American Institute of Mathematical Sciences

Advanced
April  2013, 33(4): 1333-1349. doi: 10.3934/dcds.2013.33.1333

Semigroup representations in holomorphic dynamics

Carlos Cabrera 1, , Peter Makienko 2, and  Peter Plaumann 3,

1. 

Instituto de Matemáticas., Unidad Cuernavaca. UNAM, Av. Universidad s/n. Col. Lomas de Chamilpa, C. P. 62210, Cuernavaca, Morelos, Mexico
2. 

Instituto de Matemáticas, Unidad Cuernavaca. UNAM, Av. Universidad s/n. Col. Lomas de Chamilpa, C.P. 62210, Cuernavaca, Morelos
3. 

Mathematisches Institut, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany

Received  September 2011 Revised  April 2012 Published  October 2012

We use semigroup theory to describe the group of automorphisms of some semigroups of interest in holomorphic dynamical systems. We show, with some examples, that representation theory of semigroups is related to usual constructions in holomorphic dynamics. The main tool for our discussion is a theorem due to Schreier. We extend this theorem, and our results in semigroups, to the setting of correspondences and holomorphic correspondences.
Keywords: Semigroup representations, complex polynomials, holomorphic dynamics, holomorphic correspondences., rational maps.
Mathematics Subject Classification: 37F05, 37F10, 20M3.
Citation: 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
References:
[1]

A. F. Beardon and T. W. Ng, On Ritt's factorization of polynomials,, J. London Math. Soc. (2), 62 (2000), 127.  doi: 10.1093/rpc/2000rpc587.  Google Scholar

[2]

C. Cabrera and P. Makienko, On dynamical Teichmüller spaces,, Conf. Geom and Dyn., 14 (2010), 256.  doi: 10.1090/S1088-4173-2010-00214-6.  Google Scholar

[3]

A. Douady, Systèmes dynamiques holomorphes,, Bourbaki seminar, 1982/83 (1983), 39.   Google Scholar

[4]

A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions,, Acta Math., 171 (1993), 263.  doi: 10.1007/BF02392534.  Google Scholar

[5]

A. Eremenko, On the characterization of a Riemann surface by its semigroup of endomorphisms,, Trans. Amer. Math. Soc., 338 (1993), 123.  doi: 10.2307/2154447.  Google Scholar

[6]

A. Hinkkanen, Functions conjugating entire functions to entire functions and semigroups of analytic endomorphisms,, Complex Variables and Elliptic Equations, 18 (1992), 149.  doi: 10.1080/17476939208814541.  Google Scholar

[7]

M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics,, J. Diff. Geom., 47 (1997), 17.   Google Scholar

[8]

R. Mañè, P. Sad and D. Sullivan, On the dynamics of rational maps,, Ann. Scien. Ec. Norm. Sup. Paris(4), 16 (1983), 193.   Google Scholar

[9]

K. D. Magill, Jr., A survey of semigroups of continous self maps,, Semigroup Forum, 11 (): 189.  doi: 10.1007/BF02195270.  Google Scholar

[10]

C. McMullen, "Complex Dynamics and Renormalization,", Annals of Mathematics Studies, (1994).   Google Scholar

[11]

______, "Renormalization and 3-Manifolds Which Fiber Over the Circle,", Annals of Mathematics Studies, (1996).   Google Scholar

[12]

J. Milnor, "Dynamics of One Complex Variable,", Friedr. Vieweg & Sohn, (1999).   Google Scholar

[13]

J. F. Ritt, Prime and composite polynomials,, Trans. Amer. Math. Soc., 23 (1922), 51.  doi: 10.1090/S0002-9947-1922-1501205-4.  Google Scholar

[14]

J. Schreier, Uber Abbildungen einer abstrakten Menge auf ihre Teilmengen,, Fund. Math., (1937), 261.   Google Scholar

show all references

References:
[1]

A. F. Beardon and T. W. Ng, On Ritt's factorization of polynomials,, J. London Math. Soc. (2), 62 (2000), 127.  doi: 10.1093/rpc/2000rpc587.  Google Scholar

[2]

C. Cabrera and P. Makienko, On dynamical Teichmüller spaces,, Conf. Geom and Dyn., 14 (2010), 256.  doi: 10.1090/S1088-4173-2010-00214-6.  Google Scholar

[3]

A. Douady, Systèmes dynamiques holomorphes,, Bourbaki seminar, 1982/83 (1983), 39.   Google Scholar

[4]

A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions,, Acta Math., 171 (1993), 263.  doi: 10.1007/BF02392534.  Google Scholar

[5]

A. Eremenko, On the characterization of a Riemann surface by its semigroup of endomorphisms,, Trans. Amer. Math. Soc., 338 (1993), 123.  doi: 10.2307/2154447.  Google Scholar

[6]

A. Hinkkanen, Functions conjugating entire functions to entire functions and semigroups of analytic endomorphisms,, Complex Variables and Elliptic Equations, 18 (1992), 149.  doi: 10.1080/17476939208814541.  Google Scholar

[7]

M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics,, J. Diff. Geom., 47 (1997), 17.   Google Scholar

[8]

R. Mañè, P. Sad and D. Sullivan, On the dynamics of rational maps,, Ann. Scien. Ec. Norm. Sup. Paris(4), 16 (1983), 193.   Google Scholar

[9]

K. D. Magill, Jr., A survey of semigroups of continous self maps,, Semigroup Forum, 11 (): 189.  doi: 10.1007/BF02195270.  Google Scholar

[10]

C. McMullen, "Complex Dynamics and Renormalization,", Annals of Mathematics Studies, (1994).   Google Scholar

[11]

______, "Renormalization and 3-Manifolds Which Fiber Over the Circle,", Annals of Mathematics Studies, (1996).   Google Scholar

[12]

J. Milnor, "Dynamics of One Complex Variable,", Friedr. Vieweg & Sohn, (1999).   Google Scholar

[13]

J. F. Ritt, Prime and composite polynomials,, Trans. Amer. Math. Soc., 23 (1922), 51.  doi: 10.1090/S0002-9947-1922-1501205-4.  Google Scholar

[14]

J. Schreier, Uber Abbildungen einer abstrakten Menge auf ihre Teilmengen,, Fund. Math., (1937), 261.   Google Scholar

[1]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[2]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377
[3]

Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015
[4]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363
[5]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342
[6]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316
[7]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[8]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339
[9]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116
[10]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426
[11]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376
[12]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences