A Fixed Point Theorem for Bounded Dynamical Systems

We show that a continuous map or a continuous flow on Rn with a certain recurrence relation must have a fixed point. Specifically, if there is a compact set W with the property that the forward orbit of every point in R n intersects W then there is a fixed point in W . Consequently, if the omega limit set of every point is nonempty and uniformly bounded then there is a fixed point. In this note we will prove a fixed point theorem that holds for both discrete dynamical systems (f is a continuous map) and continuous dynamical systems (φ is a continuous flow). We will show that if every point in R returns to a compact set, then there must be a fixed point. This investigation began in an attempt to answer a related question about smooth flows posed by Richard Schwartz. We will prove the theorem for maps, and derive the theorem for flows as a consequence. In fact, many of the definitions and proofs follow analogously for both cases. When that is the case we will just refer to the “dynamical system,” with the recognition that the statement applies to both flows and maps. Where necessary, separate definitions and proofs will be included. We now introduce the notion of a window; a compact set W is a window for a dynamical system on X if the forward orbit of every point x ∈ X intersects W . If a dynamical system has a window then we will say that it is bounded. We will prove the following fixed point theorem. Theorem 1. Every bounded dynamical system on R has a fixed point. The following corollaries are elementary applications of Theorem 1. Corollary 2. If W is a window for a dynamical system on R, then there is a fixed point in W . Corollary 3. If there is a compact set K such that ∅ 6= ω(x) ⊂ K for all x ∈ R, then the dynamical system has a fixed point. We will need the following theorem. Theorem 4 (Lefschetz Fixed Point Theorem). Let f : M → M be a continuous map of an n-dimensional manifold (with or without boundary), and let fk : Hk(M ;R) → Hk(M ;R) be the induced map on homology. If ∑n k=0(−1) k tr(fk) 6= 0, then f has a fixed point. The main result in this paper concerns dynamical systems on R, but some of the results hold more generally. Unless otherwise stated, our dynamical system is defined on a locally compact topological space. Date: March 6, 2008. 1991 Mathematics Subject Classification. [.