Real quadratic family.

In 1990--1998, my main focus was the dynamics in the real quadratic family, which I approached by complex methods.
The main results are:

  • No wild attractors [2] (solution of Milnor's Problem in the quadratic case);
  • Density of hyperbolic maps [3];
  • Proof of the Feigenbaum-Coullet-Tresser Renormalization Conjecture [5];
  • Typical non-regular and non-renormalizable maps are stochastic (satisfying the Martens-Nowicki criterion) [6];
  • Construction of the Full Renormalization Horseshoe [7].

    The last two papers imply the Regular or Stochastic Theorem : Almost any real quadratic map is either regular or stochastic.

    Note [8] is a survey on the subject based on my plenary talk at the AMS meeting in Washington, DC (January 2000).

    Measure and dimension of some Julia sets and parameter sets

    While studying Fatou's memoirs in the graduate school, I got interested in the problem of the area of Julia sets:
    Paper [1] contains some observations in this direction -- it is one of the earliest applications of the Koebe Distortion Theorem to dynamics.
    It became the starting point of the series of papers with Blokh on the Smooth Ergodic Theory on the interval.

    Smooth Ergodic Theory on the interval

    The series of papers [1-6] contains results obtianed in 1985-1988 in collaboration with Sasha Blokh (with final touches put later in [7]).
    They were largely motivated by Milnor's Problem on Attractors.

    Papers [4-5] prove real a priori bounds for infinitely renormalizable maps.
    They were indepedently established by Dennis Sullivan as the first step towards the Feigenbaum-Coullet-Tresser Renormalization Conjecture.
    [The doubling case had been earlier handled by Guckenheimer.]

    Transcendental Dynamics

    The results of [2,3] were obtained in the fall 1983 in collaboration with Alex Eremenko in Kharkov (Ukraine).
    They were announced in note [1] and appeared in the preprint version in Russian.
    It took long time to get the detailed English version published.

    Structural stability of rational endomorphisms

    The results of [1,2] were obtained in the fall 1981 on the cotton fields near Tashkent (Uzbekistan) and became a chapter of my PhD thesis.
    At about the same time, these results (and more) were obtained by Mane, Sad and Sullivan --
    this work has become classics of Holomorphic Dynamics.

    Entropy properties of rational endomorphisms

    The result of [1] (entropy of a rational map is equal to the log of its degree) constitutes my Master thesis (Kharkov, May 1980).
    About one year later I learned about preprint by Gromov (undated and never published but eventually influential, particularly in several variables)
    that proves the same result for the complex projective spaces of any dimension.

    Soon afterwards (in the fall 1980), I obtained the results of [3] (announced in [2]) on existence and uniqueness of the measure of maximal entropy.
    They became part of my PhD thesis (Tashkent, March 1984).
    By the time [3] appeared, the work of Mane, Friere and Lopez on the same subject emerged.