Toby Hall

The creation of horseshoes

We present results which describe constraints on the order in which periodic orbits can appear when a horseshoe is created. We associate two rational numbers q(R) and r(R) to each periodic orbit R of the horseshoe, which have the property that if r(R)<q(S) then the orbit R must appear after the orbit S; while if r(S)<r(R) and q(R)<q(S) then either orbit can appear before the other. The time required to compute these quantities is bounded by a linear function of the period of R. We also present an algorithm for determining the rotation interval of a horseshoe orbit, and describe techniques for obtaining lower bounds on the topological entropy of a horseshoe orbit.

Zeros of the kneading invariant and topological entropy for Lorenz maps

If is a unimodal map, then its topological entropy is related to the smallest positive zero s of a certain power series (the kneading invariant of f) by . Moreover, it is implicit in the results of Jonker and Rand that for each positive entropy basic set in the renormalization decomposition of the non-wandering set of f, there is a real zero of the kneading invariant such that . Here we prove a similar result for Lorenz maps. In contrast to the unimodal case, it is possible for two basic sets in the renormalization decomposition of the non-wandering set of a Lorenz map to have the same entropy, and we show that in this case there is a corresponding double zero of the kneading invariant.

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.

Unremovable periodic orbits of homeomorphisms

In [1], Asimov and Franks give conditions under which a collection of periodic orbits of a diffeomorphism f : M → M of a compact manifold persists under arbitrary isotopy of f . Together with the Nielsen–Thurston theory, their result has been of pivotal importance in recent work on the periodic orbit structure of surface automorphisms (for example [ 3, 4, 7, 8, 9, 12, 13 ]). However, their proof uses bifurcation theory and as such depends crucially upon the differentiability of f . The periodic orbit results which make use of the Asimov–Franks theorem are therefore applicable only in the differentiable case, a limitation which belies their topological character. In this paper we shall use classical Nielsen-theoretic methods to prove the analogue of the Asimov–Franks result for homeomorphisms.

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.

Fat one-dimensional representatives of pseudo-Anosov isotopy classes with minimal periodic orbit structure

We consider isotopy classes of homeomorphisms of the disc relative to a periodic orbit. Representatives of such isotopy classes are constructed which yield piecewise linear maps of the interval on identification along stable leaves: this means that their periodic orbit structures are easily determined. In the case where the isotopy class is of pseudo-Anosov type, necessary and sufficient conditions are given for this periodic orbit structure to be minimal in the isotopy class.

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.

Symbol ratio minimax sequences in the lexicographic order

Consider the space of sequences of k letters ordered lexicographically. We study the set M({\alpha}) of all maximal sequences for which the asymptotic proportions {\alpha} of the letters are prescribed, where a sequence is said to be maximal if it is at least as great as all of its tails. The infimum of M({\alpha}) is called the {\alpha}-infimax sequence, or the {\alpha}-minimax sequence if the infimum is a minimum. We give an algorithm which yields all infimax sequences, and show that the infimax is not a minimax if and only if it is the {\alpha}-infimax for every {\alpha} in a simplex of dimension 1 or greater. These results have applications to the theory of rotation sets of beta-shifts and torus homeomorphisms.