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].


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