Measures related to $(\epsilon,n)$-complexity functions
Valentin Afraimovich Lev Glebsky
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.
keywords: separability. Topological entropy complexity functions
Measures related to metric complexity
Valentin Afraimovich Lev Glebsky Rosendo Vazquez
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 $n$.
keywords: Complexity functions separability invariant measures.
Directional complexity and entropy for lift mappings
Valentin Afraimovich Maurice Courbage Lev Glebsky
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.
keywords: directional entropy. space-time window Rotation interval directional complexity

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