Symbolic Dynamics of Order-Preserving Orbits
Physica D: Nonlinear Phenomena
The maps we consider are roughly those that can be obtained by truncating non-invertible maps to weakly monotonic maps (they have a flat piece). The binary sequences that correspond to order-preserving orbits are shown to satisfy a minimax principle (which was already known for order-preserving orbits with rational rotation number). The converse is also proven: all minimax orbits are order-preserving with respect to some rotation number.
For certain families of such circle maps one can solve exactly for the parameter values for which the map has a specified rotation number rho. For rho rational we obtain the endpoints of the resonance intervals recursively. These parameter values can be organized in a natural way as the nodes of a Farey tree. We give some applications of the ideas discussed.
Copyright © 1987 Published by Elsevier B.V.
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Veerman, P. (1987). Symbolic dynamics of order-preserving orbits. Physica D: Nonlinear Phenomena, 29(1-2), 191-201.