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A continuity result on quadratic matings with respect to parameters of odd denominator rationals

Published online by Cambridge University Press:  01 June 2018

LIANGANG MA
LIANGANG MA*
Affiliation:
Dept. of Mathematics, Binzhou University, Huanghe 5th road No. 391, Binzhou 256600, Shandong, P. R. China. e-mail: maliangang000@163.com
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Abstract

In this paper we prove a continuity result on matings of quadratic lamination maps sp depending on odd denominator rationals p ∈(0,1). One of the two mating components is fixed in the result. Note that our result has its implication on continuity of matings of quadratic hyperbolic polynomials fc(z)=z2 + c, c ∈ M the Mandelbrot set with respect to the usual parameters c. This is because every quadratic hyperbolic polynomial in M is contained in a bounded hyperbolic component. Its center is Thurston equivalent to some quadratic lamination map sp, and there are bounds on sizes of limbs of M and on sizes of limbs of the mating components on the quadratic parameter slice Perm′(0).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 2 , September 2019 , pp. 369 - 388
Copyright
Copyright © Cambridge Philosophical Society 2018 

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Footnotes

†Supported by 2016Y28 from Binzhou University and ZR2018MF015 from SNSF

References

REFERENCES

[AY] Aspenberg, M. and Yampolsky, M. Mating non-renormalisable quadratic polynomials. Commun. Math. Phys. 287 (2009), 1–40.Google Scholar
[BC] Buff, X. and Chéritat, A. Quadratic Julia sets with positive area. Annals of Math. 176 (2012), 673–746.Google Scholar
[BEE] Buff, X., Écalle, J. and Epstein, A. Limits of degenerate parabolic quadratic rational maps. Geom. Funct. Anal. 23 (2013), 42–95.Google Scholar
[BKM] Bonifant, A., Kiwi, J. and Milnor, J. Cubic polynomial maps with periodic critical orbit, Part II: Escape regions. Conform. Geom. Dyn. 14 (2010), 68-112. Errata, Conform. Geom. Dyn. 14 (2010), 190–193.Google Scholar
[Bul1] Bullett, S. A Combination Theorem for covering correspondences and an application to mating polynomial maps with Kleinian groups. Conform. Geom. Dyn. 4 (2000), 75–96.Google Scholar
[Bul2] Bullett, S. Matings in holomorphic dynamics, Geometry of Riemann Surfaces, edited by Gardiner, Frederick P., Gonzalez–Diez, Gabino and Kourouniotis, Christos London Math. Soc. Lecture Notes No. 368 (Cambridge University Press January 2010), 88–119.Google Scholar
[Che] Chéritat, A. Tan Lei and Shishikura's example of non-mateable degree 3 polynomials without a levy cycle http://arxiv.org/pdf/1202.4188v1.pdf.Google Scholar
[CT] Cui, G. and Lei, T. Hyperbolic-parabolic deformations of rational maps, manuscript (2014).Google Scholar
[DH] Douady, A. and Hubbard, J. Exploring the Mandelbrot set. The Orsay Notes. http://www.math.cornell.edu/~hubbard/OrsayEnglish.pdf.Google Scholar
[Dou] Douady, A. Systémes dynamiques holomorphes. Seminar Bourbaki. Astérisque 105-106 (1983), 39–63.Google Scholar
[Dud] Dudko, D. Matings with laminations. http://arxiv.org/abs/1112.4780 (2011).Google Scholar
[Eps] Epstein, A. Counter examples to the quadratic mating conjecture. manuscript (1997).Google Scholar
[Hai] Haïssinsky, P. Chirurgie parabolique. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998), 195–198.Google Scholar
[HT] Haïssinsky, P. and Lei, T. Convergence of pinching deformations and matings of geometrically finite polynomials. Fund. Math. 181 (2004), 143–188.Google Scholar
[Hub] Hubbard, J. H. Local connectivity of Julia sets and bifurcation loci: three theorems of J. C. Yoccoz, Topological methods in modern mathematics (Publish or Perish, Houston, TX, 1993) 467–551.Google Scholar
[Jun] Jung, W. Quadratic matings and ray connections. https://arxiv.org/pdf/1707.00630.pdf (2017).Google Scholar
