# American Institute of Mathematical Sciences

December  2010, 28(4): 1299-1309. doi: 10.3934/dcds.2010.28.1299

## Measures related to metric complexity

 1 Av. Karakorum 1470, Lomas 4, San Luis Potosi, C.P.78210, SLP, Mexico, Mexico, Mexico

Received  October 2009 Revised  February 2010 Published  June 2010

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 $n$.
Citation: Valentin Afraimovich, Lev Glebsky, Rosendo Vazquez. Measures related to metric complexity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1299-1309. doi: 10.3934/dcds.2010.28.1299
 [1] Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167 [2] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444 [3] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048 [4] Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169 [5] Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $(n, m)$-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117 [6] Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015

2019 Impact Factor: 1.338