Eugen Mihailescu


Principal Researcher II
Institute of Mathematics of the Romanian Academy

Address InformationEducation and Employment ResearchPublications
CVTeaching Mathematical linksOther links

Address Information

Education

Academic Employment/Experience

  • Main Research Directions:
  • Shorter Research Visits:
  • Awards/Fellowships/Grants:

    Research

    I am interested in the relations between Dimension Theory and Thermodynamic Formalism with the Dynamics of smooth transformations, especially in the case of endomorphisms. I am also interested in Smooth Ergodic Theory and Hyperbolic Dynamics on Fractals (saddle basic sets, various iterated function systems, invariant sets from one - or several complex variable dynamics, Lyapunov exponents, equilibrium measures, hyperbolic flows, etc.).

    Here is a short description of my research. Currently I am interested in a relatively recent field, that of applications of ergodic theory and thermodynamical formalism to the dynamics of higher dimensional dynamical systems.
    In one variable, complex dynamics was born when Fatou and Julia studied, at the begining of the XX-th century, normal families of holomorphic functions and what are now called the Fatou components and Julia set. Given the essential geometric differences that appear when one considers several complex variables it is justified to expect new and interesting phenomena occuring in this case. My thesis studied actions of tori on Stein manifolds and consequences on the dynamics of periodic points, and on the other hand, the "opposite" case of strongly hyperbolic endomorphisms. One can remark that among examples of Stein manifolds with actions of circles we find the Fatou components of a holomorphic endomorphism in a projective complex space.
    At the moment I am working on applications of thermodynamic formalism in higher dimensional conformal dynamics of non-invertible maps. Notions from thermodynamics have found recently powerful applications also to complex dynamics, especially regarding questions about Hausdorff dimension of invariant sets and equilibrium measures. Here also hyperbolicity plays a very important role.
    I have studied the case of Axiom A non-invertible maps on projective spaces and their stable dimension. Here generalizations of the usual notions of topological entropy and pressure are in order, which will take into consideration the prehistories of points.
    The dynamics of hyperbolic non-invertible maps is different than the one of diffeomorphisms, although there are some common trends. In this case, the unstable tangent spaces (and hence also the unstable manifolds) depend on entire prehistories; therefore the unstable manifolds do not form a nice (Lipschitz) lamination near the basic set and the stable dimension cannot be written in general as the solution of a Bowen equation involving the usual (forward) pressure. Still, for conformal maps, the stable dimension is equal in many cases to the zero of the inverse pressure. The notion of inverse pressure (introduced by me and M. Urbanski) is better suited to deal with some aspects of non-invertible maps, than the classical topological pressure. The dimension theory for such maps is very rich and beautiful, and has connections with many fields. This direction involves the study of equilibrium and conformal measures, attractors/repellors/saddle basic sets, transfer operators, examples from one or higher dimensional complex dynamics, and symbolic dynamics.

