By Pilgrim K.M.
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Additional resources for Motivation from dynamics - a brief sketch
Thurston linear map. A multicurve in S − Y is a ﬁnite collection of simple, closed, essential, nonperipheral curves in S − Y, no two of which are homotopic in S − Y. We shall need to require a slightly stronger deﬁnition of invariance, however. 13 (Invariant multicurve). Let (F, S, Y) be an embellished map of spheres and Γ a multicurve in S − Y. Γ is F-invariant if 1. 9 Tameness assumptions 33 2. for each γi ∈ Γ , there exists a γj ∈ Γ such that γi is homotopic in S − Y to a component of F −1 (γj ).
The second invariance condition is equivalent to the condition that the matrix (FΓ ) has a nonzero entry in each row. Remark: It is not always the case that a nonempty multicurve satisfying the ﬁrst condition contains a proper subset which is a sub-multicurve satisfying both conditions–a priori it is possible for a multicurve to consist of a collection of curves, each of whose preimages are inessential or peripheral. This happens if and only if the matrix for the Thurston linear map is the zero matrix.
We say admissible V, V are ambiently homeomorphic if there is a homeomorphism h : (S, Y) → (S, Y) which is the identity on Y and sending V to V . The requirement is the following: If V, V are ambiently homeomorphic, then the homeomorphism h can be taken to be tame and in the same isotopy class as h rel Y. Said loosely another way, the equivalence relation of “ambiently homeomorphic”on the set of admissible subsurfaces should be the same in the tame and topological categories. Remarks: 1. The converse implication in (2) holds trivially using (1), since a pair of tame homeomorphisms yielding a combinatorial equivalence between F and F deﬁnes a path between F and a conjugate of F , which in turn is joined to F by a path deﬁned by an isotopy joining the conjugating map to the identity.
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