## Building blocks for quadratic Julia sets

HTML articles powered by AMS MathViewer

- by Joachim Grispolakis, John C. Mayer and Lex G. Oversteegen PDF
- Trans. Amer. Math. Soc.
**351**(1999), 1171-1201 Request permission

## Abstract:

We obtain results on the structure of the Julia set of a quadratic polynomial $P$ with an irrationally indifferent fixed point $z_0$ in the iterative dynamics of $P$. In the Cremer point case, under the assumption that the Julia set is a decomposable continuum, we obtain a*building block*structure theorem for the corresponding Julia set $J=J(P)$: there exists a nowhere dense subcontinuum $B\subset J$ such that $P(B)=B$, $B$ is the union of the impressions of a minimally invariant Cantor set $A$ of external rays, $B$ contains the critical point, and $B$ contains both the Cremer point $z_0$ and its preimage. In the Siegel disk case, under the assumption that no impression of an external ray contains the boundary of the Siegel disk, we obtain a similar result. In this case $B$ contains the boundary of the Siegel disk, properly if the critical point is not in the boundary, and $B$ contains no periodic points. In both cases, the Julia set $J$ is the closure of a

*skeleton*$S$ which is the increasing union of countably many copies of the building block $B$ joined along preimages of copies of a

*critical continuum*$C$ containing the critical point. In addition, we prove that if $P$ is any polynomial of degree $d\ge 2$ with a Siegel disk which contains no critical point on its boundary, then the Julia set $J(P)$ is not locally connected. We also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected Julia set have homeomorphic Julia sets.

