Summability condition and rigidity for finite type maps. (English) Zbl 1504.37059
Author’s abstract: We give a bound on the dimension of the Teichmüller space of a finite type map in terms of the number of summable singular values and finite singular orbits. This generalizes rigidity results due to P. Domínguez et al. [Discrete Contin. Dyn. Syst. 12, No. 4, 773–789 (2005; Zbl 1070.37024)], and M. Urbański and A. Zdunik [Isr. J. Math. 161, 347–371 (2007; Zbl 1186.37054)]. We also recover a shorter proof of a transversality theorem due to G. Levin [in: Frontiers in complex dynamics. In celebration of John Milnor’s 80th birthday. Based on a conference, Banff, Canada, February 2011. Princeton, NJ: Princeton University Press. 163–196 (2014; Zbl 1348.37075)]. Our methods are based on the deformation theory introduced by A. L. Epstein [Towers of Finite Type Complex Analytic Maps, PhD thesis, City University of New York (1993)].
Reviewer: Alexander Felshtyn (Szczecin)
MSC:
| 37F34 | Teichmüller theory; moduli spaces of holomorphic dynamical systems |
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 37C30 | Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. |
| 57K20 | 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) |
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