Genadi Levin

About 35 years ago Dennis Sullivan proved a precompactness property ("complex bounds") for infinitely renormalizable real quadratic polynomials with bounded combinatorics. We present a simple "soft" proof of this remarkable result.

The modulus of a polynomial-like (PL) map is an important invariant that controls distortion of the straightening map and, hence, geometry of the corresponding PL Julia set. Lower bounds on the modulus, called complex a priori bounds, are known in a great variety of contexts. For any rational function we complement this by an upper bound for moduli of PL maps in the satellite case that depends only on the relative period and the degree of the PL map. This rules out a priori bounds in the satellite case with unbounded relative periods. We also apply our tools to obtain lower bounds for hyperbolic lengths of geodesics in the infinitely renormalizable case, and to show that moduli of annuli must converge to 0 for a sequence of arbitrary renormalizations, under several conditions all of which are shown to be necessary.

We apply some methods and techniques of complex dynamics to study the set of symmetries of attractors of holomorphic iterated function systems (IFS), as well as relations between IFSs sharing the same attractor.

Adopting the approach of [7] we study rational function carrying invariant line fields on the Julia set. In particular, we show that under certain weak conditions all possible measurable invariant line fields of a rational function on its Julia set are determined, in a precise sense, by a finite list of canonically defined holomorphic line fields.

Let f(z)=z^2+c be an infinitely renormalizable quadratic polynomial and J_\infty be the intersection of forward orbits of "small" Julia sets of its simple renormalizations. We prove that if f admits an infinite sequence of satellite renormalizations, then every invariant measure of f: J_\infty\to J_\infty is supported on the postcritical set and has zero Lyapunov exponent. Coupled with [G. Levin, F. Przytycki, W. Shen, The Lyapunov exponent of holomorphic maps. Invent. Math. 205 (2016), 363-382], this implies that the Lyapunov exponent of such f at c is equal to zero, which answers partly a question posed by Weixiao Shen.

Let f be an infinitely-renormalizable quadratic polynomial and J_\infty be the intersection of forward orbits of "small" Julia sets of simple renormalizations of f. We prove that J_\infty contains no hyperbolic sets.

We give necessary and sufficient conditions for a function in a naturally appearing function space to be a fixed point of the Ruelle–Thurston operator associated to a rational function (see Lemma 2.1). The proof uses essentially a 2020 paper of the a

We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to under the dynamics of the tower for corresponding . That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when tends to . We also prove the convergence of the drifts to a finite limit, which can be expressed purely in terms of the limiting tower, which corresponds to a Feigenbaum map with a flat critical point.

Motivated by a problem in holomorphic dynamics, we present a certain generalization of the celebrated F. and M. Riesz Theorem.