the plane π. We define open sets P, called pruning fronts after the work of Cvitanovic (C), for which it is possible to construct an isotopy H : π ×(0,1) → π with open support contained in ( n2Z f n (P) such that H(� ,0) = f(� ) and H(� ,1) = fP(� ), where fP is a homeomorphism under which every point of P is wandering. Applying this construction with f being Smales horseshoe, it is possible to obtain an uncountable family of homeomorphisms, depending on infinitely many parameters, going from trivial to chaotic dynamic behaviour. This family is a 2-dimensional analog of a 1-dimensional universal family.

How to prune a horseshoe

Let F : 2→2 be a homeomorphism. An open F-invariant subset U of 2 is a pruning region for F if it is possible to deform F continuously to a homeomorphism FU for which every point of U is wandering, but which has the same dynamics as F outside of U. This concept is motivated by the Pruning Front Conjecture (PFC) introduced by Cvitanovic, which claims that every Henon map can be understood as a pruned horseshoe. This paper contains recent results in pruning theory, concentrating on prunings of the horseshoe. We describe conditions on a disk D which ensure that the orbit of its interior is a pruning region; explain how prunings of the horseshoe can be understood in terms of underlying tree maps; discuss the connection between pruning and Thurstons classification theorem for surface homeomorphisms; motivate a conjecture describing the forcing relation on horseshoe braid types; and use this theory to give a precise statement of the PFC.

Braid forcing and star-shaped train tracks

Abstract Global results are proved about the way in which Boylands forcing partial order organizes a set of braid types: those of periodic orbits of Smales horseshoe map for which the associated train track is a star. This is a special case of a conjecture introduced in de Carvalho and Hall (Exp. Math. 11(2) (2002) 271), which claims that forcing organizes all horseshoe braid types into linearly ordered families which are, in turn, parameterized by homoclinic orbits to the fixed point of code 0.

Unimodal generalized pseudo-Anosov maps

An infinite family of generalized pseudo-Anosov homeomorphisms of the sphere S is constructed, and their invariant foliations and singular orbits are described explicitly by means of generalized train tracks. The complex strucure induced by the invariant foliations is described, and is shown to make S into a complex sphere. The generalized pseudo-Anosovs thus become quasiconformal automorphisms of the Riemann sphere, providing a complexification of the unimodal family which differs from that of the Fatou/Julia theory.

Conjugacies between horseshoe braids

A proof is given of part of a conjecture about the way in which the set of horseshoe braid types is organized by the forcing partial order. Horseshoe periodic orbits are labelled by their height (a rational number) and decoration (a word in the symbols 0 and 1), and it is shown that, with the exception of the so-called limit cases, periodic orbits of the same height and decoration have the same braid type.

Piecewise linear model for tree maps

We generalize to tree maps the theorems of Parry and Milnor–Thurston about the semi-conjugacy of a continuous piecewise monotone map f to a continuous piecewise linear map with constant slope, equal to the exponential of the entropy of f.

Inverse limits as attractors in parameterized families

We show how a parameterized family of maps of the spine of a manifold can be used to construct a family of homeomorphisms of the ambient manifold which have the inverse limits of the spine maps as global attractors. We describe applications to unimodal families of interval maps, to rotation sets, and to the standard family of circle maps.

Paper folding, Riemann surfaces and convergence of pseudo-Anosov sequences

A method is presented for constructing closed surfaces out of Euclidean polygons with infinitely many segment identifications along the boundary. The metric on the quotient is identified. A sufficient condition is presented which guarantees that the Euclidean structure on the polygons induces a unique conformal structure on the quotient surface, making it into a closed Riemann surface. In this case, a modulus of continuity for uniformising coordinates is found which depends only on the geometry of the polygons and on the identifications. An application is presented in which a uniform modulus of continuity is obtained for a family of pseudo-Anosov homeomorphisms, making it possible to prove that they converge to a Teichmuller mapping on the Riemann sphere. 30C35, 30F10, 37E30; 30C62, 30F45, 37F30 This article addresses the classical problem of constructing surfaces out of subsets of the plane by making identifications along the boundary: in contrast to the usual discussion arising in the classification of surfaces, however, infinitely many identifications are allowed. The topological structure of the identification space S is studied and conditions are given which guarantee that S is a closed surface. The quotient metric on S induced by the Euclidean metric on the plane is identified, and the question of whether or not this metric induces a unique complex structure on S is discussed. A sufficient condition for uniqueness of the complex structure is given and, when it holds, a modulus of continuity for uniformising coordinates is obtained. The interplay between the metric and conformal structures is central to the paper, promoting the topological structure to a Riemann surface structure and providing quantitative control over the quotient map. This analytic control is then used to prove convergence of a certain sequence of pseudo-Anosov homeomorphisms to a generalised pseudo-Anosov homeomorphism. Let P be a finite collection of disjoint polygons in the (complex) plane. A paper-folding scheme is an equivalence relation which glues together segments — possibly infinitely

Paper surfaces and dynamical limits

It is very common in mathematics to construct surfaces by identifying the sides of a polygon together in pairs: For example, identifying opposite sides of a square yields a torus. In this article the construction is considered in the case where infinitely many pairs of segments around the boundary of the polygon are identified. The topological, metric, and complex structures of the resulting surfaces are discussed: In particular, a condition is given under which the surface has a global complex structure (i.e., is a Riemann surface). In this case, a modulus of continuity for a uniformizing map is given. The motivation for considering this construction comes from dynamical systems theory: If the modulus of continuity is uniform across a family of such constructions, each with an iteration defined on it, then it is possible to take limits in the family and hence to complete it. Such an application is briefly discussed.

Itineraries for inverse limits of tent maps: A backward view

Abstract Previously published admissibility conditions for an element of { 0 , 1 } Z to be the itinerary of a point of the inverse limit of a tent map are expressed in terms of forward orbits. We give necessary and sufficient conditions in terms of backward orbits, which is more natural for inverse limits. These backward admissibility conditions are not symmetric versions of the forward ones: in particular, the maximum backward itinerary which can be realised by a tent map mode locks on intervals of kneading sequences.