Orthogonality and the Hausdorff dimension of the maximal measure

Lopes, Artur Oscar

doi:10.1090/S0002-9939-1986-0848874-7

Orthogonality and the Hausdorff dimension of the maximal measure
HTML articles powered by AMS MathViewer

by Artur Oscar Lopes PDF

Proc. Amer. Math. Soc. 98 (1986), 51-55 Request permission

Abstract:

In this paper the orthogonality properties of iterated polynomials are shown to remain valid in some cases for rational maps. Using a functional equation fulfilled by the generating function, the author shows that the Hausdorff dimension of the maximal measure is a real analytical function of the coefficients of an Axiom A rational map satisfying the property that all poles of $f$ and zeros of $f’(z)$ have multiplicity one.

References

Hans Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103–144 (1965). MR 194595, DOI 10.1007/BF02591353
M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Orthogonal polynomials associated with invariant measures on Julia sets, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 381–384. MR 663789, DOI 10.1090/S0273-0979-1982-15043-1
M. F. Barnsley and A. N. Harrington, Moments of balanced measures on Julia sets, Trans. Amer. Math. Soc. 284 (1984), no. 1, 271–280. MR 742425, DOI 10.1090/S0002-9947-1984-0742425-6
D. Bessis, M. L. Mehta, and P. Moussa, Orthogonal polynomials on a family of Cantor sets and the problem of iterations of quadratic mappings, Lett. Math. Phys. 6 (1982), no. 2, 123–140. MR 651127, DOI 10.1007/BF00401737
D. Bessis and P. Moussa, Orthogonality properties of iterated polynomial mappings, Comm. Math. Phys. 88 (1983), no. 4, 503–529. MR 702566
Ricardo Mañé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), no. 1, 27–43. MR 736567, DOI 10.1007/BF02584743

Analytic function theory

M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), no. 3, 351–385. MR 741393, DOI 10.1017/S0143385700002030
Artur Oscar Lopes, Equilibrium measures for rational maps, Ergodic Theory Dynam. Systems 6 (1986), no. 3, 393–399. MR 863202, DOI 10.1017/S0143385700003576
Ricardo Mañé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), no. 1, 27–43. MR 736567, DOI 10.1007/BF02584743
Anthony Manning, The dimension of the maximal measure for a polynomial map, Ann. of Math. (2) 119 (1984), no. 2, 425–430. MR 740898, DOI 10.2307/2007044
David Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 99–107. MR 684247, DOI 10.1017/s0143385700009603