On an analytic approach to the Fatou conjecture

Abstract

Abstract. Let f be a quadratic map (more generally, f (z) = z d + c, d > 1) of the complex plane. We give sufficient conditions for f to have no measurable invariant linefields on its Julia set. We also prove that if the series n≥0 1/(f n ) (c) converges absolutely, then its sum is non-zero. In the proof we use analytic tools, such as integral and transfer (Ruelle-type) operators and approximation theorems.

Multipliers of periodic orbits in spaces of rational maps

“…Then it boils down to the fact that T has no fixed points of a certain form (which is roughly a linear combination of functions B O for non-repelling orbits), and this follows from the contraction property of T . The latter idea goes back to Thurston's work mentioned above and has been applied, among others, in [6], [33], [36], [9], [17], [13], [27]. See end of Sect.…”

“…If ρ j = 1, then, by the assumption, (f n j )"(b j k ) = 0, and henceB j is not zero in this case as well. In this notation, the connections (10), (17) read as follows:…”

“…For applications to the problems of rigidity in complex dynamics, see [6], [33], [9], [26], [17], [27], [28], [37], [14], [15]. In particular, in [17] we use the operator T for proving the absence of invariant linefield on some Julia sets. The scheme of [17] is applied in [27], [28], [37].…”

“…In particular, in [17] we use the operator T for proving the absence of invariant linefield on some Julia sets. The scheme of [17] is applied in [27], [28], [37].…”

Quadratic dynamics in binary number systems

“…Since a is Collet-Eckmann, the series P k¼1 k¼1 ½1=Dðf k21 a ÞðaÞ converges absolutely at a geometric rate. By a recent result of Levin [8], the absolute convergence of this series guarantees its convergence to a nonzero value. Hence which is unbounded.…”