Eugen Mihailescu


Associate Professor
Institute of Mathematics of the Romanian Academy

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Research

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    Publications

    1. Equilibrium measures on saddle sets of holomorphic maps on P^2, joint with J. E Fornaess, Mathematische Annalen, Online First 2013.
    2. Entropy production for a class of inverse SRB measures, joint with M. Urbanski, Journal of Statistical Physics, Online First 2012.
    3. Upper estimates for the stable dimensions of fractal sets with variable number of foldings, joint with B. Stratmann, preprint 2012.
    4. Hausdorff dimension of limit sets of countable conformal iterated function systems with overlaps, joint with M. Urbanski, to appear Contemporary Mathematics, AMS, 2013.
    5. Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms, Discrete and Continuous Dynamical Systems, vol. 32, no. 7, 2485-2502, 2012. (ISI journal)
    6. On some coding and mixing properties on folded fractals, Monatshefte fuer Mathematik, vol. 167, no. 2, 241-255, 2012, (ISI journal)
    7. Higher dimensional expanding maps and toral extensions, to appear Proceedings of the American Mathematical Society, 2013.
    8. Hyperbolic dynamics on folded saddle sets, Thermodynamic Formalism, Geometry and Stochastics, MF Oberwolfach Reports 3/2012, 31-34, DOI: 10.4171/OWR/2012/03.
    9. Approximations of Gibbs states of arbitrary Holder potentials on hyperbolic folded sets , Discrete and Continuous Dynamical Systems, vol. 32, no. 3, 2012, 961-975. (ISI journal)
    10. Inverse limits and statistical properties for chaotic implicitly defined economic models, Journal of Mathematical Analysis and Applications, vol. 394, no.2, 2012, 517-528. (ISI journal)
    11. On a class of stable conditional measures, Ergodic Theory and Dynamical Systems, vol. 31, no. 5, 1499-1515, 2011. (ISI journal)
    12. Local geometry and dynamical behavior on folded basic sets, Journal of Statistical Physics, vol. 142, 2011, 154-167. (ISI journal)
    13. Unstable directions and fractal dimensions for a family of skew products with overlaps in fibers, Mathematische Zeitschrift, vol. 269, 2011, 733-750. (ISI journal)
    14. Hausdorff dimension of the limit set for conformal iterated function systems with overlaps, joint with M. Urbanski, Proceedings of the American Mathematical Society, vol. 139, 2011, 2767-2775. (ISI journal)
    15. Asymptotic distributions of preimages for endomorphisms, Ergodic Theory and Dynamical Systems, vol. 31, 2011, 911-935. (ISI journal)
    16. Physical measures for multivalued inverse iterates near hyperbolic repellors, Journal of Statistical Physics, vol. 139, no. 5, 2010, 800-819 (ISI journal).
    17. Metric properties of some fractal sets and applications of inverse pressure, Mathematical Proceedings Cambridge Phil. Soc., vol. 148, no.3, May 2010, 553-572. (ISI journal)
    18. Relations between stable dimension and the preimage counting function on basic sets with overlaps, joint with M. Urbanski, Bulletin of the London Mathematical Society, vol. 42, 2010, 15-27. (ISI journal)
    19. Ergodic properties for some non-expanding non-reversible systems, Nonlinear Analysis: Theory, Methods & Applications, vol. 73, 2010, 3779-3787. (ISI journal)
    20. Dynamics on higher dimensional real or complex fractals, Revue Roumaine de Mathematiques Pures et Appliquees, vol. LIV, no.5-6, 2009, 513-524.
    21. Metric properties and dynamics for conformal maps, Proceedings of the International Congress of Romanian Mathematicians Bucharest 2007, pg. 161-169, Ed. Academiei 2009.
    22. Transversal families of hyperbolic skew products , joint with M. Urbanski, Discrete and Continuous Dynamical Systems, vol. 21, no.3, 907-928, 2008. (ISI journal)
    23. 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. (ISI journal)
    24. Unstable manifolds and Holder structures associated with non-invertible maps, Discrete and Continuous Dynamical Systems, 14, 2006, no 3, 419-446. (ISI journal)
    25. Estimates for the stable dimension for holomorphic maps, joint with M. Urbanski, Houston Journal of Mathematics, 31 (2), 2005, 367-389. (ISI journal)
    26. 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. (ISI journal)
    27. Holomorphic maps for which the unstable manifolds depend on prehistories, joint with M. Urbanski, Discrete and Continuous Dynamical Systems, vol.9, no.2, 2003. (ISI journal)
    28. The set K- for hyperbolic non-invertible maps, Ergodic Theory and Dynamical Systems, June 2002, 3, 873-888. (ISI journal)
    29. Applications of thermodynamic formalism in complex dynamics on P2, Discrete and Continuous Dynamical Systems, vol. 7, 4, October 2001, 821-836. (ISI journal)
    30. Periodic points for actions of tori in Stein manifolds, Mathematische Annalen 314, 1999, no 1, 39-52. (ISI journal)
    31. PH.D Thesis, Univ. of Michigan Press 1999.
    32. A class of functions which are continuous but nowhere monotonous, Annals of the University of Bucharest, 42/43, 1993/94, 63-71.
    33. On some topologies on the space of analytic coherent subsheaves, Revista Matematica TUNTCD, 1994, Bucuresti

    Curriculum Vitae

    Full version of my CV.

    Research description
    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 Chaos Theory, Ergodic Theory and Dynamics on Fractals (saddle basic sets, hyperbolicity, chaotic behavior, complex dynamics in one or several variables, Lyapunov exponents, equilibrium measures, iterated function systems, hyperbolic flows, attractors, etc.).
    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.

    Teaching

    During the academic year 2009-2010 I taught 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 at the email address above, 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).
    At SNSB I directed several theses of students, and have been also on several PH.D Thesis Committees.
    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.

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    Created initially August 2000.