``Chaos unimportant'' claims topologist (Q1343662)

The object of this brief paper is to point out two different perspectives on chaos which may be held by an analyst and a topologist. Suppose that a discrete time dynamical system is modeled by iterating a continuous map \(F : X \to X\), where \(X\) is a metric space. The author notes that problems of chaos are essentially only asymptotic concerns, and may be deferred arbitrarily to infinity by replacing the metric on \(X\) by one which gives an equivalent topology but postpones chaotic behavior indefinitely. Of course, the main issue is the appropriateness of the equivalent metric to the analyst. In most cases, the appropriate notion of equivalence is likely to be local Lipschitz or local Hölder equivalence, and postponing chaos by topological slight-of-hand is simply not useful.