Closure measures for coarse-graining of the tent map

We quantify the relationship between the dynamics of a time-discrete dynamical system, the tent map T and its iterations T-m, and the induced dynamics at a symbolical level in information theoretical terms. The symbol dynamics, given by a binary string s of length m, is obtained by choosing a partition point alpha is an element of [0,1] and lumping together the points x is an element of [0,1] s.t. T-i (x) concurs with the i - 1th digit of s-i.e., we apply a so called threshold crossing technique. Interpreting the original dynamics and the symbolic one as different levels, this allows us to quantitatively evaluate and compare various closure measures that have been proposed for identifying emergent macro-levels of a dynamical system. In particular, we can see how these measures depend on the choice of the partition point alpha. As main benefit of this new information theoretical approach, we get all Markov partitions with full support of the time-discrete dynamical system induced by the tent map. Furthermore, we could derive an example of a Markovian symbol dynamics whose underlying partition is not Markovian at all, and even a whole hierarchy of Markovian symbol dynamics. (C) 2014 AIP Publishing LLC.