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| For a topological dynamical system $(X,f)$ we consider the structure of the set $\mathcal F(f)$ of probability distribution functions of the distances between two trajectories. If $f$ has the weak specification property then $\mathcal F(f)$ is closed and convex with a unique minimal element, and it can contain all nondecreasing functions $[0,{\rm diam}X]\to [0.1]$. Based on this we propose a new notion of chaos. Note that these problems cannot be easily studied by standard tools like the ergodic theorem. |
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