Mikhail Lyubich

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Mikhail Lyubich, Oberwolfach 2008

Mikhail Lyubich ( Russian Михаил Юрьевич Любич , transcription Michail Jurjewitsch Lyubitsch ; born  February 25, 1959 in Kharkiv ) is a Russian - American mathematician who deals with dynamic systems .

Lyubich is the son of Yuri Ilyich Lyubitsch, a mathematics professor at Kharkiv University. He studied at the University of Kharkiv from 1975 to 1980 (with the diploma degree entropy of rational maps ), had to go to the University of Tashkent ( Dynamics of rational maps and their invariants ) for his doctorate in 1984 because of the anti-Semitic educational policy in the former Soviet Union . In 1989 he left the Soviet Union with his family and went to the State University of New York at Stony Brook (SUNY) at the invitation of John Milnor . He became an assistant professor there in 1990 and later a professor. From 2002 to 2008 he was also a professor at the University of Toronto (on a Canada Research Chair ). Since 2007 he has been director of the Institute of Mathematical Sciences (IMS) at SUNY.

He was already working in the Soviet Union for his dissertation on dynamics in a complex variable, especially the ergodic theory and stability of mapping with rational functions. He proved the existence of a measure for the maximum entropy of a rational mapping (Lyubich measure).

At the end of the 1990s, Lyubich proved the phenomenon of the universality of the period doubling cascades of quadratic mappings of the unit interval in an analytical way, discovered by Mitchell Feigenbaum , Pierre Coullet and Tresser in the late 1970s (computer evidence had existed since 1982 by Oscar Lanford ). More precisely, he proved the existence of a hyperbolic fixed point of the associated renormalization group transformation, assumed by Feigenbaum, not only for period doubling, but in general for renormalization operators of limited type . Rigorous evidence in Feigenbaum's renormalization group theory had previously been given by Dennis Sullivan and Curtis McMullen in the context of the complex dynamics , and Lyubich, in a way, set a keystone. Lyubich also demonstrated the self-similarity around certain points of the Mandelbrot set (suggested by Milnor).

At the end of the 1990s, Lyubich showed that hyperbolicity is densely distributed in the class of mappings through quadratic real functions (which was also independently proven by Grzegorz Swiatek and J. Graczyk), a long open conjecture. In 1998 he proved that almost all quadratic real mappings are regular or stochastic.

With Jeremy Kahn he is pursuing a program to prove the local connection of the Mandelbrot set (an important open conjecture in the complex dynamics).

In 1994 he was invited speaker at the International Congress of Mathematicians in Zurich ( On the borderline of real and complex dynamics ). In 1987 he received the Leningrad Mathematical Society Prize . In 1991 he was a Sloan Research Fellow and in 2002 a Guggenheim Fellow . In 2010 he received the Jeffery Williams Prize . He is a fellow of the American Mathematical Society . He was selected as plenary speaker at the International Congress of Mathematicians 2014 in Seoul (Analytic Low-Dimensional Dynamics: from dimension one to two). In 2019 Lyubich was elected to the American Academy of Arts and Sciences .

Fonts

  • Feigenbaum-Coullet-Tresser Universality and Milnor's Hairiness Conjecture , Annals of Mathematics, Volume 149, 1999, pp. 319-420.
  • Almost every quadratic map is either regular or stochastic , Annals of Mathematics, Volume 156, 2002, pp. 1-78
  • Dynamics of quadratic polynomials , I-II. Acta Mathematica, Volume 178, 1997, pp 185-297, Part 3 Asterisque, Volume 261, 2000, pp 173-200 (Douady Volume)
  • with M. Yampolski Dynamics of quadratic polynomials: complex bounds for real maps , Annales Institut Fourier, Volume 47, 1997, pp. 1219-1255
  • Combinatorics, geometry and attractors of quasi-quadratic maps , Annals of Mathematics, Volume 140, 1994, pp. 347-404.
  • with Artur Avila , Welington de Melo Regular or stochastic dynamics in real analytic families of unimodal maps , Inventiones Mathematicae, Volume 154, 2003, pp. 451-550.
  • Regular and stochastic dynamics in the real quadratic family , Proc. Nat. Acad. Sciences USA, Vol. 95, 1998, pp. 14025-14027, pdf
  • The quadratic family as a qualitatively solvable model of chaos , Notices of the American Math. Society, October 2000, Online
  • The measure of maximal entropy of a rational endomorphism of the Riemann sphere , Functional Analysis and Applications, Volume 16, 1982, pp. 78-79.
  • The dynamics of rational transforms: the topological picture , Russian Mathematical Surveys, Volume 41, 1986, pp. 43-117.

Web links

Individual evidence

  1. ^ Mathematics Genealogy Project
  2. ↑ more general unimodular mappings, which first increase monotonically up to a maximum and then decrease
  3. ↑ iterated maps with
  4. ↑ An attractive limit cycle exists
  5. ↑ that is, there is an absolutely continuous invariant measure
personal data
SURNAME Lyubich, Mikhail
ALTERNATIVE NAMES Любич, Михаил Юрьевич (Russian spelling)
BRIEF DESCRIPTION Russian-American mathematician
DATE OF BIRTH February 25, 1959
PLACE OF BIRTH Kharkiv