Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T19:55:34.209Z Has data issue: false hasContentIssue false
  • English
  • Français

Distribution of polynomials with cycles of a given multiplier

Published online by Cambridge University Press:  11 January 2016

Giovanni Bassanelli  and
François Berteloot
Giovanni Bassanelli
Affiliation:
Université Paul Sabatier MIG Institut de Mathématiques, de Toulouse 31062 Toulouse Cedex 9, France, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the space of degree d polynomials, the hypersurfaces defined by the existence of a cycle of period n and multiplier eiθ are known to be contained in the bifurcation locus. We prove that these hypersurfaces equidistribute the bifurcation current. This is a new result, even for the space of quadratic polynomials.

Keywords

37F4532U15
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 201 , March 2011 , pp. 23 - 43
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2011

References

[1] Bassanelli, G. and Berteloot, F., Bifurcation currents in holomorphic dynamics on Pk , J. Reine Angew. Math. 608 (2007), 201–235.Google Scholar
[2] Bassanelli, G. and Berteloot, F., Lyapunov exponents, bifurcation currents, and laminations in bifurcation loci, Math. Ann. 345 (2009), 1–23.Google Scholar
[3] Berteloot, F., Lyapunov exponent of a rational map and multipliers of repelling cycles, to appear in Riv. Mat. Univ. Parma.Google Scholar
[4] Berteloot, F., Dupont, C., and Molino, L., Normalization of random families of holomorphic contractions and applications to dynamics, Ann. Inst. Fourier (Grenoble) 58 (2008), 2137–2168.Google Scholar
[5] Branner, B. and Hubbard, J. H., The iteration of cubic polynomials, I: The global topology of parameter space, Acta Math. 160 (1988), 143–206.Google Scholar
[6] Buff, X. and Epstein, A., “Bifurcation measure and postcritically finite rational maps” in Complex Dynamics: Families and Friends, A. K. Peters, Wellesley, Mass., 2009, 491–512.CrossRefGoogle Scholar
[7] Chirka, E. M., Complex Analytic Sets, Kluwer Academic Publishers, 1989.Google Scholar
[8] DeMarco, L., Dynamics of rational maps: A current on the bifurcation locus, Math. Res. Lett. 8 (2001), 57–66.Google Scholar
[9] DeMarco, L., Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann. 326 (2003), 43–73.CrossRefGoogle Scholar
[10] Dinh, T. C. and Sibony, N., Dynamics in several complex variables: Endomorphisms of projective spaces and polynomial-like mappings, preprint, arXiv:0810.0811 Google Scholar
[11] Dujardin, R., “Cubic polynomials: A measurable view on the parameter space” in Complex Dynamics: Families and Friends, A. K. Peters, Wellesley, Mass., 2009, 451–489.Google Scholar
[12] Dujardin, R. and Favre, C., Distribution of rational maps with a preperiodic critical point, Amer. J. Math. 130 (2008), 979–1032.Google Scholar
[13] Hörmander, L., The Analysis of Linear Partial Differential Operators, I, Springer, New York, 1983.Google Scholar
[14] Mañé, R., “The Hausdorff dimension of invariant probabilities of rational maps” in Dynamical Systems (Valparaiso, Chile, 1986), Lect. Notes in Math. 1331 (1988), 86–117.Google Scholar
[15] Mañé, R., Sad, P., and Sullivan, D., On the dynamics of rational maps, Ann. Sci. Éc. Norm. Supér. (4) (1983), 193–217.Google Scholar
[16] Milnor, J. W., Geometry and dynamics of quadratic rational maps, Experiment. Math. 2 (1993), 37–83.Google Scholar
[17] Silverman, J. H., The Arithmetic of Dynamical Systems, Grad. Texts in Math. 241, Springer, New York, 2007.Google Scholar