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**1 - 3**of**3**### Small dilatation pseudo-Anosov mapping classes coming from the simplest hyperbolic braid

, 2009

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### The dynamics of mapping classes on surfaces

, 2012

"... 9 Penner examples and quasi-periodicity. In this lecture, we will study a sequence of pseudo-Anosov mapping classes defined by R. Penner [Pen], and describe how they correspond to points on a fibered face. Figure 1: Penner’s sequence Consider the family of mapping classes φg: Sg → Sg shown in Figure ..."

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9 Penner examples and quasi-periodicity. In this lecture, we will study a sequence of pseudo-Anosov mapping classes defined by R. Penner [Pen], and describe how they correspond to points on a fibered face. Figure 1: Penner’s sequence Consider the family of mapping classes φg: Sg → Sg shown in Figure 1. Here Sg is a compact oriented surface of genus g with two boundary components, and φg is the composition φg = rg ◦ dcg ◦ d −1 bg ◦ dag. Lemma 1 (Penner) λ(φg) g ≤ 11. Penner constructed this example and proved Lemma 1 as a key step in proving that the minimum dilatations δg for pseudo-Anosov mapping classes on closed genus g surfaces behaves asymptotically like log(δg) ≃ 1 g. 1 Penner’s Lemma can be generalized to other similar examples (see [Bau], [Val]). Let (S, α) be a pair with Σ a compact oriented surface with boundary, and α a relatively closed curve on S. Let Σ = S \ α be the closure where α is replaced by two copies α + and α −. Now construct surfaces Yn by gluing together Σ in an circular chain as in Figure 2 The surfaces Yn are n-cyclic coverings of S, Figure 2: Circular chain of surfaces ρn: Yn → S, and have covering automorphism group Z/nZ = 〈rn〉. Fix k> 0, and assume n ≥ k. Let γ be a relative multi-curve on S whose algebraic intersection with α is zero, and let γn be a lift to Yn that is contained in Σ1 ∪ Σ2 ∪ · · · ∪ Σk, and is compatibly chosen for all n. Let dn be the Dehn twist defined on γn. We call (S, α) a wedge and γ a connecting curve. For any mapping class η: Σ → Σ, let ηn: Yn → Yn be the mapping class defined by η(x) if x ∈ Σ1 ηn(x) = x otherwise Theorem 2 ([Hir]) Let (S, α) be a wedge, γ a connecting curve. Let η: Σ → Σ be any mapping class such that dγ ◦ η is pseudo-Anosov. Then 1. fn = rn ◦ dn ◦ ηn is pseudo-Anosov for each n, and 2. there is a constant C independent from n such that λ(φn) |χ(Yn) | ≤ C.

### Quotient families of mapping classes

, 2012

"... Thurston’s fibered face theory allows us to partition the set of pseudo-Anosov mapping classes on different compact oriented surfaces into subclasses with related dynamical behavior. This is done via a correspondence between the rational points on fibered faces in the first cohomology of a hyperboli ..."

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Thurston’s fibered face theory allows us to partition the set of pseudo-Anosov mapping classes on different compact oriented surfaces into subclasses with related dynamical behavior. This is done via a correspondence between the rational points on fibered faces in the first cohomology of a hyperbolic 3-manifold and the monodromies of fibrations of the 3-manifold over the circle. In this paper, we generalize examples of Penner, and define quotient families of mapping classes. We show that these mapping classes correspond to open linear sections of fibered faces. The construction gives a simple way to produce families of pseudo-Anosov mapping classes with bounded normalized dilatation and computable invariants, and gives concrete examples of mapping classes associated to sequences of points tending to interior and to the boundary of fibered faces. As an additional aid to calculations, we also develop the notion of Teichmüller polynomials for families of digraphs.