A general approach to stability and sensitivity in dynamical systems

Suppose $(X,d)$ is a metric space and $f:X\to X$ a continuous map. Let $\sum = X^{\N}$ denote the set of all sequences of elements of $X$. $E_f: X\to\sum$ is given by $E_f(x) = (x,f(x),f^2(x),\ldots).$ $E_f(x)$ is called the trajectory or time evolution of $f$ at $x$. Let $\mathcal T$ be a toplogy on $\sum$. We define $f$ to be $\mathcal T$-stable ($\mathcal T$-sensitive) at $x$ if $E_f$ is $\mathcal T$-continuous at $x$ (if $E_f$ is $\mathcal T$-discontinuous at $x$). We construct topologies on $\sum$ by using a generalised notion of a metric on $\sum$ which we call a sensitivity function. We show that the classical notions of stability due to Liapunov, Birkhoff, Lefschetz and Poisson can be expressed in terms of suitably chosen sensitivity functions. This approach unifies old ideas and suggests new notions of sensitivity all stronger than that due to Liapunov. The mutual implications of these various notions are discussed in detail. Throughout, the analysis is in terms of elementary topology.