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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
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References:
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