
ВИДЕОТЕКА 
Конференция «Hyperbolic Dynamics and Structural Stability», посвященная 85летию Д. В. Аносова



Conditional shadowing property S. Yu. Pilyugin^{} ^{} St. Petersburg State University, Mathematics and Mechanics Faculty 

Аннотация: The main property of dynamical systems studied by the shadowing theory can be stated as follows. Consider a homeomorphism \begin{equation} \mathrm{dist}(f(y_n),y_{n+1})<d \end{equation} hold. One says that $$ \mathrm{dist}(f^n(x),y_n)<\varepsilon. $$ Usually, the shadowing property is a corollary of some kind of hyperbolicity of Let us mention, for example, the limit shadowing property [4]; in this case, inequalities (1) are replaced by the relations $$ \mathrm{dist}(f(y_n),y_{n+1})\to 0,\quad n\to\infty, $$ and one looks for a point $$ \mathrm{dist}(f^n(x),y_n)\to 0,\quad n\to\infty. $$ Let us mention one more example of "conditional" shadowing (here the term "conditional" means that the uniform estimate (1) is replaced by particular conditions on the smallness of the values In the paper [5], the authors studied shadowing of pseudotrajectories near a nonisolated fixed point Finally, let us mention the research of [6] devoted to conditional shadowing for a nonautonomous system whose linear part satisfies some conditions generalizing nonuniform hyperbolicity. In this talk, we study conditional shadowing for a nonautonomous system in a Banach space assuming that the linear part admits a family of invariant subspaces (scale) with different behavior of trajectories. Conditions of shadowing are formulated in terms of smallness of the projections of onestep errors to the scale and of smallness of Lipschitz constants of the projections of nonlinear terms. We also give conditions under which a system has the conditional property of inverse shadowing (dual to the shadowing property). The main results of the talk are published in [7]. [1] S.Yu. Pilyugin, Shadowing in Dynamical Systems, Lect. Notes Math., Vol. 1706, Springer (1999). [2] K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Kluwer (2000). [3] S.Yu. Pilyugin, K. Sakai, Shadowing and Hyperbolicity, Lect. Notes Math., Vol. 2193, Springer (2017). [4] T. Eirola, O. Nevanlinna, S.Yu. Pilyugin, Limit shadowing property. Numer. Funct. Anal. Optim., 18 (1997), 75–92. [5] A.A.Petrov, S.Yu. Pilyugin, Shadowing near nonhyperbolic fixed points. Discrete Contin. Dyn. Syst., 34 (2014), 3761–3772. [6] L. Backes, D. Dragicevic, A general approach to nonautonomous shadowing for nonlinear dynamics. Bull. Sci. Math., 170 (2021). [7] S. Yu. Pilyugin, Multiscale conditional shadowing. J. of Dynamics and Diff. Equations (2021). Язык доклада: английский 