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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.
The further text contains the history of the results from this part of dynamics as well as the contemporary state.

MSC:

37B25 Stability of topological dynamical systems
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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