Small dilatation pseudo-Anosov mapping classes (Intelligence of Low-dimensional Topology)

1 Minimum dilatation problem Let φ : S → S be a pseudo-Anosov mapping class on an oriented surface S = S g,n of genus g and n punctures. The dilatation λ(φ) is the expansion factor of φ along the stable transverse measured singular foliation associated to φ, and is a Perron algebraic unit greater than one. The set of dilatations for a fixed S is discrete [17]. Let P(S) be the set of all pseudo-Anosov mapping classes on S. Let δ(S) be the minimum dilatation for φ ∈ P(S). Let P g,n be the set of pseudo-Anosov mapping classes on S g,n with dilatation equal to δ(S g,n). The minimum dilatation problem (cf. [15, 14, 3]) can be stated as follows. Problem 1 (Minimum Dilatation Problem I) What is the behavior of δ(S g,n) as a function of g and n? The exact value of δ(S g,n) is not known except for very small cases (for example, for closed surfaces, the answer is only known for g = 2 [6]). However, more is known about the asymptotic behavior of δ(S g,n) as a function of g and n, and the topological Euler characteristic χ(S g,n).