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**Open questions leading to a global perspective in dynamics.**
*(English)*
Zbl 1147.37010

Nonlinearity 21, No. 4, T37-T43 (2008); corrigendum ibid. 28, No. 3, C1 (2015).

This useful overview paper contains the author’s meditation about the following question: can we describe the behaviour in the long run of typical trajectories for the “majority” of systems? By the system, the author means \(C^r\) \((r\geq1)\) flows, diffeomorphisms or transformations defined on a smooth compact, boundaryless manifold \(M\). To this aim the author wrote the following so called Main global conjecture:

- [1
- ] There is a dense set \(D\) of dynamics such that any element of \(D\) has finitely many attractors whose union of basins of attraction has total probability.
- [2
- ] The attractors of the elements in \(D\) support a physical (SRB) measure.
- [3
- ] For any element in \(D\) and any of its attractors, for almost all small perturbations in generic \(k\)-parameter families of dynamics, \(k\in\mathbb{N}\), there are finitely many attractors whose union of basins is nearly (Lebesgue) equal to the basins of the initial attractors; each such perturbed attractor supports a physical measure.
- [4
- ] Stochastic stability of attractors: the attractors of elements in \(D\) are stochastically stable in their basins of attraction.
- [5
- ] For generic finite-dimensional families of dynamics, with total probability in parameter space, the corresponding systems display attractors satisfying the properties above.

Reviewer: Alois Klíč (Praha)