×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps, Nonlinearity 4 (1991), no. 1, 103 – 134. · Zbl 0718.58035
[2] M. Denker and M. Urbański, On Sullivan’s conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), no. 2, 365 – 384. · Zbl 0722.58028
[3] Jacek Graczyk and Stanislav Smirnov, Non-uniform hyperbolicity in complex dynamics, Invent. Math. 175 (2009), no. 2, 335 – 415. · Zbl 1163.37008
[4] François Ledrappier, Some properties of absolutely continuous invariant measures on an interval, Ergodic Theory Dynamical Systems 1 (1981), no. 1, 77 – 93. · Zbl 0487.28015
[5] François Ledrappier, Quelques propriétés ergodiques des applications rationnelles, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 1, 37 – 40 (French, with English summary). · Zbl 0567.58016
[6] N. Makarov and S. Smirnov, On thermodynamics of rational maps. II. Non-recurrent maps, J. London Math. Soc. (2) 67 (2003), no. 2, 417 – 432. · Zbl 1050.37014
[7] Ricardo Mañé, On the Bernoulli property for rational maps, Ergodic Theory Dynam. Systems 5 (1985), no. 1, 71 – 88. · Zbl 0605.28011
[8] William Parry, Topics in ergodic theory, Cambridge Tracts in Mathematics, vol. 75, Cambridge University Press, Cambridge-New York, 1981. · Zbl 0449.28016
[9] Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1989. Corrected reprint of the 1983 original. · Zbl 0676.28008
[10] Feliks Przytycki, On the Perron-Frobenius-Ruelle operator for rational maps on the Riemann sphere and for Hölder continuous functions, Bol. Soc. Brasil. Mat. (N.S.) 20 (1990), no. 2, 95 – 125. · Zbl 0723.58030
[11] Feliks Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 (1993), no. 1, 309 – 317. · Zbl 0787.58037
[12] Feliks Przytycki, On the hyperbolic Hausdorff dimension of the boundary of a basin of attraction for a holomorphic map and of quasirepellers, Bull. Pol. Acad. Sci. Math. 54 (2006), no. 1, 41 – 52. · Zbl 1130.37375
[13] Feliks Przytycki and Juan Rivera-Letelier, Statistical properties of topological Collet-Eckmann maps, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 1, 135 – 178 (English, with English and French summaries). · Zbl 1115.37048
[14] Feliks Przytycki, Juan Rivera-Letelier, and Stanislav Smirnov, Equality of pressures for rational functions, Ergodic Theory Dynam. Systems 24 (2004), no. 3, 891 – 914. · Zbl 1058.37032
[15] Juan Rivera-Letelier, A connecting lemma for rational maps satisfying a no-growth condition, Ergodic Theory Dynam. Systems 27 (2007), no. 2, 595 – 636. · Zbl 1110.37037
[16] V. A. Rohlin. Exact endomorphisms of a Lebesgue space. Amer. Math. Soc. Transl. (2), 39:1-36, 1964.
[17] Dennis Sullivan, Conformal dynamical systems, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 725 – 752.
[18] Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. · Zbl 0475.28009
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.