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MiS Preprint

Non-generating Partitions of Unimodular Maps

Oliver Pfante and Jürgen Jost


We quantify the relationship between the dynamics of a time-discrete dynamical system, driven by a unimodular map $T:[0,1] \rightarrow [0,1]$ on the unit interval and its iterations $T^{m}$, and the induced dynamics at a symbolic level in information theoretical terms. The symbolic dynamics are obtained by a threshold crossing technique. A binary string $s$ of length $m$ is obtained by choosing a partition point $\alpha \in [ 0,1 ]$ and putting $s^{i}=1 $ or $0$ depending on whether $T^{i}(x)$ is larger or smaller than $\alpha$. First, we investigate how the choice of the partition point $\alpha$ determines which symbolic sequences are forbidden, that is, cannot occur in the symbolic dynamics. The periodic points of $T$ mark the choices of $\alpha$ where the set of those forbidden sequences changes. Second, we interpret the original dynamics and the symbolic ones as different levels of a complex system. This allows us to quantitatively evaluate a closure measure that has been proposed for identifying emergent macro-levels of a dynamical system. In particular, we see that this measure necessarily has its local minima at those choices of $\alpha$ where also the set of forbidden sequences changes. Third, we study the limit case of infinite binary strings and interpret them as a series of coin tosses. These coin tosses are not i.i.d. but exhibit memory effects which depend on $\alpha$ and can be quantified in terms of the closure measure.

MSC Codes:
70-XX, 37-XX, 37B10
chaotic dynamics, information theory, aggregation

Related publications

2015 Repository Open Access
Oliver Pfante and Jürgen Jost

Non-generating partitions of unimodular maps