Pruning theory and Thurston's classification of surface homeomorphisms (Q5953555)

New techniques for studying the dynamics of families of surface homeomorphisms are introduced. Namely, two dynamical deformation theories are presented -- one for surface homeomorphisms, called pruning, and another one for graph endomorphisms, called kneading. Both theories give conditions under which all of the dynamics in an open set can be destroyed, while leaving the dynamics unchanged elsewhere. These two theories are related to each other and to Thurston's classification of surface homeomorphisms up to isotopy.NEWLINENEWLINEA more comprehensive description of the results obtained is as follows.NEWLINENEWLINE1. Pruning Theory. The initial surface homeomorphism \(F:S\to S\) is taken to be one whose dynamics is well understood, in the sense that all of the nontrivial dynamics is contained in an \(F\)-invariant thick graph \(\mathbb G\), which has the property that \(F:\mathbb G\to \mathbb G\) induces a graph endomorphism \(f:G\to G\). Conditions are given under which all of the dynamics in a topological disk \(D\) (called a pruning disk) can be destroyed by an isotopy supported in \(D\) (and an arbitrarily small region in \(S\setminus \mathbb G\)).NEWLINENEWLINEThe Pruning Family. An uncountable family \(\mathcal P(F)\) of homeomorphisms of \(S\) is constructed. The dynamics of maps in \(\mathcal P(F)\) can be understood as the dynamics of \(F\) less that which is pruned away from a sequence of pruning disks.NEWLINENEWLINE2. Kneading Theory. Given a graph endomorphism \(f:G\to G\), conditions are given on open subsets \(K\) of \(G\) (called kneading sets) for which it is possible to modify both \(G\) and \(f\) to obtain a graph endomorphism \(f_K:G_K\to G_K\) whose dynamics can be understood as being the dynamics of \(f\) less the dynamics which is contained in \(K\).NEWLINENEWLINEThe Kneading Family. An uncountable family \(\mathcal K(f)\) of endomorphisms of one-dimensional spaces can be constructed. The dynamics of maps in \(\mathcal K(f)\) can be understood as the dynamics of \(f\) less that which is pulled away from a sequence of kneading sets.NEWLINENEWLINE3. Both families. If \(f:G\to G\) is induced by a thick graph map \(F:\mathbb G\to \mathbb G\), the kneading family \(\mathcal K(f)\) and the pruning family \(\mathcal P(F)\) are in one-to-one correspondence: collapsing segments of stable sets take \(\mathcal P(F)\) to \(\mathcal K(f)\) and the inverse limit takes \(\mathcal K(f)\) back to \(\mathcal P(F)\).NEWLINENEWLINEAs a consequence the following results are obtained.NEWLINENEWLINEThe pruning family is big. The pruning family of Smale's horseshoe map \(F\) contains a subfamily whose dynamics mimics that of a full family of unimodal maps of the interval. In particular, \(\mathcal P(F)\) contains uncountably many distinct dynamical systems.NEWLINENEWLINEPruning and Thurston Classification. For any pseudo-Anosov map isotopic to \(F\) relative to some finite \(F\)-invariant set, there is a map in the pruning family \(\mathcal P(F)\) with the same (non-trivial) dynamics.