April  2019, 39(4): 2157-2172. doi: 10.3934/dcds.2019090

Quasiregular semigroups with examples

Department of Mathematical Sciences, Northern Illinois University, Dekalb, IL 60115, USA

I would like to thank the anonymous referee for eliminating some errors and for providing suggestions which improved the readability of the paper

Received  May 2018 Revised  August 2018 Published  January 2019

Fund Project: This work was supported by a grant from the Simons Foundation (#352034, Alastair Fletcher)

Rational semigroups were introduced by Hinkkanen and Martin as a generalization of the iteration of a single rational map. There has subsequently been much interest in the study of rational semigroups. Quasiregular semigroups were introduced shortly after rational semigroups as analogues in higher real dimensions, but have received far less attention. Each map in a quasiregular semigroup must necessarily be a uniformly quasiregular map. While there is a completely viable theory for the iteration of uniformly quasiregular maps, it is a highly non-trivial matter to construct them. In this paper, we study properties of the Julia and Fatou sets of quasiregular semigroups and, equally as importantly, give several families of examples illustrating some of the behaviours that can arise.

Citation: Alastair Fletcher. Quasiregular semigroups with examples. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2157-2172. doi: 10.3934/dcds.2019090
References:
[1]

A. F. Beardon, Iteration of Rational Functions, Springer, 1991. Google Scholar

[2]

W. Bergweiler, Iteration of quasiregular mappings, Comput. Methods Funct. Theory, 10 (2010), 455-481. doi: 10.1007/BF03321776. Google Scholar

[3]

A. Fletcher and D. Macclure, Strongly automorphic mappings and their uniformly quasiregular Julia sets, to appear in J. Anal. Math..Google Scholar

[4]

A. Fletcher and D. Nicks, Quasiregular dynamics on the n-sphere, Erg. Th. Dyn. Sys., 31 (2011), 23-31. doi: 10.1017/S0143385709001072. Google Scholar

[5]

A. Fletcher and D. Nicks, Julia sets of uniformly quasiregular mappings are uniformly perfect, Math. Proc. Cam. Phil. Soc., 151 (2011), 541-550. doi: 10.1017/S0305004111000478. Google Scholar

[6]

A. Fletcher and J.-M. Wu, Julia sets and wild Cantor sets, Geometriae Dedicata, 174 (2015), 169-176. doi: 10.1007/s10711-014-0010-3. Google Scholar

[7]

A. Hinkkanen and G. J. Martin, The dynamics of semigroups of rational functions I, Proc. London Math. Soc., 73 (1996), 358-384. doi: 10.1112/plms/s3-73.2.358. Google Scholar

[8]

A. HinkkanenG. Martin and V. Mayer, Local dynamics of uniformly quasiregular mappings, Math. Scand., 95 (2004), 80-100. doi: 10.7146/math.scand.a-14450. Google Scholar

[9]

T. Iwaniec and G. J. Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn., 21 (1996), 241-254. Google Scholar

[10] T. Iwaniec and G. Martin, Geometric Function Theory and Non-linear Analysis, Oxford University Press, 2001. Google Scholar
[11]

J. Jaerisch and H. Sumi, Dynamics of infinitely generated nicely expanding rational semigroups and the inducing method, Trans. Amer. Math. Soc., 369 (2017), 6147-6187. doi: 10.1090/tran/6862. Google Scholar

[12]

G. J. Martin, Quasiconformal and affine groups, J. Diff. Geom., 29 (1989), 427-448. doi: 10.4310/jdg/1214442884. Google Scholar

[13]

G. J. Martin and K. Peltonen, Stoilow factorization for quasiregular mappings in all dimensions, Proc. Amer. Math. Soc., 138 (2010), 147-151. doi: 10.1090/S0002-9939-09-10056-4. Google Scholar

[14]

V. Mayer, Uniformly Quasiregular Mappings of Lattès type, Conform. Geom. Dyn., 1 (1997), 104-111. doi: 10.1090/S1088-4173-97-00013-1. Google Scholar

[15]

V. Mayer, Quasiregular analogues of critically finite rational functions with parabolic orbifold, J. Anal. Math., 75 (1998), 105-119. doi: 10.1007/BF02788694. Google Scholar

[16]

R. Miniowitz, Normal families of quasimeromorphic mappings, Proc. Amer. Math. Soc., 84 (1982), 35-43. doi: 10.1090/S0002-9939-1982-0633273-X. Google Scholar

[17]

S. Morosawa, Semigroups whose Julia sets are Cantor targets, Comp. Var. Ell. Eq., 2018. doi: 10.1080/17476933.2018.1481834. Google Scholar

[18]

S. Rickman, Quasiregular Mappings, Springer-Verlag, 1993. doi: 10.1007/978-3-642-78201-5. Google Scholar

[19]

H. Siebert, Fixpunkte und normale Familien quasiregul ärer Abbildungen, Dissertation, CAU Kiel, 2004.Google Scholar

[20]

R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575. doi: 10.1090/S0002-9939-00-05313-2. Google Scholar

