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, 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

[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

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

[1]

John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047
[2]

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
[3]

Eugen Mihailescu, Mariusz Urbański. Holomorphic maps for which the unstable manifolds depend on prehistories. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 443-450. doi: 10.3934/dcds.2003.9.443
[4]

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
[5]

Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481
[6]

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

Cezar Joiţa, William O. Nowell, Pantelimon Stănică. Chaotic dynamics of some rational maps. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 363-375. doi: 10.3934/dcds.2005.12.363
[8]

Toshikazu Ito, Bruno Scárdua. Holomorphic foliations transverse to manifolds with corners. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 537-544. doi: 10.3934/dcds.2009.25.537
[9]

Jeffrey Diller, Han Liu, Roland K. W. Roeder. Typical dynamics of plane rational maps with equal degrees. Journal of Modern Dynamics, 2016, 10: 353-377. doi: 10.3934/jmd.2016.10.353
[10]

David DeLatte. Diophantine conditions for the linearization of commuting holomorphic functions. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 317-332. doi: 10.3934/dcds.1997.3.317
[11]

Marco Abate, Jasmin Raissy. Formal Poincaré-Dulac renormalization for holomorphic germs. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1773-1807. doi: 10.3934/dcds.2013.33.1773
[12]

Xu Zhang, Guanrong Chen. Polynomial maps with hidden complex dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2941-2954. doi: 10.3934/dcdsb.2018293
[13]

Lisa C. Jeffrey and Frances C. Kirwan. Intersection pairings in moduli spaces of holomorphic bundles on a Riemann surface. Electronic Research Announcements, 1995, 1: 57-71.

[14]

Andrzej Biś. Entropies of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 639-648. doi: 10.3934/dcds.2004.11.639
[15]

Jun Hu, Oleg Muzician, Yingqing Xiao. Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3189-3221. doi: 10.3934/dcds.2018139
[16]

Guizhen Cui, Yan Gao. Wandering continua for rational maps. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1321-1329. doi: 10.3934/dcds.2016.36.1321
[17]

Eriko Hironaka, Sarah Koch. A disconnected deformation space of rational maps. Journal of Modern Dynamics, 2017, 11: 409-423. doi: 10.3934/jmd.2017016
[18]

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
[19]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[20]

Nathaniel D. Emerson. Dynamics of polynomials with disconnected Julia sets. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 801-834. doi: 10.3934/dcds.2003.9.801

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences