Markov Partitions and Driven IFS

The s = 4.0 logistic map and the s = 2.0 tent map have driven IFS determined entirely by their forbidden pairs because of a particular relation between the graphs and bins.

For both graphs, note each bin on the bottom of the unit square goes to a union of complete bins on the side of the square. For the logistic map L(X), note

L(bin1) = bin1 U bin2 U bin3

L(bin2) = bin4

L(bin3) = bin4

L(bin4) = bin1 U bin2 U bin3

and for the tent map T(x)

T(bin1) = bin1 U bin2

T(bin2) = bin3 U bin4

T(bin3) = bin3 U bin4

T(bin4) = bin1 U bin2

This relation between graphs and bins is called a Markov partition of the trapping square. For a Markov partition, the driven IFS is always determined by its empty length 2 subsquares. In this way, these are the simplest of driven IFS. One way to measure the complexity of a dynamical process is to find the smallest collection of empty subsquares needed to generate the driven IFS. This "grammatical complexity" of processes is an area of considerable research.

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