Measures related to $(\epsilon,n)$-complexity functions

Valentin Afraimovich; Lev Glebsky

doi:10.3934/dcds.2008.22.23

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2008, Volume 22, Issue 1&2: 23-34. Doi: 10.3934/dcds.2008.22.23

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Measures related to $(\epsilon,n)$-complexity functions

Valentin Afraimovich^1, and
Lev Glebsky^1,

1.
IICO-UASLP, Karakorum 1470, Lomas 4a, 78210, San Luis Potosi, S.L.P.

Received: May 2007

Revised: September 2007

Published: March 2008

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Abstract

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.

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Mathematics Subject Classification: 28C15, 37C99.

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