abstr2
This work deals with dynamical systems given by mappings, i.e., with the iteration behavior of mappings. Such discrete-time systems can
emerge directly as models for physical or biological phenomena. The flows or semiflows generated by ordinary differential equations
(ODEs), functional differential equations (FDEs) and parabolic partial differential equations (parabolic PDEs) are models with continuous
time. Mappings arise naturally also in the study of continuous time systems; we mention the most frequently occurring examples: The time-T
map for an ODE where the right hand side is T-periodic in time, and the return map (Poincaré map) associated with a periodic orbit of a
(semi-)flow and a hyperplane transversal to this orbit.
In the ODE case, such mappings are finite-dimensional diffeomorphisms. In the case of FDEs and parabolic PDEs, the state space is infinite
dimensional and the mappings are typically compact, so they cannot have a continuous inverse, even if they are one-to-one. It is not difficult
to construct examples of FDEs, e.g., of the type x(t) = g(x(t-1)) (g: R --> R ), where different
initial states lead to forward solutions that coincide after a finite time. They are most easily obtained taking g
constant on some interval. These examples show that mappings related to infinite-dimensional semiflows
typically fail to be diffeomorphic and may even be noninjective.
The purpose of the present work ist to extend important ideas from the mathematical field `Hyperbolic Sets' to noninvertible mappings in
Banach spaces.