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Complexity of injective piecewise contracting interval maps

Published online by Cambridge University Press:  04 June 2018

E. CATSIGERAS ,
P. GUIRAUD  and
A. MEYRONEINC
E. CATSIGERAS
Affiliation:
Instituto de Matemática y Estadística Rafael Laguardia, Universidad de la República, Montevideo, Uruguay email eleonora@fing.edu.uy
P. GUIRAUD
Affiliation:
Centro de Investigación y Modelamiento de Fenómenos Aleatorios Valparaíso, Facultad de Ingeniería, Universidad de Valparaíso, Valparaíso, Chile email pierre.guiraud@uv.cl
A. MEYRONEINC
Affiliation:
Departamento de Matemáticas, Instituto Venezolano de Investigaciones Científicas, Apartado 20632, Caracas 1020A, Venezuela email ameyrone@ivic.gob.ve
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Abstract

We study the complexity of the itineraries of injective piecewise contracting maps on the interval. We prove that for any such map the complexity function of any itinerary is eventually affine. We also prove that the growth rate of the complexity is bounded from above by the number, $N-1$, of discontinuities of the map. To show that this bound is optimal, we construct piecewise affine contracting maps whose itineraries all have the complexity $(N-1)n+1$. In these examples, the asymptotic dynamics take place in a minimal Cantor set containing all the discontinuities.

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Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 1 , January 2020 , pp. 64 - 88
Copyright
© Cambridge University Press, 2018 

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