[21]

H. Sumi, On dynamics of hyperbolic rational semigroups, J. Math. Kyoto Univ., 37 (1997), 717-733. doi: 10.1215/kjm/1250518211. Google Scholar

[22]

P. Tukia and J. Väisäla, Lipschitz and quasiconformal approximation and extension, Ann. Acad. Sci. Fenn. Ser. A I Math., 6 (1981), 303-342. doi: 10.5186/aasfm.1981.0626. Google Scholar

[23]

M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Springer-Verlag, 1988. doi: 10.1007/BFb0077904. Google Scholar

show all references

References:
[1]

A. F. Beardon, Iteration of Rational Functions, Springer, 1991. Google Scholar

[2]

W. Bergweiler, Iteration of quasiregular mappings, Comput. Methods Funct. Theory, 10 (2010), 455-481. doi: 10.1007/BF03321776. Google Scholar

[3]

A. Fletcher and D. Macclure, Strongly automorphic mappings and their uniformly quasiregular Julia sets, to appear in J. Anal. Math..Google Scholar

[4]

A. Fletcher and D. Nicks, Quasiregular dynamics on the n-sphere, Erg. Th. Dyn. Sys., 31 (2011), 23-31. doi: 10.1017/S0143385709001072. Google Scholar

[5]

A. Fletcher and D. Nicks, Julia sets of uniformly quasiregular mappings are uniformly perfect, Math. Proc. Cam. Phil. Soc., 151 (2011), 541-550. doi: 10.1017/S0305004111000478. Google Scholar

[6]

A. Fletcher and J.-M. Wu, Julia sets and wild Cantor sets, Geometriae Dedicata, 174 (2015), 169-176. doi: 10.1007/s10711-014-0010-3. Google Scholar

[7]

A. Hinkkanen and G. J. Martin, The dynamics of semigroups of rational functions I, Proc. London Math. Soc., 73 (1996), 358-384. doi: 10.1112/plms/s3-73.2.358. Google Scholar

[8]

A. HinkkanenG. Martin and V. Mayer, Local dynamics of uniformly quasiregular mappings, Math. Scand., 95 (2004), 80-100. doi: 10.7146/math.scand.a-14450. Google Scholar

[9]

T. Iwaniec and G. J. Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn., 21 (1996), 241-254. Google Scholar

[10] T. Iwaniec and G. Martin, Geometric Function Theory and Non-linear Analysis, Oxford University Press, 2001. Google Scholar
[11]

J. Jaerisch and H. Sumi, Dynamics of infinitely generated nicely expanding rational semigroups and the inducing method, Trans. Amer. Math. Soc., 369 (2017), 6147-6187. doi: 10.1090/tran/6862. Google Scholar

[12]

G. J. Martin, Quasiconformal and affine groups, J. Diff. Geom., 29 (1989), 427-448. doi: 10.4310/jdg/1214442884. Google Scholar

[13]

G. J. Martin and K. Peltonen, Stoilow factorization for quasiregular mappings in all dimensions, Proc. Amer. Math. Soc., 138 (2010), 147-151. doi: 10.1090/S0002-9939-09-10056-4. Google Scholar

[14]

V. Mayer, Uniformly Quasiregular Mappings of Lattès type, Conform. Geom. Dyn., 1 (1997), 104-111. doi: 10.1090/S1088-4173-97-00013-1. Google Scholar

[15]

V. Mayer, Quasiregular analogues of critically finite rational functions with parabolic orbifold, J. Anal. Math., 75 (1998), 105-119. doi: 10.1007/BF02788694. Google Scholar

[16]

R. Miniowitz, Normal families of quasimeromorphic mappings, Proc. Amer. Math. Soc., 84 (1982), 35-43. doi: 10.1090/S0002-9939-1982-0633273-X. Google Scholar

[17]

S. Morosawa, Semigroups whose Julia sets are Cantor targets, Comp. Var. Ell. Eq., 2018. doi: 10.1080/17476933.2018.1481834. Google Scholar

[18]

S. Rickman, Quasiregular Mappings, Springer-Verlag, 1993. doi: 10.1007/978-3-642-78201-5. Google Scholar

[19]

H. Siebert, Fixpunkte und normale Familien quasiregul ärer Abbildungen, Dissertation, CAU Kiel, 2004.Google Scholar

[20]

R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575. doi: 10.1090/S0002-9939-00-05313-2. Google Scholar

[21]

H. Sumi, On dynamics of hyperbolic rational semigroups, J. Math. Kyoto Univ., 37 (1997), 717-733. doi: 10.1215/kjm/1250518211. Google Scholar

[22]

P. Tukia and J. Väisäla, Lipschitz and quasiconformal approximation and extension, Ann. Acad. Sci. Fenn. Ser. A I Math., 6 (1981), 303-342. doi: 10.5186/aasfm.1981.0626. Google Scholar

[23]

M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Springer-Verlag, 1988. doi: 10.1007/BFb0077904. Google Scholar

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