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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, 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), 127-138. doi: 10.1093/rpc/2000rpc587.  Google Scholar

[2]

C. Cabrera and P. Makienko, On dynamical Teichmüller spaces, Conf. Geom and Dyn., 14 (2010), 256-268. doi: 10.1090/S1088-4173-2010-00214-6.  Google Scholar

[3]

A. Douady, Systèmes dynamiques holomorphes, Bourbaki seminar, 1982/83, Astèrisque, 105, Soc. Math. France, Paris, (1983), 39-63.  Google Scholar

[4]

A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993), 263-297. 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), 123-131. 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), 149-154. doi: 10.1080/17476939208814541.  Google Scholar

[7]

M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Diff. Geom., 47 (1997), 17-94.  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), 193-217.  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, vol. 135, Princeton University Press, Princeton, NJ, 1994.  Google Scholar

[11]

______, "Renormalization and 3-Manifolds Which Fiber Over the Circle," Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 1996.  Google Scholar

[12]

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

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J. F. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc., 23 (1922), 51-66. doi: 10.1090/S0002-9947-1922-1501205-4.  Google Scholar

[14]

J. Schreier, Uber Abbildungen einer abstrakten Menge auf ihre Teilmengen, Fund. Math., (1937), 261-264. 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), 127-138. doi: 10.1093/rpc/2000rpc587.  Google Scholar

[2]

C. Cabrera and P. Makienko, On dynamical Teichmüller spaces, Conf. Geom and Dyn., 14 (2010), 256-268. doi: 10.1090/S1088-4173-2010-00214-6.  Google Scholar

[3]

A. Douady, Systèmes dynamiques holomorphes, Bourbaki seminar, 1982/83, Astèrisque, 105, Soc. Math. France, Paris, (1983), 39-63.  Google Scholar

[4]

A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993), 263-297. 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), 123-131. 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), 149-154. doi: 10.1080/17476939208814541.  Google Scholar

[7]

M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Diff. Geom., 47 (1997), 17-94.  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), 193-217.  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, vol. 135, Princeton University Press, Princeton, NJ, 1994.  Google Scholar

[11]

______, "Renormalization and 3-Manifolds Which Fiber Over the Circle," Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 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), 51-66. doi: 10.1090/S0002-9947-1922-1501205-4.  Google Scholar

[14]

J. Schreier, Uber Abbildungen einer abstrakten Menge auf ihre Teilmengen, Fund. Math., (1937), 261-264. Google Scholar

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