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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.