[Kam] Kameyama, A. On Julia sets of postcritically finite branched coverings Part II–S 1-parametrisation of Julia sets. J. Math. Soc. Japan 55 No. 2 (2003), 455–468.Google Scholar
[Lav] Lavaurs, P. Systémes dynamiques holomorphes: explosion de points périodique paraboliques. Thèse de doctorat de l'Université de Paris-Sud, Orsay, France (1989).Google Scholar
[Luo] Jiaqi Luo. Combinatorics and holomorphic dynamics: captures, matings, Newton's method. Thesis, Cornell University (1995).Google Scholar
[Ma] Ma, L. Continuity of quadratic matings. PhD thesis. University of Liverpool, (2015).Google Scholar
[Mey1] Meyer, D. Unmating of rational maps, sufficient criteria and examples, https://arxiv.org/abs/1110.6784 (2011).Google Scholar
[Mey2] Meyer, D. Expanding Thurston maps as quotients, https://arxiv.org/abs/0910 (2003).Google Scholar
[Mey3] Meyer, D. Invariant Peano curves of expanding Thurston maps. Acta. Math. 210 No. 1 (2013), 95–171.Google Scholar
[Mil1] Milnor, J. Dynamics in one complex variable. Ann. of Mathematics stud. 160 (2006).Google Scholar
[Mil2] Milnor, J. Pasting together Julia sets. Exp. Math. 13 No. 1 (2004), 55–92.Google Scholar
[Mil3] Milnor, J. Cubic polynomial maps with periodic critical orbit, Part I Complex Dynamics, Families and Friends in Honor of Hubbard, John Hamal, Schleicher, D., editor (Peters, A. K., Wellesley, MA), pages 333–411.Google Scholar
[MP] Meyer, D. and Petersen, C. On the notions of mating. Annales de la Faculté des Sciences de Toulouse : Mathématiques, Série 6 : Tome 21 (2012), no. S5, p. 839–876.Google Scholar
[Ree1] Rees, S. Realization of matings of polynomials as rational maps of degree two. Manuscript (1986).Google Scholar
[Ree2] Rees, S. A partial description of parameter space of rational maps of degree two: Part I. Acta Math. 168 (1992) 11–87.Google Scholar
[Ree3] Rees, S. A partial description of the parameter space of rational maps of degree two, part II. Proc London Math. Soc. (1995), s3-70 (3) 644–690.Google Scholar
[Sch1] Schleicher, D. On fibers and local connectivity of Mandelbrot and Multibrot sets. http://arxiv.org/abs/math/9902155 (1999).Google Scholar
[Sch2] Schleicher, D. On fibers and local connectivity of compact sets in ℂ. http://arxiv.org/abs/math/9902154 (1999).Google Scholar
[Shi1] Shishikura, M. On a theorem of M. Rees for matings of polynomials. London Math. Soc. Lecture Note Ser., 274 CMP (2000) 14.Google Scholar
[Shi2] Shishikura, M. The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Ann. of Math. Second Series. 147 No. 2 (1998), pp. 225–267.Google Scholar
[Shi3] Shishikura, M. Bifurcation of parabolic fixed points, the Mandelbrot set, Theme and variations, London Math. Soc. 274 (2000).Google Scholar
[ST] Shishikura, M. and Tan, L. A family of cubic rational maps and matings of cubic polynomials. Experiment. Math. 9 (1) (2000), 29–53.Google Scholar
[Tan1] Lei, T. Matings of quadratic polynomials. Ergodic Theory Dynam. Syst. 12 (1992), pp. 589–620.Google Scholar
[Tan2] Lei, Tan Local properties of the Mandelbrot set at parabolic points. The Mandelbrot Set, theme and Variations. London Math. Soc. 274, pp. 133–160, 2000.Google Scholar
[Thu] Thurston, W. P. On the geometry and dynamics of iterated rational maps, edited by Dierk Schleicher with the help of Nikita Selinger. Preprint (Princeton university).Google Scholar
[Tim1] Timorin, V. External boundary of M 2. Fields Institute Communications Vol. 53: Holomorphic Dynamics and Renormalisation, A Volume in Honour of John Milnor's 75th Birthday.Google Scholar
[Tim2] Timorin, V. Parameter slices of quadratic rational functions. http://math.jacobs-university.de/timorin/perk/perk.html.Google Scholar
[Wit] Wittner, B. On the bifurcation loci of rational maps of degree two. PhD Thesis (Cornell University, 1988).Google Scholar
[Yoc] Yoccoz, J. C. Sur la connectivité locale de M. unpublished (1989).Google Scholar
[YZ] Yampolsky, M. and Zakeri, S. Mating Siegel quadratic polynomials. J. Amer. Math. Soc. 14 (2001), no. 1, 25–78.Google Scholar