Abstract
This is an exposition of our resent results contained in Kozlovski et al. (Rigidity for real polynomials, preprint, 2003; Density of hyperbolicity, preprint, 2003) and Kozlovski and van Strien (Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, preprint, 2006) where we prove the density of hyperbolicity for one dimensional real maps and non-renormalizable complex polynomials. The proofs of these results are very technical, so in this paper we try to show the main ideas on some simplified examples and also give some outlines of the proofs.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Blokh and M. Misiurewicz, Typical limit sets of critical points for smooth interval maps. Ergod. Theory Dyn. Syst. 20(1), 15–45 (2000)
J. Graczyk and G. Światek, The Real Fatou Conjecture. Ann. Math. Stud., vol. 144. Princeton University Press, Princeton (1998)
M.V. Jakobson, Smooth mappings of the circle into itself. Mat. Sb. (N.S.) 85(127), 163–188 (1971)
J. Kahn and M. Lyubich, The quasi-additivity law in conformal geometry. IMS preprint (2005)
O.S. Kozlovski, Axiom A maps are dense in the space of unimodal maps in the C k topology. Ann. Math. (2) 157(1), 1–43 (2003)
O.S. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials. Preprint (2006)
O.S. Kozlovski, W. Shen, and S. van Strien, Rigidity for real polynomials. Ann. Math. (2) 165(3), 749–841 (2007)
O.S. Kozlovski, W. Shen, and S. van Strien, Density of hyperbolicity in dimension one. Ann. Math. (2) 166(4), 900–920 (2007)
M. Lyubich, Dynamics of quadratic polynomials. I. Acta Math. 178(2), 185–247 (1997)
M. Lyubich, Dynamics of quadratic polynomials. II. Acta Math. 178(2), 247–297 (1997)
W. Shen, On the metric properties of multimodal interval maps and C 2 density of axiom A. Invent. Math. 156(2), 301–403 (2004)
S. Smale, Mathematical Problems for the Next Century. Mathematics: Frontiers and Perspectives, pp. 271–294. Am. Math. Soc., Providence (2000)
D. Sullivan, Bounds, Quadratic Differentials, and Renormalization Conjectures. American Mathematical Society Centennial Publications, vol. II, pp. 417–466. Am. Math. Soc., Providence, (1992)
J.-C. Yoccoz, Sur la conectivité locale de M. Unpublished (1989)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science+Business Media B.V.
About this paper
Cite this paper
Kozlovski, O., van Strien, S. (2009). Stable Maps are Dense in Dimensional One. In: Sidoravičius, V. (eds) New Trends in Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2810-5_27
Download citation
DOI: https://doi.org/10.1007/978-90-481-2810-5_27
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-2809-9
Online ISBN: 978-90-481-2810-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)