American Institute of Mathematical Sciences

Advanced
September  2010, 26(3): 847-856. doi: 10.3934/dcds.2010.26.847

Complete conjugacy invariants of nonlinearizable holomorphic dynamics

Kingshook Biswas 1,

1. 

Ramakrishna Mission Vivekananda University, Belur Math, WB-711202, India

Received  May 2009 Revised  August 2009 Published  December 2009

Perez-Marco proved the existence of non-trivial totally invariant connected compacts called hedgehogs near the fixed point of a nonlinearizable germ of holomorphic diffeomorphism. We show that if two nonlinearisable holomorphic germs with a common indifferent fixed point have a common hedgehog then they must commute. This allows us to establish a correspondence between hedgehogs and nonlinearizable maximal abelian subgroups of Diff($\mathbb{C},0$). We also show that two nonlinearizable germs with the same rotation number are conjugate if and only if a hedgehog of one can be mapped conformally onto a hedgehog of the other. Thus the conjugacy class of a nonlinearizable germ is completely determined by its rotation number and the conformal class of its hedgehogs.
Keywords: nonlinearizable dynamics, hedgehogs., Irrationally indifferent fixed points.
Mathematics Subject Classification: Primary: 37F5.
Citation: Kingshook Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 847-856. doi: 10.3934/dcds.2010.26.847
[1]

Marian Gidea, Yitzchak Shmalo. Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6123-6148. doi: 10.3934/dcds.2018264
[2]

Paula Kemp. Fixed points and complete lattices. Conference Publications, 2007, 2007 (Special) : 568-572. doi: 10.3934/proc.2007.2007.568
[3]

John Franks, Michael Handel, Kamlesh Parwani. Fixed points of Abelian actions. Journal of Modern Dynamics, 2007, 1 (3) : 443-464. doi: 10.3934/jmd.2007.1.443
[4]

Alexey A. Petrov, Sergei Yu. Pilyugin. Shadowing near nonhyperbolic fixed points. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3761-3772. doi: 10.3934/dcds.2014.34.3761
[5]

Juan Campos, Rafael Ortega. Location of fixed points and periodic solutions in the plane. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 517-523. doi: 10.3934/dcdsb.2008.9.517
[6]

Fabian Ziltener. Note on coisotropic Floer homology and leafwise fixed points. Electronic Research Archive, 2021, 29 (4) : 2553-2560. doi: 10.3934/era.2021001
[7]

Kingshook Biswas. Smooth combs inside hedgehogs. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 853-880. doi: 10.3934/dcds.2005.12.853
[8]

Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043
[9]

Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583
[10]

Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043
[11]

Adrian Petruşel, Radu Precup, Marcel-Adrian Şerban. On the approximation of fixed points for non-self mappings on metric spaces. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 733-747. doi: 10.3934/dcdsb.2019264
[12]

Anna Cima, Armengol Gasull, Víctor Mañosa. Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 889-904. doi: 10.3934/dcds.2018038
[13]

Inmaculada Baldomá, Ernest Fontich, Pau Martín. Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4159-4190. doi: 10.3934/dcds.2017177
[14]

Jifa Jiang, Lei Niu. On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 217-244. doi: 10.3934/dcds.2016.36.217
[15]

A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121.

[16]

Inmaculada Baldomá, Ernest Fontich, Rafael de la Llave, Pau Martín. The parameterization method for one- dimensional invariant manifolds of higher dimensional parabolic fixed points. Discrete & Continuous Dynamical Systems, 2007, 17 (4) : 835-865. doi: 10.3934/dcds.2007.17.835
[17]

Byung-Soo Lee. A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 557-565. doi: 10.3934/naco.2013.3.557
[18]

Frederic Gabern, Àngel Jorba. A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 143-182. doi: 10.3934/dcdsb.2001.1.143
[19]

Lianzhang Bao, Wenxian Shen. Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 1107-1130. doi: 10.3934/dcds.2020072
[20]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (108)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy