How does a choice of Markov partition affect the resultant symbolic dynamics?

The mutual relationship among Markov partitions is investigated for one-dimensional piecewise monotonic map. It is shown that if a Markov partition is regarded as a map-refinement of the other Markov partition, that is, a concept we newly introduce in this article, one can uniquely translate a set of symbolic sequences by one Markov partition to those by the other or vice versa. However, the set of symbolic sequences constructed using Markov partitions is not necessarily translated with each other if there exists no map-refinement relation among them. By using a roof map we demonstrate how the resultant symbolic sequences depend on the choice of Markov partitions. (C) 2010 American Institute of Physics. [doi:10.1063/1.3491097]