arXiv subjects

Toby Hall

Publications and source records attributed to Toby Hall.

21 records · Page 2Linked to original sources

arXiv2008

Decoration invariants for horseshoe braids

The Decoration Conjecture describes the structure of the set of braid types of Smale's horseshoe map ordered by forcing, providing information about the order in which periodic orbits can appear when a horseshoe is created. A proof of this conjecture is given for the class of so-called lone decorations, and it is explained how to calculate associated braid conjugacy invariants which provide additional information about forcing for horseshoe braids.

math.DS↗

arXiv2004

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.

math.DS↗

arXiv2002

Braid forcing and star-shaped train tracks

Global results are proved about the way in which Boyland's forcing partial order organizes a set of braid types: those of periodic orbits of Smale's horseshoe map for which the associated train track is a star. This is a special case of a conjecture introduced in a previous paper, 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.

math.DS↗