American Institute of Mathematical Sciences

Advanced
July  1997, 3(3): 317-332. doi: 10.3934/dcds.1997.3.317

Diophantine conditions for the linearization of commuting holomorphic functions

David DeLatte 1,

1. 

Department of Mathematics, University of North Texas, Denton, TX 76203-0116, United States

Received  January 1997 Published  April 1997

We prove the local simultaneous linearizability of a pair of commuting holomorphic functions at a shared fixed point under a very general - we conjecture optimal - diophantine condition. Let $f,g :\mathbb{C} \to \mathbb{C}$ with a common fixed point at the origin and suppose that $f(z) = \lambda z + \cdots$ and $\lambda \ne 0$. The map, $f,$ is called linearizable if there is an analytic diffeomorphism, $h$, which conjugates $f$ with its linear part in a neighborhood of the origin, i.e., $h^{-1} \circ f \circ h (z) = \lambda z$ where $\lambda = f'(0).$ Two such diffeomorphisms are simultaneously linearizable if they are linearized by the same map, $h$. If $|\lambda| = 1$ then the situation is delicate. Nonlinearizable maps are topologically abundant, i.e., for $\lambda$ in a dense co-meager set in $\mathbb{S}^1$ there exist nonlinearizable analytic maps with linear coefficient $\lambda$. In contrast there is a diophantine condition on $\lambda$ that is satisfied by a set of full measure in $\mathbb{S}^1$ which assures linearizability of the map $f$.
Keywords: Diophantine conditions, holomorphic functions..
Mathematics Subject Classification: Primary: 58F23; Secondary: 30D05, 58F3.
Citation: David DeLatte. Diophantine conditions for the linearization of commuting holomorphic functions. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 317-332. doi: 10.3934/dcds.1997.3.317
[1]

E. Muñoz Garcia, R. Pérez-Marco. Diophantine conditions in small divisors and transcendental number theory. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1401-1409. doi: 10.3934/dcds.2003.9.1401
[2]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41
[3]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75
[4]

Koichiro Naito. Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations II: Gaps of dimensions and Diophantine conditions. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 449-488. doi: 10.3934/dcds.2004.11.449
[5]

Samir Hodžić, Enes Pasalic. Generalized bent functions -sufficient conditions and related constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 549-566. doi: 10.3934/amc.2017043
[6]

Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481
[7]

Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1333-1349. doi: 10.3934/dcds.2013.33.1333
[8]

Percy Fernández-Sánchez, Jorge Mozo-Fernández, Hernán Neciosup. Dicritical nilpotent holomorphic foliations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3223-3237. doi: 10.3934/dcds.2018140
[9]

Bernard Dacorogna. Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 257-263. doi: 10.3934/dcdsb.2001.1.257
[10]

John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047
[11]

Toshikazu Ito, Bruno Scárdua. Holomorphic foliations transverse to manifolds with corners. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 537-544. doi: 10.3934/dcds.2009.25.537
[12]

Shrikrishna G. Dani. Simultaneous diophantine approximation with quadratic and linear forms. Journal of Modern Dynamics, 2008, 2 (1) : 129-138. doi: 10.3934/jmd.2008.2.129
[13]

Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43
[14]

Hans Koch, João Lopes Dias. Renormalization of diophantine skew flows, with applications to the reducibility problem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 477-500. doi: 10.3934/dcds.2008.21.477
[15]

Chao Ma, Baowei Wang, Jun Wu. Diophantine approximation of the orbits in topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2455-2471. doi: 10.3934/dcds.2019104
[16]

Kingshook Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 847-856. doi: 10.3934/dcds.2010.26.847
[17]

Marco Abate, Francesca Tovena. Formal normal forms for holomorphic maps tangent to the identity. Conference Publications, 2005, 2005 (Special) : 1-10. doi: 10.3934/proc.2005.2005.1
[18]

Marco Abate, Jasmin Raissy. Formal Poincaré-Dulac renormalization for holomorphic germs. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1773-1807. doi: 10.3934/dcds.2013.33.1773
[19]

Eugen Mihailescu, Mariusz Urbański. Holomorphic maps for which the unstable manifolds depend on prehistories. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 443-450. doi: 10.3934/dcds.2003.9.443
[20]

J. M. Peña. Refinable functions with general dilation and a stable test for generalized Routh-Hurwitz conditions. Communications on Pure & Applied Analysis, 2007, 6 (3) : 809-818. doi: 10.3934/cpaa.2007.6.809

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences