## Multivalued spatial discretization of dynamical systems.(English)Zbl 0843.54041

Martin, Gaven (ed.) et al., Proceedings of the miniconference on analysis and applications, held at the University of Queensland, Brisbane, Australia, September 20-23, 1993. Canberra: Australian National University, Centre for Mathematics and its Applications. Proc. Cent. Math. Appl. Aust. Natl. Univ. 33, 61-70 (1994).
Standard computer models of the dynamical system given by a map $$f$$ from a compact metric space $$\Omega$$ into itself are dynamical systems $$\psi$$ defined on a finite subset $$L$$ of $$\Omega$$. The behaviour of $$\psi$$ is possibly degenerate, collapsing. To eliminate this effect one can try to model the system $$f$$ by a multivalued perturbation of it.
The author considers dyamical systems generated by (multivalued) Borel mappings $$f: \Omega\to {\mathcal B}$$ where $${\mathcal B}$$ is the family of Borel subsets of $$\Omega$$. Then they consider a map $$\varphi: L\to 2^L$$ as a discretization of the system $$f$$. Results are obtained concerning the approximation of trajectories of the system $$f$$ by trajectories of the system $$\varphi$$ as well as those concerning the approximation of semi-invariant measures and semi-ergodic measures. Shadowing in multivalued systems is also studied.
For the entire collection see [Zbl 0816.00015].

### MSC:

 54H20 Topological dynamics (MSC2010) 37A99 Ergodic theory 54C60 Set-valued maps in general topology