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Measures with positive Lyapunov exponent and conformal measures in rational dynamics. (English) Zbl 1267.37042
The paper concerns rational maps on the Riemann sphere. Given such a map $$f$$, a Hölder continuous map $$\phi$$ from the sphere to itself, and a real number $$t$$, then for $$t\geq 0$$ a probability measure $$m$$ on the sphere is said to be $$(\phi, t)$$-conformal if the Julia set of $$f$$ has full measure and if for each Borel set $$A$$ on which $$f$$ is injective, $$m(f(A)) = \int_A|Df|^t\;dm$$, where $$|Df|$$ represents the spherical derivative. (The definition of $$(\phi, t)$$-conformality is slightly more involved when $$t<0$$.) Given such a measure $$m$$ satisfying an additional nondegeneracy requirement, and an $$f$$-invariant probability measure $$\mu$$ with positive Lyapunov exponent, it is demonstrated that a number of different conditions are each equivalent to $$\mu$$ being absolutely continuous with respect to $$m$$. One of these conditions is that $$f$$ induces a well-behaved system of return maps $$f^{n_i}:U_i\to U$$ where $$U$$ is an open ball with $$m(U)>0$$ and $$\{ U_i\}$$ is a partition of $$U$$ up to a set of measure $$0$$, and that this system generates $$\mu$$. When these conditions hold, $$\mu$$ is unique and $$m$$ is ergodic.
The key in the proof is to establish the existence of what is called a regularly returning cylinder $$A$$ in the natural extension $$F:Y\to Y$$. Here $$Y$$ is the set of all sequences $$(y_0, y_1, \dots)$$ such that $$f(y_i) = y_{i-1}$$ for every $$i>0$$, and $$F((y_0, y_1, \dots )) = (f(y_0), y_0, y_1, \dots )$$, and $$A$$ is a subset of $$Y$$ such that the projection of $$A\subset Y$$ onto its first component is a nice subset of the sphere and the projections of $$F^{-n}(A)$$ are well-behaved for all $$n\geq 0$$.

##### MSC:
 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
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