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Transitive maps which are not ergodic with respect to Lebesgue measure

Published online by Cambridge University Press:  19 September 2008

Sebastian Van Strien
Sebastian Van Strien
Affiliation:
Mathematics Department, University of Amsterdam, Netherlands (e-mail: [email protected])
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Abstract

In this paper we shall give examples of rational maps on the Riemann sphere and also of polynomial interval maps which are transitive but not ergodic with respect to Lebesgue measure. In fact, these maps have two disjoint compact attractors whose attractive basins are ‘intermingled’, each having a positive Lebesgue measure in every open set. In addition, we show that there exists a real bimodal polynomial with Fibonacci dynamics (of the type considered by Branner and Hubbard), whose Julia set is totally disconnected and has positive Lebesgue measure. Finally, we show that there exists a rational map associated to the Newton iteration scheme corresponding to a polynomial whose Julia set has positive Lebesgue measure.

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 4 , August 1996 , pp. 833 - 848
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[AKYZ]Alexander, J. C., Kan, I., Yorke, J. A. and You, Zhiping. Riddled basins. Int. J. Bifur. Chaos Appl. Sci. Eng. 2 (1992), 795–813.CrossRefGoogle Scholar
[AHKY]Alexander, J. C., Hunt, B. R., Kan, I. and Yorke, J. A.. Intermingled basins for the triangle map. Preprint, 1993.Google Scholar
[Ba]Barna, B.. Über die Divergenzpunkte des Newtonschen Verfahrens zur Bestimmung von Wurzels algebraischen Gleichungen II. Publ. Mathematicae Debrecen 4 (1956), 384–397.CrossRefGoogle Scholar
[Be]Beardon, A. F.. Iterations of Rational Maps (Graduate Texts). Springer, Berlin, 1991.CrossRefGoogle Scholar
[BH]Branner, B. and Hubbard, J. H.. The iteration of cubic polynomials I. Acta Math. 160 (1988), 143–206;CrossRefGoogle Scholar
The iteration of cubic polynomials II. Acta Math. 169 (1992), 229–325.CrossRefGoogle Scholar
[BL]Blokh, A. M. and Lyubich, M. Yu.. Measurable dynamics of S-unimodal maps of the interval. Ann. Sci. Ec. Norm. Sup. 4e série 24 (1991), 545–573.CrossRefGoogle Scholar
[BKNS]Bruin, H., Keller, G., Nowicki, T. and van Strien, S.. Absorbing Cantor sets in dynamical systems: Fibonacci maps. Ann. Maths. 193 (1996), 97–130.CrossRefGoogle Scholar
[CGS]Curry, J., Garnett, L. and Sullivan, D.. On the iteration of rational functions: computer experiments with Newton's method. Commun. Math. Phys. 91 (1983), 267–277.CrossRefGoogle Scholar
[CE]Collet, P. and Eckmann, J.-P.. Iterated Maps of the Interval as Dynamical Systems. Birkhauser, Boston, 1980.Google Scholar
[He]Herman, M.. Research Problems in Complex Analysis. Eds Brannan, D. A. and Hayman, W. K.Bull. London Math. Soc. 21 1989, 1–35. Problems 2.81 and 2.79.Google Scholar
[Kan]Kan, I.. Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin. Bull. Am. Math. Soc. 31 1994, 68–74.CrossRefGoogle Scholar
[Ke]Keane, M.. Non-ergodic interval exchange transformations. Israel J. Math. 26 (1977), 188–196.CrossRefGoogle Scholar
[KN]Keller, G. and Nowicki, T.. Fibonacci maps re(aℓ)visited. Preprint, 1992.Google Scholar
[LS]Levin, G. and van Strien, S.. Local connectivity of the Julia set of real polynomials. Preprint, Stony Brook 1995.Google Scholar
[Ly1]Lyubich, M. Yu.. Ergodic theory for smooth one-dimensional dynamical systems. Preprint, Stonybrook IMS 1991/1911.Google Scholar
[Ly2]Lyubich, M. Yu.. On the typical behaviour of trajectories of the exponent. Russian Math. Surveys 41 (1986), 207–208.CrossRefGoogle Scholar
[Ly3]Lyubich, M. Yu.. The measurable dynamics of the exponential map. Siberian J. Math. 28 (1987), 111–127Google Scholar
[Ly4]Lyubich, M. Yu.. Milnor's attractors, persistent recurrence and renormalization. Topological Methods in Modern Mathematics, A Symposium in Honor of John Milnor's 60th Birthday, 1992.Google Scholar
[LM]Lyubich, M. and Milnor, J.. The unimodal Fibonacci map. J. Am. Math. Soc. 6 (1993), 425–457.CrossRefGoogle Scholar
[Ma]Mañé, R.. On the instability of Hermann rings. Invent. Math. 81 (1985), 459–471.CrossRefGoogle Scholar
[McM]McMullen, C.. Complex Dynamics and Renormalizaton (Annals of Math. Studies). Princeton University Press, Princeton, NJ, 1994.Google Scholar
[MS]de Melo, W. and van Strien, S.. One-dimensional Dynamics (Ergehnisse Series 25). Springer, 1993.CrossRefGoogle Scholar
[Mi]Misiurewicz, M.. On iterates of e z. Ergod. Th. & Dynam. Sys. 1 (1981), 103–106.CrossRefGoogle Scholar
[Re1]Rees, M.. Ergodic rational maps with dense critical point forward orbit. Ergod. Th. & Dynam. Sys. 4 (1984), 311–322.CrossRefGoogle Scholar
[Re2]Rees, M.. Positive measure sets of ergodic rational maps. Ann. Sci. Ec. Norm. Sup. 4e série 19 (1986), 383–407.CrossRefGoogle Scholar
[Re3]Rees, M.. The exponential map is not recurrent. Math. Z. 191 (1986), 593–598.CrossRefGoogle Scholar
[SN]van Strien, S. and Nowicki, T.. Polynomial maps with a Julia set of positive Lebesgue measure: Fibonacci maps. Preprint, Stonybrook IMS 1994/1993. Extended version available by anonymous ftp from ftp.fwi.uva.nl in directory pub/mathematics/reports/Geometry_and_Dynamics/Dyn95-1Google Scholar