#### Jürgen Jost, Oliver Pfante

Paper #: 15-02-004

We quantify the relationship between the dynamics of a time-discrete dynamical system, driven by a uni- modular map *T* : [0,1] → [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

*α*∈ [0,1] and putting

*s*= 1 or 0 depending on whether

^{i}*T*(

^{i}*x*) is larger or smaller than

*α*.

First, we investigate how the choice of the partition point

*α*determines which symbolic sequences are forbidden, that is, cannot occur in the symbolic dynamics. The periodic points of

*T*mark the choices of

*α*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

*α*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

*α*and can be quantified in terms of the closure measure.