Mikhail Lyubich

Professor, Canada Research Chair, Department of Mathematics

Office: SS4049, (416) 978-5163; Fax (416) 978-4107

Selected recent papers.
Dynamics of quadratic maps (real and complex).

(joint with John Milnor) The Fibonacci unimodal map. J. Amer. Math. Soc., v. 6 (1993), 425-457.
Combinatorics, geometry and attractors of quasi-quadratic maps. Annals of Mathematics, v. 140 (1994), 347-404.
See also: Note on the geometry of generalized parabolic towers.
Dynamics of quadratic polynomials, I-II. Acta Mathematica, v. 178 (1997), 185-297.
(joint with Michael Yampolsky) Dynamics of quadratic polynomials: Complex bounds for real maps. Annalles de l'Institut Fourier, v. 47 (1997), \# 4, 1219 - 1255.
Dynamics of quadratic polynomials III: parapuzzle and SBR measures. Asterisque volume in honor of Adrien Douady's 60th birthday ``G\'eom\'etrie complexe et syst\'emes dynamiques'', v. 261 (2000), 173 - 200.
How big is the set of infinitely renormalizable quadratics? The volume "Voronezh Winter Mathematical Schools" in honor of 80th birthday of S.G. Krein. AMS Transl. (2), v. 184 (1998), 131 - 143.
Feigenbaum-Coullet-Tresser Universality and Milnor's Hairiness Conjecture. Annals of Mathematics, v. 149 (1999), 319 - 420.
Almost every real quadratic map is either regular or stochastic. Annals of Mathematics, v. 156 (2002), 1 - 78.
(joint with A. Avila and W. de Melo) Regular or stochastic dynamics in real analytic families of unimodal maps. Preprint IMS at Stony Brook, #2001/15.

Surveys
Milnor's attractors, persistent recurrence and renormalization, ``Topological Methods in Modern Mathematics, A Symposium in Honor of John Milnor's 60th Birthday", Publish or Perish, 1993.
On the borderline of real and complex dynamics. Proc. ICM, Z\"urich 1994. Birkh\"auser 1995, v. 2, 1203 -- 1215.
Renormalization ideas in conformal dynamics. Cambridge Seminar ``Current Developments in Math.", May 1995. International Press, 1995. Cambridge, MA, 155 - 184.
The quadratic family as a qualitatively solvable model of chaos. Notices of the American Math. Society, October 2000.
Six Lectures on Real and Complex Dynamics. (based on European Lectures given in Barcelona, Copenhagen and St Petersburg in May-June 1999).

Polynomial diffeomorphisms of ${\bf C}^2$.
(joint with Eric Bedford \& John Smillie) Polynomial diffeomorphisms of ${\bf C}^2$, IV: The measure of maximal entropy and laminar currents. Inventions Math., v. 112 (1993), 77 - 125.
(joint with Eric Bedford \& John Smillie) Distribution of periodic points of polynomial diffeomorphisms of ${\bf C}^2$. Inventions Math., v. 114 (1993), 277 - 288.

Laminations.
(joint with Yair Minsky) Laminations in holomorphic dynamics. J. Diff. Geom., v. 47 (1997), 17 - 94.
(joint with Vadim Kaimanovich) Conformal and harmonic measures on laminations associated with rational maps. Preprint IMS at Stony Brook #2001/5.
Laminations and Holomorphic Dynamics. Lecture Notes of the mini-course given at the Conference ``New Directions in Dynamical Systems" in Kyoto, August 2002.

Reviews.
Review on the books by C. McMullen ``Complex dynamics and renormalization" and ``Renormalization and 3-manifolds which fiber over the circle". Bulletin of the AMS, v. 36 (1999), 103 - 107.

Remark. The above papers (with underlined titles) are stored as compressed postscript files. To get a file click on its title. To uncompress the file, use command ``gunzip". For example, the paper ``Feigenbaum-Coullet-Tresser Universality..." is stored as ``universe.ps.gz". By means of ``gunzip universe.ps.gz" you will create a postscript file named ``universe.ps". It can be then printed out by standard means.