    Publications

    (papers are in ISI journals, except those in proceedings of conferences and the last 2 in the list)
    1. Unstable directions and fractal dimensions for a family of skew products with overlaps, preprint 2009.
    2. Metric properties of some fractal sets and applications of inverse pressure, to appear in Mathematical Proceedings Cambridge Phil. Soc., doi:10.1017/S0305004109990326, Published Online by Cambridge University Press 20 Nov 2009.
    3. Ergodic properties for some non-expanding non-reversible systems, preprint 2009.
    4. Weighted distributions of preimages for hyperbolic endomorphisms, preprint 2009.
    5. Dynamics on higher dimensional fractals, will appear in Proceedings of the International Conference on Complex Analysis and the XI-th Romanian-Finnish Seminar, Alba Iulia 2008.
    6. Relations between stable dimension and the preimage counting function on basic sets with overlaps, joint with M. Urbanski, Bulletin London Mathematical Society, vol. 42, 2010, 15-27.
    7. Asymptotic distributions, preprint 2008.
    8. Metric properties and dynamics for conformal maps, Proceedings of the International Congress of Romanian Mathematicians Bucharest 2007, pg. 161-169, Ed. Academiei 2009.
    9. Transversal families of hyperbolic skew products , joint with M. Urbanski, Discrete and Continuous Dynamical Systems, vol. 21, no.3, 907-928, 2008.
    10. Unstable manifolds and Holder structures associated with non-invertible maps, Discrete and Continuous Dynamical Systems, 14, 2006, no 3, 419-446.
    11. Inverse pressure estimates and the independence of stable dimension for non-invertible maps, joint with M. Urbanski, Canadian Journal of Mathematics, vol. 60, no. 3, 658-684, 2008.
    12. Inverse topological pressure with applications to holomorphic dynamics in several variables, joint with Mariusz Urbanski, Communications in Contemporary Mathematics, vol.6, no.4, 2004, 653-682.
    13. Estimates for the stable dimension for holomorphic maps, joint with M. Urbanski, Houston Journal of Mathematics, 31 (2), 2005, 367-389.
    14. Holomorphic maps for which the unstable manifolds depend on prehistories, joint with M. Urbanski, Discrete and Continuous Dynamical Systems, vol.9, no.2, 2003.
    15. The set K- for hyperbolic non-invertible maps, Ergodic Theory and Dynamical Systems, June 2002, 3, 873-888.
    16. Applications of thermodynamic formalism in complex dynamics on P2, Discrete and Continuous Dynamical Systems, vol. 7, 4, October 2001, 821-836.
    17. Periodic points for actions of tori in Stein manifolds, Mathematische Annalen 314, 1999, no 1, 39-52
    18. PH.D Thesis, Univ. of Michigan Press 1999.
    19. A class of functions which are continuous but nowhere monotonous, Annals of the University of Bucharest, 42/43, 1993/94, 63-71.
    20. On some topologies on the space of analytic coherent subsheaves, Revista Matematica, 1994, Bucuresti

    Curriculum Vitae

    A copy of my CV

    Teaching

    During the academic year 2009-2010 I will teach a Masters course on "Analysis on fractals" at SNSB (Normal Superior School of Bucharest). Students at SNSB that are interested in dynamical systems, ergodic theory, measure theory or complex analysis/dynamics can contact me for further study and/or problems in these areas. In 2008-2009 I taught a Masters course on "Differentiable Dynamics" at SNSB (Normal Superior School of Bucharest). In 2007-2008 I taught a Masters course on "Topology and dynamics for hyperbolic maps" at SNSB. In 2006 I taught two advanced courses at SNSB, for some selected best students of Romanian universities. The first course was "Introduction to dynamical systems" (Spring 2006), and the second is a Masters course "Ergodic theory with applications to dynamics" (Fall 2006). My teaching experience consists also in the diverse courses I taught in the mathematics departments of the University of Michigan, University of North Texas, Texas A & M University, Universitatea Politehnica Bucuresti, University of South Carolina, etc.
    I taught various courses and seminars, like Probability (Univ. of North Texas 2002), Calculus 115 (Univ. of Michigan 1995-1997), Calculus II-- Math 215 with Maple (Univ.of Michigan 1998-1999), Differential Equations III (Texas A& M Univ. 2000-2001), Real Analysis, and Advanced Mathematics (Univ. Politehnica Bucuresti, 2001-2002), Business Calculus (Univ. North Texas 2002), etc.
    I participated in the elaboration of teaching curriculae and exam problems and the training of teaching assistants at Univ. of Michigan, Texas A& M Univ. and Univ. of North Texas. Most of them were classical proof courses; a few others were following the Harvard Calculus Reform of emphasizing real life applications of mathematics and the involvement of computers in the teaching process (Mathlab, Maple, etc.)
    During my two-year Visiting Professorship at Texas A& M University, I supervised three teaching assistants and two grading assistants for the Calculus and Differential Equations courses that I taught. At Univ. of North Texas I also supervised a grading assistant.

    Mathematical Links

    Other Links

    Created initially August 2000.