# American Institute of Mathematical Sciences

December  2015, 20(10): 3385-3401. doi: 10.3934/dcdsb.2015.20.3385

## Directional complexity and entropy for lift mappings

 1 Instituto de Investigación en Comunicación Óptica, Universidad Autónoma de San Luis Potosí, Karakorum 1470, Lomas 4a 78220, San Luis Potosi, S.L.P, Mexico 2 Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS et Université Paris 7-Denis Diderot, 10, rue Alice Domon et Léonie Duquet 75205 Paris Cedex 13, France 3 Instituto de Investigación en Comunicación Óptica, Universidad Autónoma de San Luis Potos, Karakorum 1470, Lomas 4a 78220, San Luis Potosi, S.L.P, Mexico

Received  December 2014 Revised  March 2015 Published  September 2015

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.
Citation: Valentin Afraimovich, Maurice Courbage, Lev Glebsky. Directional complexity and entropy for lift mappings. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3385-3401. doi: 10.3934/dcdsb.2015.20.3385
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