# zbMATH — the first resource for mathematics

Positive measure sets of ergodic rational maps. (English) Zbl 0611.58038
Let $$\{f_{\lambda}:\lambda\in \Lambda \}$$ be an analytic family of rational maps of degree d. It is shown that there exists a $$\lambda$$-set of positive measure for which $$f_{\lambda}$$ is ergodic with respect to Lebesgue measure and has an equivalent invariant measure. This theorem holds under two conditions: 1) For some $$\lambda_ 0\in \Lambda$$ $$f_{\lambda_ 0}$$ has all critical point forward orbits finite and critical points non-periodic. 2) $$DF_ i(\lambda_ 0)\neq 0$$ for $$1\leq i\leq 3d-2$$ where $$F_ i(\lambda)=f_{\lambda}^{r_ i+s_ i}(x_ i(\lambda))-f_{\lambda}^{r_ i}(x_ i(\lambda))$$, $$x_ i(\lambda)$$ the critical points of $$f_{\lambda}$$ and $$f^{r_ i}_{\lambda_ 0}(x_ i(\lambda_ 0))$$ is periodic with period $$s_ i$$.
Reviewer: M.Denker

##### MSC:
 37A99 Ergodic theory 28D05 Measure-preserving transformations
Full Text:
##### References:
 [1] M. BENEDICKS and L. CARLESON , On Iterations of l-ax2 on (-1, 1) , Institut Mittag-Leffler, Report No. 3, 1983 . [2] H. BROLIN , Invariant Sets Under Iteration of Rational Functions , (Arkiv for Mathmatik, Vol. 6, 1965 , pp. 103-144). MR 33 #2805 | Zbl 0127.03401 · Zbl 0127.03401 [3] P. L. DUREN , Univalent Functions , New York, Springer, 1983 . MR 85j:30034 | Zbl 0514.30001 · Zbl 0514.30001 [4] P. FATOU , Sur les équations fonctionnelles (Bull. Soc. Math. Fr., Vol. 47, 1919 , pp. 161-271 and Vol. 48, 1920 , pp. 33-94 and pp. 208-314). Numdam | JFM 47.0921.02 · JFM 47.0921.02 [5] M. R. HERMAN , Construction d’un difféomorphisme minimal d’entropie topologique non nulle (Erg. Theory and Dynam. Syst., Vol. 1, 1981 , pp. 65-76). MR 83c:58046 | Zbl 0469.58008 · Zbl 0469.58008 [6] M. R. HERMAN , Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2 (to appear). · Zbl 0554.58034 [7] M. V. JABOBSON , Absolutely Continuous Invariant Measures for One-Parameter Families of One-Dimensional Maps (Comm. in Math. Phys., Vol. 81, 1981 , pp. 39-88). Article | MR 83j:58070 | Zbl 0497.58017 · Zbl 0497.58017 [8] R. MANÉ , P. SAD and D. SULLIVAN , On the Dynamics of Rational Maps (Ann. Éc. Norm. Sup., Vol. 16, 1983 , pp. 193-217). Numdam | MR 85j:58089 | Zbl 0524.58025 · Zbl 0524.58025 [9] M. REES , Ergodic Rational Maps with Dense Critical Point Forward Orbit (Erg. Theory and Dynam. Syst., Vol. 4, 1984 , pp. 311-322). MR 85m:58111 | Zbl 0553.58008 · Zbl 0553.58008 [10] D. SULLIVAN , Conformal Dynamical Systems, Geometric Dynamics , (Lecture Notes in Math., No. 1007, 1981 , pp. 725-752). MR 85m:58112 | Zbl 0524.58024 · Zbl 0524.58024 [11] D. SULLIVAN , Quasi-Conformal Homeomorphisms and Dynamics I (to appear). [12] W. SZLENK , Some Dynamical Properties of Certain Differentiable Mappings of an Interval (Bol. Soc. Mat. Mex., Vol. 24, No. 2, 1979 , pp. 57-82). MR 83c:58048 | Zbl 0487.58013 · Zbl 0487.58013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.