MiS Preprint Repository

We quantify the relationship between the dynamics of a time-discrete dynamical system, driven by a unimodular map $T:[0,1]\to [0,1]$ 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.