Measures with positive Lyapunov exponent and conformal measures in rational dynamics.

*(English)*Zbl 1267.37042The 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 \(n\geq 0\).

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 \(n\geq 0\).

Reviewer: Mike Hurley (Cleveland)

##### 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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