Semigroup representations in holomorphic dynamics

Previous Article
Entropy of endomorphisms of Lie groups
DCDS Home
This Issue
Next Article
Dynamics of continued fractions and kneading sequences of unimodal maps

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

Abstract
Related Papers

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 and 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.
[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.
[3]	A. Douady, Systèmes dynamiques holomorphes, Bourbaki seminar, 1982/83, Astèrisque, 105, Soc. Math. France, Paris, (1983), 39-63.
[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.
[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.
[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.
[7]	M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Diff. Geom., 47 (1997), 17-94.
[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.
[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, vol. 135, Princeton University Press, Princeton, NJ, 1994.
[11]	______, "Renormalization and 3-Manifolds Which Fiber Over the Circle," Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 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), 51-66. doi: 10.1090/S0002-9947-1922-1501205-4.
[14]	J. Schreier, Uber Abbildungen einer abstrakten Menge auf ihre Teilmengen, Fund. Math., (1937), 261-264.

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.
[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.
[3]	A. Douady, Systèmes dynamiques holomorphes, Bourbaki seminar, 1982/83, Astèrisque, 105, Soc. Math. France, Paris, (1983), 39-63.
[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.
[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.
[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.
[7]	M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Diff. Geom., 47 (1997), 17-94.
[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.
[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, vol. 135, Princeton University Press, Princeton, NJ, 1994.
[11]	______, "Renormalization and 3-Manifolds Which Fiber Over the Circle," Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 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), 51-66. doi: 10.1090/S0002-9947-1922-1501205-4.
[14]	J. Schreier, Uber Abbildungen einer abstrakten Menge auf ihre Teilmengen, Fund. Math., (1937), 261-264.

[1]	John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2018, 38 (7) : 3223-3237. doi: 10.3934/dcds.2018140
[7]	Toshikazu Ito, Bruno Scárdua. Holomorphic foliations transverse to manifolds with corners. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 537-544. doi: 10.3934/dcds.2009.25.537
[8]	Cezar Joiţa, William O. Nowell, Pantelimon Stănică. Chaotic dynamics of some rational maps. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 363-375. doi: 10.3934/dcds.2005.12.363
[9]	David DeLatte. Diophantine conditions for the linearization of commuting holomorphic functions. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 317-332. doi: 10.3934/dcds.1997.3.317
[10]	Marco Abate, Jasmin Raissy. Formal Poincaré-Dulac renormalization for holomorphic germs. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1773-1807. doi: 10.3934/dcds.2013.33.1773
[11]	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
[12]	Xu Zhang, Guanrong Chen. Polynomial maps with hidden complex dynamics. Discrete and 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 and 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 and Continuous Dynamical Systems, 2018, 38 (7) : 3189-3221. doi: 10.3934/dcds.2018139
[16]	Guizhen Cui, Yan Gao. Wandering continua for rational maps. Discrete and 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]	Richard Sharp, Anastasios Stylianou. Statistics of multipliers for hyperbolic rational maps. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1225-1241. doi: 10.3934/dcds.2021153
[19]	Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583
[20]	Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 545-557 . doi: 10.3934/dcds.2011.31.545

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Carlos Cabrera Peter Makienko Peter Plaumann
on Google Scholar
Carlos Cabrera Peter Makienko Peter Plaumann

[Back to Top]