Conditional shadowing property

The main property of dynamical systems studied by the shadowing theory can be stated as follows. Consider a homeomorphism $f$ of a metric space $(X,\mathrm{dist})$. Let $d>0$. A sequence $\{y_n\in X\}$ is called a $d$-pseudotrajectory of $f$ if the inequalities
\begin{equation} \mathrm{dist}(f(y_n),y_{n+1})<d \end{equation}
hold.
One says that $f$ has the (standard) shadowing property if for any $\varepsilon>0$ there is a $d>0$ such that for any $d$-pseudotrajectory $\{y_n\in X\}$ of $f$ there is a point $x\in X$ for which
$$ \mathrm{dist}(f^n(x),y_n)<\varepsilon. $$

Usually, the shadowing property is a corollary of some kind of hyperbolicity of $f$ (see [1-3]). At the same time, the shadowing theory studies many properties different from the standard shadowing property that are not closely related to hyperbolicity.
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 $x$ such that
$$ \mathrm{dist}(f^n(x),y_n)\to 0,\quad n\to\infty. $$

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 $\mathrm{dist}(f(y_n),y_{n+1})$).
In the paper [5], the authors studied shadowing of pseudotrajectories near a nonisolated fixed point $p$ of a diffeomorphism $f$; in this case, the smallness of the values $\mathrm{dist}(f(y_n),y_{n+1})$ had been related to the values $\mathrm{dist}(y_n,p)$.
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 one-step 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).