Motions in semidynamical systems. (Russian) Zbl 0567.54023

Let (X,\(\rho)\) be a metric space and C(X) the collection of all nonempty compact subsets of X. For A,B\(\in C(X)\) the following distances are defined: \[ \rho (a,B)=\inf \{\rho (a,b);\quad b\in B\}, \]
\[ \beta (A,B)=\sup \{\rho (a,B);\quad a\in A\} \] and \[ \alpha (A,B)=\max \{\beta (A,B);\quad \beta (B,A)\}. \] A mapping \(f: X\times R^+\to C(X)\) is said to define a quasi-semidynamical system on X if it satisfies the axioms: (1) \(f(x,0)=x\) for all \(x\in X\) and (2) \(f(f(x,t_ 1),t_ 2)=f(x,t_ 1+t_ 2)\) for all \(x\in X\); \(t_ 1,t_ 2\in R^+\). It is called a semidynamical system if additionally \((4)\quad \lim_{t\to t_ 0}\alpha (f(x,t),f(x,t_ 0))=0.\) A (quasi-) semidynamical system is called semicontinuous with respect to x if \((3)\quad \lim_{x\to x_ 0}\beta (f(x,t),f(x_ 0,t))=0.\) For these types of systems the author studies problems related to the existence and the continuity of motions or the classification of maximal motions.
Reviewer: Gh.Toader


54H20 Topological dynamics (MSC2010)
Full Text: EuDML