Authors: José María Amigó ¹; Sergi Elizalde ²; Matthew B. Kennel ³

• 1 Centro de Investigacion Operativa
• 2 Department of Mathematics [Dartmouth]
• 3 Institute for Nonlinear Science

Orbits generated by discrete-time dynamical systems have some interesting combinatorial properties. In this paper we address the existence of forbidden order patterns when the dynamics is generated by piecewise monotone maps on one-dimensional closed intervals. This means that the points belonging to a sufficiently long orbit cannot appear in any arbitrary order. The admissible patterns are then (the inverses of) those permutations avoiding the so-called forbidden root patterns in consecutive positions. The last part of the paper studies and enumerates forbidden order patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. In spite of their simple structure, shift systems exhibit all important features of low-dimensional chaos, allowing to export the results to other dynamical systems via order-isomorphisms. This paper summarizes some results from [1] and [11].