Recurrent and gradient dynamics of Iterated Function Systems

Abstract

We study the qualitative dynamics of general Iterated Function Systems (IFSs) through chain recurrence and a decomposition into recurrent and gradient-like behavior. At the same time, we move the focus from the topological property of the phase space to the dynamical properties of the system and we say that an IFS has compact dynamics if it has an invariant compact set (global attractor) that attracts every compact subset of the phase space. For an IFS with compact dynamics, we introduce a distinction that has no counterpart for a single map: ordinary chain recurrence, which may be realized along a suitable itinerary, and orbitwise chain recurrence, which requires the whole semigroup orbit of a point to remain in its chain component. This leads naturally to two directed graphs encoding the recurrent components and the downstream relations between them. We prove that the chain structure of an IFS with compact dynamics over a locally compact space is completely determined by the restriction of the IFS to its compact global attractor. We then obtain a Conley-type trichotomy: every point of the global attractor has either orbitwise recurrent dynamics, mixed recurrent-gradient dynamics, or purely gradient dynamics. The three alternatives admit equivalent descriptions in terms of bitrajectories and strict Lyapunov functions. We also give conditions guaranteeing connectedness of the chain graph and of the orbitwise chain graph. Finally, we compare the point dynamics of the IFS with two associated single-map systems and establish a localization principle extending several compact-space results from Conley theory for closed relations to IFSs whose global attractor has a compact neighborhood. This is a second and highly modified version of the original manuscript. Most of the examples present in the first version have been removed and will appear soon in a second manuscript.