Linking Combinatorial and Classical Dynamics: Conley Index and Morse Decompositions

We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex. Moreover, F may be chosen in such a way that the isolated invariant sets, Conley indices, Morse decompositions and Conley–Morse graphs of the combinatorial vector field give rise to isomorphic objects in the multivalued map case.

On One 2-Valued Transformation: Its Invariant Measure and Application to Masked Dynamical Systems

We consider one familySof 2-valued transformations on the interval [0, 1] with measureμ, endowed with a set of weight functions. We construct invariant measureμS=μfor this multivalued dynamical system with weights and show the interplay between such systems and masked dynamical systems, which leads to image processing.

Attractors for Nonautonomous Multivalued Evolution Systems Generated by Time-Dependent Subdifferentials

In a real separable Hilbert space, we consider nonautonomous evolution equations including time-dependent subdifferentials and their nonmonotone multivalued perturbations. In this paper, we treat the multivalued dynamical systems associated with time-dependent subdifferentials, in which the solution is not unique for a given initial state. In particular, we discuss the asymptotic behaviour of our multivalued semiflows from the viewpoint of attractors. In fact, assuming that the time-dependent subdifferential converges asymptotically to a time-independent one (in a sense) as time goes to infinity, we construct global attractors for nonautonomous multivalued dynamical systems and its limiting autonomous multivalued dynamical system. Moreover, we discuss the relationship between them.