Unimodal generalized pseudo-Anosov maps. (English) Zbl 1057.37041

de Carvalho, André; Hall, Toby

Geom. Topol. 8, 1127-1188 (2004).

This extended paper consists of 8 sections. The introductory Section 1 contains the overview on pseudo-Anosov homeomorphisms and some basis definitions and notation. In Section 2 – Markov thick interval maps – a brief description of thick interval maps, where the surface is the sphere and the graph is an interval, is given. The Section 3 – Generalized train tracks – contains a description of generalized train tracks associated to thick interval maps. The Section 4 – Unimodal maps, symbolic dynamics and the horseshoe – contains a summary of the theory of unimodal maps, symbolic dynamics and the Smale horseshoe which will be used later. In Section 5 – The outside dynamics of a unimodal map – a description of those orbits of a unimodal map which are never lost “inside” the fold is provided. In Section 6 – Invariant unimodal generalized train tracks – the invariant generalized train tracks corresponding to elements of the set of periodic or preperiodic kneading sequences, whose associated transition matrices are irreducible and aperiodic, are described explicitly. In Section 7 – Unimodal generalized pseudo-Anosov maps – the construction of generalized pseudo-Anosov maps starting from the invariant train tracks of Section 6 is described. In Section 8 – The complex structure – it is shown that the topological sphere \(S\) constructed above carries a natural complex structure which makes it into a complex sphere. With respect to this structure the \(\Phi\)-invariant foliations become the horizontal and vertical trajectories of an integrable quadratic differential which is meromorphic away from finitely many essential singularities, with respect to which \(\Phi\) is a Teichmüller mapping.
The text of the paper is accompanied by many interesting and useful figures.

Reviewer: Alois Klíč (Praha)

MSC:

37E30	Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
57M50	General geometric structures on low-dimensional manifolds
37F50	Small divisors, rotation domains and linearization in holomorphic dynamics
37F75	Dynamical aspects of holomorphic foliations and vector fields
37D20	Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37E05	Dynamical systems involving maps of the interval
37B10	Symbolic dynamics

Keywords:

Pseudo-Anosov homeomorphisms; train tracks; unimodal maps; horseshoe; interval maps; symbolic dynamics; complex structure; foliations

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