Statistical properties of quadratic polynomials with a neutral fixed point

Abstract

We describe the statistical properties of the dynamics of the quadratic polynomials P α (z) = e 2παi z + z 2 on the complex plane, with α of high return times. In particular, we show that these maps are uniquely ergodic on their measure theoretic attractors, and the unique invariant probability is a physical measure describing the statistical behavior of typical orbits in the Julia set. This confirms a conjecture of Perez-Marco on the unique ergodicity of hedgehog dynamics, in this class of maps.2010 Mathematics Subject Classification. Primary: 37F50, Secondary: 37F25, 58F11. The second author acknowledges funding from Leverhulme Trust while carrying out this research. 1The domain U contains 0 and cp P , but not the other critical point of P at −1.Following [IS06], we define the class of mapsand ϕ has a quasi-conformal extension onto C. .Every map in IS has a fixed point of multiplier one at 0, and a unique critical point at cp h = ϕ(−1/3) ∈ U h which is mapped to cv h = −4/27. Elements of IS have the same covering structure as the one of P on U.

Typical orbits of quadratic polynomials with a neutral fixed point: Non-Brjuno type

“…Indeed, we show that given a compact class of germs f (in the compact open topology), the L 1 norm of the error (over infinite vertical strips) is uniformly bounded by a constant divided by the rotation of f at 0. This estimate allows us to prove the results under the sharp Brjuno condition on α here, and will be used to describe the statistical properties of the quadratic maps P α of high type in [AC12]. Moreover, one may use these estimates to answer some major optimality questions, like Herman's condition, in the class of quadratic maps of high type.…”

Dimension paradox of irrationally indifferent attractors

“…, where U n is the domain of definition of f n . By the precompactness of the class IS α ∪ {Q α } with α ∈ HT N , there exists a constant δ 1 > 0 independent of n (actually independent of f ∈ IS 0 ∪ {Q 0 }) such that the δ 1 -neighborhood of these sets B δ1 ( 16 We would like to mention that the definitions of χn in this paper and in [AC18], [Che19] are different. In this paper we require that χn(1) = 1 but in the latter two literatures χn(1) = k 0 for some k 0 ≥ 1.…”

“…These are locally invariant compact sets where both f and f −1 are injective on a neighbourhood of f . It turns out that when f ∈ F with f (0) = e 2πiα and α ∈ HT N , every hedgehog of f is contained in Λ(f ), see [AC18] for details. For instance, this holds for the quadratic polynomials e 2πiα z + z 2 .…”

Topology of irrationally indifferent attractors

“…It has been recently shown by the author that Λ(f ) is a connected and measure zero subset of the plane, and the restriction of f : Λ(f ) → Λ(f ) is a homeomorphism, [Che10,Che13]. Moreover, f : Λ(f ) → Λ(f ) supports a unique invariant probablity measure, [AC12]. These results have been achieved without precise knowledge of the topological structure of Λ(f ).…”