Route to chaotic synchronisation in coupled map lattices: Rigorous results
B. Fernandez P. Guiraud
Discrete & Continuous Dynamical Systems - B 2004, 4(2): 435-456 doi: 10.3934/dcdsb.2004.4.435
Two-dimensional mappings obtained by coupling two piecewise increasing expanding maps are considered. Their dynamics is described when the coupling parameter increases in the expanding domain. By introducing a coding and by analysing an admissibility condition, upper and lower bounds of the corresponding symbolic systems are obtained. As a consequence, the topological entropy is located between two decreasing step functions of the coupling parameter. The analysis firstly applies to mappings with piecewise affine local maps which allow explicit expressions and, in a second step, is extended by continuity to mappings with piecewise smooth local maps.
keywords: Symbolic systems topological entropy.
Spectrum of dimensions for Poincaré recurrences of Markov maps
B. Fernandez E. Ugalde J. Urías
Discrete & Continuous Dynamical Systems - A 2002, 8(4): 835-849 doi: 10.3934/dcds.2002.8.835
The spectrum of dimensions for Poincaré recurrences of Markov maps is obtained by constructing a sequence of approximating maps whose spectra are known to be solution of non-homogeneous Bowen equations. We prove that the spectrum of the Markov map also satisfies such an equation.
keywords: Markov maps Poincaré recurrences spectra of dimensions.

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