Symmetric rigidity for circle endomorphisms having bounded geometry

Adamski, John; Hu, Yunchun; Jiang, Yunping; Wang, Zhe

doi:10.1090/proc/15921

Symmetric rigidity for circle endomorphisms having bounded geometry
HTML articles powered by AMS MathViewer

by John Adamski, Yunchun Hu, Yunping Jiang and Zhe Wang PDF

Proc. Amer. Math. Soc. 150 (2022), 3581-3593 Request permission

Abstract:

Let $f$ and $g$ be two circle endomorphisms of degree $d\geq 2$ such that each has bounded geometry, preserves the Lebesgue measure, and fixes $1$. Let $h$ fixing $1$ be the topological conjugacy from $f$ to $g$. That is, $h\circ f=g\circ h$. We prove that $h$ is a symmetric circle homeomorphism if and only if $h=Id$.

References

John Adamski, Symmetric Rigidity for Circle Endomorphisms with Bounded Geometry and Their Dual Maps, ProQuest LLC, Ann Arbor, MI, 2020. Thesis (Ph.D.)–City University of New York. MR 4119915
Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
Frederick P. Gardiner and Yunping Jiang, Asymptotically affine and asymptotically conformal circle endomorphisms, Infinite dimensional Teichmüller spaces and moduli spaces, RIMS Kôkyûroku Bessatsu, B17, Res. Inst. Math. Sci. (RIMS), Kyoto, 2010, pp. 37–53. MR 2560683
Frederick P. Gardiner and Dennis P. Sullivan, Symmetric structures on a closed curve, Amer. J. Math. 114 (1992), no. 4, 683–736. MR 1175689, DOI 10.2307/2374795
Yunchun Hu, Martingales for Uniformly Quasisymmetric Circle Endomorphisms, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–City University of New York. MR 3271885
Yunchun Hu, Yunping Jiang, and Zhe Wang, Martingales for quasisymmetric systems and complex manifold structures, Ann. Acad. Sci. Fenn. Math. 38 (2013), no. 1, 115–140. MR 3076801, DOI 10.5186/aasfm.2013.3812
Yunping Jiang, Renormalization and geometry in one-dimensional and complex dynamics, Advanced Series in Nonlinear Dynamics, vol. 10, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1442953, DOI 10.1142/9789814350105
Yun Ping Jiang, Geometry of geometrically finite one-dimensional maps, Comm. Math. Phys. 156 (1993), no. 3, 639–647. MR 1240589, DOI 10.1007/BF02096866
Yunping Jiang, Lecture Notes in Dynamical Systems and Quasiconformal Mappings: A Course Given in Department of Mathematics at CUNY Graduate Center, Spring Semester of 2009.
Yunping Jiang, An introduction to geometric Gibbs theory, Dynamics, games and science, CIM Ser. Math. Sci., vol. 1, Springer, Cham, 2015, pp. 327–339. MR 3644921
Yunping Jiang, Geometric Gibbs theory, Sci. China Math. 63 (2020), no. 9, 1777–1824. MR 4145919, DOI 10.1007/s11425-019-1638-6
Yunping Jiang, Symmetric invariant measures, Quasiconformal mappings, Riemann surfaces, and Teichmüller spaces, Contemp. Math., vol. 575, Amer. Math. Soc., Providence, RI, 2012, pp. 211–218. MR 2933901, DOI 10.1090/conm/575/11399
G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR 0385004