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Abstract
Metric complexity functions measure an amount of instability of trajectories
in dynamical systems acting on metric spaces. They reflect an ability
of trajectories to diverge by the distance of $\epsilon$
during the time interval $n$. This ability depends on the position of
initial points in the phase space, so, there are some distributions of initial
points with respect to these features that present themselves in the form of
Borel measures. There are two approaches to deal with metric complexities:
the one based on the notion of $\epsilon$-nets ($\epsilon$-spanning) and
the other one defined through $\epsilon$-separability. The last one has been
studied in [1, 2]. In the present article we concentrate on the
former. In particular, we prove that the measure is invariant if
the complexity function grows subexponentially in $n$.
Mathematics Subject Classification: 28C15, 37C99.
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