2010/03/02 by Eiko Kin, Kin, Eiko, Mitsuhiko Takasawa +1
Mathematics · #37E30 #57M27 (Primary) 57M50 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #math.DS #math.GT #msc:37E30 #msc:57M27 #msc:57M50
paper · pdf · doi:10.48550/arxiv.1003.0545
24 pages, 7 figures; v3: minor modification
arxiv created 2011/10/03 · arxiv updated 2011/10/04
Let δg be the minimal dilatation for pseudo-Anosovs on a closed surface Σg of genus g and let δg+ be the minimal dilatation for pseudo-Anosovs on Σg with orientable invariant foliations. This paper concerns the pseudo-Anosovs which occur as the monodromies on closed fibers for Dehn fillings of N(r) for each r ∈ \-3/2, -1/2, 2\ of the magic manifold N. The manifold N(-3/2) is homeomorphic to the Whitehead sister link exterior. We consider the set Λg(r) (resp. Λg+(r)) which consists of the dilatations of all monodromies (resp. monodromies having orientable invariant foliations) on a closed fiber of genus g for Dehn fillings of N(r), where the fillings are on the boundary slopes of fibers of N(r). Hironaka obtained upper bounds of δg and δg+ by computing min Λg(-1/2) and min Λ+g(-1/2) respectively. We prove that min Λg(-3/2)< min Λg(-1/2) for g ≡ 0,1,5,6,7,9 \pmod10 and min Λ+g(-3/2)< min Λ+g(-1/2) for g ≡ 1,5,7,9 \pmod10. These inequalities improve the previous upper bounds of δg and δg+ for these g. We prove that for each r ∈ \-3/2, -1/2, 2\ and each g ≥ 3, there exists a monodromy Φg(r) on a closed fiber of genus g for a Dehn filling of N(r) such that its dilatation λ(Φg(r)) satisfies limg → ∞ |χ(Σg)| log λ(Φg(r)) = 2 log((3+√(5))/2).