## References

- Jan M. Aarts and Lex G. Oversteegen,
*The geometry of Julia sets*, Trans. Amer. Math. Soc.**338**(1993), no. 2, 897–918. MR**1182980**, DOI 10.1090/S0002-9947-1993-1182980-3 - Alan F. Beardon,
*Iteration of rational functions*, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR**1128089**, DOI 10.1007/978-1-4612-4422-6 - A. D. Brjuno,
*On convergence of transforms of differential equations to the normal form*, Dokl. Akad. Nauk SSSR**165**(1965), 987–989 (Russian). MR**0192098** - A. D. Brjuno,
*Analytic form of differential equations. I, II*, Trudy Moskov. Mat. Obšč.**25**(1971), 119–262; ibid. 26 (1972), 199–239 (Russian). MR**0377192** - A. D. Brjuno,
*Analytic form of differential equations. I, II*, Trudy Moskov. Mat. Obšč.**25**(1971), 119–262; ibid. 26 (1972), 199–239 (Russian). MR**0377192** - Witold D. Bula and Lex G. Oversteegen,
*A characterization of smooth Cantor bouquets*, Proc. Amer. Math. Soc.**108**(1990), no. 2, 529–534. MR**991691**, DOI 10.1090/S0002-9939-1990-0991691-9 - Shaun Bullett and Pierrette Sentenac,
*Ordered orbits of the shift, square roots, and the devil’s staircase*, Math. Proc. Cambridge Philos. Soc.**115**(1994), no. 3, 451–481 (English, with English and French summaries). MR**1269932**, DOI 10.1017/S0305004100072236 - Lennart Carleson and Theodore W. Gamelin,
*Complex dynamics*, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR**1230383**, DOI 10.1007/978-1-4612-4364-9 - H. Cremer. Zum zentrumproblem.
*Math. Ann.*, 98:151–163, 1928. - Robert L. Devaney and Lisa R. Goldberg,
*Uniformization of attracting basins for exponential maps*, Duke Math. J.**55**(1987), no. 2, 253–266. MR**894579**, DOI 10.1215/S0012-7094-87-05513-X - Robert L. Devaney and MichałKrych,
*Dynamics of $\textrm {exp}(z)$*, Ergodic Theory Dynam. Systems**4**(1984), no. 1, 35–52. MR**758892**, DOI 10.1017/S014338570000225X - Adrien Douady,
*Systèmes dynamiques holomorphes*, Bourbaki seminar, Vol. 1982/83, Astérisque, vol. 105, Soc. Math. France, Paris, 1983, pp. 39–63 (French). MR**728980** - Adrien Douady,
*Disques de Siegel et anneaux de Herman*, Astérisque**152-153**(1987), 4, 151–172 (1988) (French). Séminaire Bourbaki, Vol. 1986/87. MR**936853** - Sam Perlis,
*Maximal orders in rational cyclic algebras of composite degree*, Trans. Amer. Math. Soc.**46**(1939), 82–96. MR**15**, DOI 10.1090/S0002-9947-1939-0000015-X - Lisa R. Goldberg and John Milnor,
*Fixed points of polynomial maps. II. Fixed point portraits*, Ann. Sci. École Norm. Sup. (4)**26**(1993), no. 1, 51–98. MR**1209913** - Michael-R. Herman,
*Recent results and some open questions on Siegel’s linearization theorem of germs of complex analytic diffeomorphisms of $\textbf {C}^n$ near a fixed point*, VIIIth international congress on mathematical physics (Marseille, 1986) World Sci. Publishing, Singapore, 1987, pp. 138–184. MR**915567** - J. Kiwi. Non-accessible critical points of Cremer polynomials. Preprint.
- K. Kuratowski,
*Topology. Vol. II*, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR**0259835** - John C. Mayer and James T. Rogers Jr.,
*Indecomposable continua and the Julia sets of polynomials*, Proc. Amer. Math. Soc.**117**(1993), no. 3, 795–802. MR**1145423**, DOI 10.1090/S0002-9939-1993-1145423-7 - J. Milnor. Dynamics in one complex variable: introductory lectures. Technical Report 5, SUNY–Stony Brook, 1990. Institute for Mathematical Sciences.
- J. Milnor. Locally connected Julia sets: Expository lectures. Technical Report 11, SUNY–Stony Brook, 1992. Institute for Mathematical Sciences.
- N. T. Varopoulos,
*Potential theory and diffusion on Riemannian manifolds*, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 821–837. MR**730112** - R. Perez-Marco. Fixed points and circle maps. Acta Math. 179:243–294, 1997.
- R. Perez-Marco. Topology of Julia sets and hedgehogs. Technical Report 48, Universite de Paris-Sud, 1994.
- C. L. Petersen. Local connectivity of some Julia sets containing a circle with an irrational rotation. IHES/M/94/26, 1994. Acta Math. 177:163–224, 1996.
- James T. Rogers Jr.,
*Singularities in the boundaries of local Siegel disks*, Ergodic Theory Dynam. Systems**12**(1992), no. 4, 803–821. MR**1200345**, DOI 10.1017/S0143385700007112 - James T. Rogers Jr.,
*Critical points on the boundaries of Siegel disks*, Bull. Amer. Math. Soc. (N.S.)**32**(1995), no. 3, 317–321. MR**1316499**, DOI 10.1090/S0273-0979-1995-00600-2 - J. T. Rogers, Jr. Diophantine conditions imply critical points on the boundaries of Siegel disks of polynomials.
*Comm. Math Phys.*, To appear. - N. E. Rutt. Prime ends and indecomposability.
*Bull. A. M. S.*, 41:265–273, 1935. - J. J. Corliss,
*Upper limits to the real roots of a real algebraic equation*, Amer. Math. Monthly**46**(1939), 334–338. MR**4** - D. Sørensen. Local connectivity of quadratic Julia sets. Master’s thesis, Mathematical Institute, The Technical University of Denmark, 1992.
- D. Sørensen.
*Complex dynamical systems*. PhD thesis, Mathematical Institute, The Technical University of Denmark, 1994. - Dennis Sullivan,
*Conformal dynamical systems*, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 725–752. MR**730296**, DOI 10.1007/BFb0061443 - Dennis Sullivan,
*Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains*, Ann. of Math. (2)**122**(1985), no. 3, 401–418. MR**819553**, DOI 10.2307/1971308 - Jean-Christophe Yoccoz,
*Linéarisation des germes de difféomorphismes holomorphes de $(\textbf {C}, 0)$*, C. R. Acad. Sci. Paris Sér. I Math.**306**(1988), no. 1, 55–58 (French, with English summary). MR**929279** - Jean-Christophe Yoccoz,
*Théorème de Siegel, nombres de Bruno et polynômes quadratiques*, Astérisque**231**(1995), 3–88 (French). Petits diviseurs en dimension $1$. MR**1367353**

## Additional Information

**Joachim Grispolakis**- Affiliation: Technical University of Crete, Chania, Greece
- Email: mgrysp@euclid.aml.tuc.gr
**John C. Mayer**- Email: mayer@math.uab.edu
**Lex G. Oversteegen**- Affiliation: University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
- MR Author ID: 134850
- Email: overstee@math.uab.edu
- Received by editor(s): August 30, 1995
- Additional Notes: Portions of this paper were presented at the Spring Topology Conference in Auburn, Alabama, March 1994, and in the special session on Geometry of Dynamical Systems at the AMS meeting in Orlando, Florida, March 1995.
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 1171-1201 - MSC (1991): Primary 30C35, 54F20
- DOI: https://doi.org/10.1090/S0002-9947-99-02346-6
- MathSciNet review: 1615975