The research: Differential Equations and Dynamical Systems

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Dynamical Systems were pioneered in Warsaw by Krzyzewski and Szlenk (a former student of Mazur). Courses taught by Yakov Sinai during his visits to Warsaw in the late 60's had a big impact on the emerging community. Also Misiurewicz, Swiatek and Wojtkowski - invited speakers at ICM's - must be mentioned. Now the research group (Baranski, Przytycki, Rams, Zdunik) is focused on 1-dimensional dynamics, real and complex (iteration of holomorphic maps), higher dimension non-conformal features, small scale geometry of limit sets of iterated function systems, as well as non-uniform hyperbolicity problems. This involves understanding of the topological structure and geometry of invariant subsets for these maps as well as dynamically defined subsets in the parameter space. Related areas under consideration are random dynamical systems and their invariant sets, structure of deterministic and random repellers (attractors), and fractal geometry. Methods used belong to ergodic theory, thermodynamical formalism, and geometric measure theory. This research group (jointly with groups in probability and analysis) ran in 2006-2010 two EU Marie Curie programs hosted at IMPAN in cooperation with MIMUW: Deterministic and Stochastic Dynamics Fractals, Turbulence (SPADE2 within the Transfer of Knowledge) and the Warsaw node of the network Conformal Structures and Dynamics (CODY).

A very broad range of research topics is considered by researchers dealing with differential equations - here the group owes a lot to the broad expertise of its creator, Bojarski - and related fields. The current topics include nonlinear elliptic systems and geometric analysis (Strzelecki), Gagliardo-Nirenberg and Hardy inequalities (Kalamajska, Pietruska-Paluba) as well as generating functions for multiple zeroes of zeta function, geometry of complex affine plane, and - last but definitely not least - the study of limit cycles in various systems (Zoladek, Bobienski). For example, one of the research directions concerns Hilbertís 16-th problem about the number and location of limit cycles of a planar polynomial vector field. In the neighbourhood of an integrable system, the problem reduces to the investigation of zeroes of Abelian integrals, i.e., integrals of a rational 1-form along a cycle which is contained in a leaf of an integrable foliation. The methods used in this study are algebraic properties of the holonomy group along algebraic invariant curves and linear ordinary differential equation satisfied by these integrals.