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See also ibid. pp. ) G. Levin and F. , Proc. A. M. S. 125 (1997) 2179–2190. M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Th. and Dy. Sys. 3 (1983) 351–385. R. Man˜e´, On the uniqueness of the maximizing measure for rational maps. Bol. Soc. Brasil. Mat. 14 (1983) 27–43. R. Man˜e´, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. E´cole Norm. Sup. 16 (1983) 193–217. C. McMullen, Families of rational maps and iterative root-finding algorithms, Ann.

The construction in this case is identical, except that we take L(t) = at + 1/2. If a is even, this construction yields nothing new. 1 is then trivial, and it follows that every Latte`s map with L0 = a must have a postcritical fixed point. However, when a is odd, the period two orbits 0 $ 1/2, γ/2 $ (γ + 1)/2 in T (γ) map to ramified period two orbits in T (γ)/G2 , and there is no postcritical fixed point. Caution: In this last case, we can no longer realize every conjugacy class of maps by restricting γ to the Siegel region.

M. Herman, Examples de fractions rationnelles ayant une orbite dense sur la sphe`re de Riemann, Bull. Sci. Math. France 112 (1984) 93–142. G. K'nigs, Recherches sur les integrals de certains equations fonctionelles, Ann. Sci. E´c. Norm. Sup. 1 (1884) suppl. 1–41. S. Latte`s, Sur l’iteration des substitutions rationnelles et les fonctions de Poincare´, C. R. Acad. Sci. Paris 166 (1918) 26–28. (See also ibid. pp. ) G. Levin and F. , Proc. A. M. S. 125 (1997) 2179–2190. M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Th.

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Analysis of manifolds by Munkres J.R.

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