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### Open Access Journals

DCDS

The $(\epsilon,n)$-complexity functions describe total instability
of trajectories in dynamical systems. They reflect an ability of
trajectories going through a Borel set to diverge on the distance
$\epsilon$ during the time interval $n$. Behavior of the $(\epsilon,
n)$-complexity functions as $n\to\infty$ is reflected in the
properties of special measures. These measures are constructed as
limits of atomic measures supported at points of
$(\epsilon,n)$-separated sets. We study such measures. In
particular, we prove that they are invariant if the
$(\epsilon,n)$-complexity function grows subexponentially.

DCDS

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 $n$.

DCDS-B

We introduce and study the notion of a directional complexity and entropy for maps of degree $1$
on the circle. For piecewise affine Markov maps we use symbolic dynamics to relate this complexity to the symbolic complexity. We apply a combinatorial machinery to obtain exact
formulas for the directional entropy, to find the maximal directional entropy, and to show that
it equals the topological entropy of the map.

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