# American Institue of Mathematical Sciences

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

## Semigroup representations in holomorphic dynamics

 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.
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. [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. [3] A. Douady, Systèmes dynamiques holomorphes,, Bourbaki seminar, 1982/83 (1983), 39. [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. [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. [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. [7] M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics,, J. Diff. Geom., 47 (1997), 17. [8] R. Mañè, P. Sad and D. Sullivan, On the dynamics of rational maps,, Ann. Scien. Ec. Norm. Sup. Paris(4), 16 (1983), 193. [9] K. D. Magill, Jr., A survey of semigroups of continous self maps,, Semigroup Forum, 11 (): 189. doi: 10.1007/BF02195270. [10] C. McMullen, "Complex Dynamics and Renormalization,", Annals of Mathematics Studies, (1994). [11] ______, "Renormalization and 3-Manifolds Which Fiber Over the Circle,", Annals of Mathematics Studies, (1996). [12] J. Milnor, "Dynamics of One Complex Variable,", Friedr. Vieweg & Sohn, (1999). [13] J. F. Ritt, Prime and composite polynomials,, Trans. Amer. Math. Soc., 23 (1922), 51. doi: 10.1090/S0002-9947-1922-1501205-4. [14] J. Schreier, Uber Abbildungen einer abstrakten Menge auf ihre Teilmengen,, Fund. Math., (1937), 261.

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##### 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. [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. [3] A. Douady, Systèmes dynamiques holomorphes,, Bourbaki seminar, 1982/83 (1983), 39. [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. [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. [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. [7] M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics,, J. Diff. Geom., 47 (1997), 17. [8] R. Mañè, P. Sad and D. Sullivan, On the dynamics of rational maps,, Ann. Scien. Ec. Norm. Sup. Paris(4), 16 (1983), 193. [9] K. D. Magill, Jr., A survey of semigroups of continous self maps,, Semigroup Forum, 11 (): 189. doi: 10.1007/BF02195270. [10] C. McMullen, "Complex Dynamics and Renormalization,", Annals of Mathematics Studies, (1994). [11] ______, "Renormalization and 3-Manifolds Which Fiber Over the Circle,", Annals of Mathematics Studies, (1996). [12] J. Milnor, "Dynamics of One Complex Variable,", Friedr. Vieweg & Sohn, (1999). [13] J. F. Ritt, Prime and composite polynomials,, Trans. Amer. Math. Soc., 23 (1922), 51. doi: 10.1090/S0002-9947-1922-1501205-4. [14] J. Schreier, Uber Abbildungen einer abstrakten Menge auf ihre Teilmengen,, Fund. Math., (1937), 261.
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