Forty years of unimodal dynamics: on the occasion of Artur Avila winning the Brin Prize
In 2011 Artur Avila won the Brin Prize for his fundamental contributions in three domains of dynamical systems: one-dimensional dynamics; quasi-periodic Schrödinger operators; Teichmüller dynamics and interval exchange transformations.NEWLINENEWLINEThe paper under review is written on the occasion of his prize winning. It is an excellent survey of the development of one-dimensional dynamics from 1970s till nowadays, when real analytic unimodal dynamics has became fully probabilistically understood. Its most part concerns the unimodal dynamics (dynamics of a mapping of a segment to itself with one critical point that is an extremum). It makes an emphasis on works of Avila, who has been the key player in the final stage of the story.NEWLINENEWLINEThe paper is very well-written and can be read by non-specialists: all the definitions and basic results are recalled, and key ideas of proofs are explained. It starts with the basic theory: Milnor-Thurston kneading combinatorial invariant, Feigenbaum map and universality phenomenon, stochastic maps (i.e., having an absolutely continuous invariant probability measure) and their abundance (Jakobson's theorem). The next sections give a survey of the development of complex methods: famous conjectures on density of hyperbolicity in quadratic family and local connectivity of the Mandelbrot set; results of Douady-Hubbard on quadratic-like maps; Yoccoz's results on renormalization and puzzles. The density of hyperbolicity in the family of real quadratic polynomials was proved separately by the author of the survey and in the joint paper with J. Graczyk and G. Swiatek. Both papers combine real and complex methods based on a renormalization theory presented in Sections 1.4 and 1.9. A renormalizable map is a unimodal map that has a periodic interval; its combinatorics is the order of intermediate images of the latter interval. The Feigenbaum-Coullet-Tresser renormalization conjecture asserts that for every given combinatorics the renormalization operator has a unique fixed point: a renormalizable map with given combinatorics; moreover this fixed point is hyperbolic with one-dimensional unstable direction. This conjecture was proved by the author of the survey for stationary combinatorics in the space of quadratic-like maps.NEWLINENEWLINEOne of the main ingredient used in the proof of density of hyperbolicity in the real quadratic family were complex a priori bounds for real infinitely renormalizable maps obtained by G. Levin, S. van Strien, the author of the survey and M. Yampolsky: the fundamental annuli of the renormalizations have moduli uniformly bounded from below. These bounds are the main point needing reality of the maps.NEWLINENEWLINELater on it was proved by the author of the survey that almost every real quadratic map is either regular (i.e., hyperbolic), or stochastic. The proof uses complex methods together with the proof of the existence of a hyperbolic horseshoe for the renormalization operator: the renormalization horseshoe. This regular or stochastic theorem was the first occasion to confirm a general conjecture suggested by J. Palis on attractors of typical dissipative dynamical systems.NEWLINENEWLINEThe next sections (Sections 2 and 3) present an overview of Artur Avila's main results in one-dimensional dynamics. The joint paper by \textit{A. Avila, W. de Melo} and the author published in [Invent. Math. 154, No. 3, 451--550 (2003; Zbl 1050.37018)] extends the above regular or stochastic theorem to arbitrary nontrivial real analytic family of unimodal maps with quadratic critical point. The strategy of the proof was to transfer the theorem for quadratic polynomials to more general families through the holonomy along the hybrid classes. A joint paper of \textit{A. Avila} and \textit{C. G. Moreira} published in [Ann. Math. (2) 161, No. 2, 831--881 (2005; Zbl. 1078.37029)] concerns statistical properties of typical stochastic quadratic maps. They showed that almost every stochastic quadratic map is Collet-Eckmann, has torrential (iterated exponential) decay of geometry and linearly slow recurrence of the critical point (confirming a conjecture of Sinai from 1970s). Another joint paper [\textit{A. Avila} and \textit{C. G. Moreira}, Publ. Math., Inst. Hautes Étud. Sci. 101, 1-67 (2005; Zbl 1078.37030)] concerns stochastic maps in real analytic families of unimodal maps. They have obtained an important and delicate result on the equidistribution of the critical orbit with respect to the physical measure. It implies important corollaries, one of them says how the multipliers of periodic orbits can be read off the kneading sequence for typical stochastic maps. Another (unpublished) joint result says that in the real quadratic family, the set of infinitely renormalizable parameters has Hausdorff dimension strictly less than one. Next, the author presents the following results concerning the case of unimodal maps \(f_{c,d}(z)=z^d+c\) with a higher-degree critical point:NEWLINENEWLINE- A joint result of \textit{A. Avila} et al., [Ann. Math. (2) 170, No. 2, 783--797 (2009; Zbl 1204.37047)] on the Multibrot set (higher-degree analogue of the Mandelbrot set): its local connectivity at non-infinitely-renormalizable non-critically-periodic parameters. This generalizes Yoccoz's theorem, which was originally proved for quadratic maps and whose methods do not work for higher degrees.NEWLINENEWLINE- A joint result of \textit{A. Avila, M. Lyubich} and \textit{W. Shen} saying that a typical map \(f_{c,d}\) of mixing type is stochastic [J. Eur. Math. Soc. (JEMS) 13, No. 1, 27--56 (2011; Zbl 1213.37076)]. NEWLINENEWLINE- A joint result of \textit{A. Avila} and \textit{M. Lyubich} on the full renormalization horseshoe and exponential contraction along hybrid classes [Publ. Math., Inst. Hautes Étud. Sci. 114, 171--223 (2011; Zbl 1286.37047)]. NEWLINENEWLINEIn Section 3, the author presents his joint results with A. Avila on the geometry of Feigenbaum maps. They proved the existence of Feigenbaum Julia sets of Hausdorff dimension less than two with stationary combinatorics and a conditional result on the existence of Feigenbaum Julia sets of Hausdorff dimension 2 with zero area, provided that those with positive area (``black holes'') exist [J. Am. Math. Soc. 21, No. 2, 305--363 (2008; Zbl 1205.37058)]. Afterwards they showed, in an unpublished manuscript, that the black hole Feigenbaum Julia sets actually exist. Section 4 provides basic bibliographic notes, including history and textbooks on complex dynamics.
scientific article; zbMATH DE number 6094023