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Thanks to a recent result by Jean-Marc Schlenker, we establish an explicit linear inequality between the normalized entropies of pseudo-Anosov automorphisms and the hyperbolic volumes of their mapping tori. As its corollaries, we give an improved lower bound for values of entropies of pseudo-Anosovs on a surface with fixed topology, and a proof of a slightly weaker version of the result by Farb, Leininger and Margalit first, and by Agol later, on finiteness of cusped manifolds generating surface automorphisms with small normalized entropies. Also, we present an analogous linear inequality between the Weil-Petersson translation distance of a pseudo-Anosov map (normalized by multiplying the square root of the area of a surface) and the volume of its mapping torus, which leads to a better bound.Σ × [0, 1]/(x, 1) ∼ (h(x), 0).Since the topology of the mapping torus depends only on the mapping class ϕ, we denote its topological type by N ϕ A celebrated theorem by Thurston [24] asserts that N ϕ admits a hyperbolic structure iff ϕ is pseudo-Anosov. By Mostow-Prasad rigidity a hyperbolic structure of finite volume in dimension 3 is unique and geometric invariants are in fact topological invariants. In [12], Kin, Takasawa and the first named author compared the hyperbolic volume of N ϕ , denoted by vol N ϕ , with the entropy of ϕ, denoted by ent ϕ. By entropy we mean the infimum of the topological entropy of automorphisms isotopic to ϕ. In particular, they proved that there is a constant C(g, m) > 0 depending only on the topology of Σ such that ent ϕ ≥ C(g, m) volN ϕ .

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Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Finally, this analysis is applied to investigate the nature of the Markoff conjecture.

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