This monograph explores the very general question of the dynamics of generic homeomorphisms on certain compact metric spaces \(X\). The group \(H(X)\) of homeomorphisms on \(X\) is a topological group with a separable and completely metrizable topology, and hence has the Baire property that any countable intersections of open dense sets is dense (i.e., is a residual set). The authors say that a property \(P\) is satisfied by a generic homeomorphism if the set of \(f\in H(X)\) that satisfy \(P\) contains a residual set.
It is perhaps unsurprising that the dynamics arising in this general situation is complex, given that the category of continuous functions is so much broader than the category of smooth functions. It is surprising than the dynamics is as complicated as it turns out to be. For example, if \(X\) is a compact, smooth manifold without boundary of dimension two or greater, and if \(f\) is a generic map on \(X\) with attractor \(A\), then: (a) \(A\) contains infinitely many repellors for \(f\); (b) the interior of \(A\) is non-empty; (c) \(A\) is a union of the basins of repulsion for the repellors contained in \(A\); (d) the topological boundary \(\partial A\) is a quasi-attractor (the intersection of a sequence of attractors, but not an attractor), and analogously for repellors; and (e) there are uncountably many distinct sequences \(A_1\supset R_1\supset A_2\supset R_2\supset \dots\) with \(A=A_1\) where the \(A_i\) are attractors and the \(R_i\)’s are repellors.
The authors’ approach and techniques are essentially topological; they build tools to characterize generic homeomorphisms, and in the process identity properties that are immune to the fairly violent perturbations possible in the category of continuous functions. The authors include a section that applies their analysis to the circle \(\widehat{S}'\) where the generic dynamics are not as complicated as in the more general case, and more intuitively accessible.