Alexey Osipov – Saint Petersburg State University
Abstract: The theory of shadowing studies the problem of closeness of approximate and exact trajectories of dynamical systems. A dynamical system has some shadowing property if any sufficiently precise approximate trajectory is, in some sense, close to some exact trajectory. Since the notions of ''an approximate trajectory'' and ''being close'' can be formalized in several ways, various shadowing properties can be introduced. We will try to describe state-of-the-art of theory of shadowing mostly concentrating on the case of discrete time invertible dynamical systems. We will talk about the problem of genericity (in Baire's sense) of discrete time dynamical systems having some shadowing property (the standard shadowing property, orbital shadowing property, or weak shadowing properties). In particular, we will sketch the proof of $C^1$-nondensity of orbital shadowing property. We will study relations between hyperbolicity and various shadowing properties. For example, we will outline the proof of equivalence of $Omega$-stability and so-called Lipschitz periodic shadowing property. It the end of the talk, we will probably discuss discretisations of discrete time dynamical systems or shadowing properties for the case of continuous time dynamical systems with finite time blow-up.
PhD in Mathematics Applied in Economics and Management
CEMAPRE, ISEG, Technical University of Lisbon
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