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Summary: DISTRIBUTIONS OF ORDER PATTERNS OF INTERVAL MAPS
AARON ABRAMS, ERIC BABSON, HENRY LANDAU, ZEPH LANDAU, JAMES
POMMERSHEIM
Abstract. A permutation describing the relative orders of the first n iter-
ates of a point x under a self-map f of the interval I = [0, 1] is called an order
pattern. For fixed f and n, measuring the points x I (according to Lebesgue
measure) that generate the order pattern gives a probability distribution
µn(f) on the set of length n permutations. We study the distributions that
arise this way for various classes of functions f.
Our main results treat the class of measure preserving functions. We obtain
an exact description of the set of realizable distributions in this case: for each
n this set is a union of open faces of the polytope of flows on a certain digraph,
and a simple combinatorial criterion determines which faces are included. We
also show that for general f, apart from an obvious compatibility condition,
there is no restriction on the sequence {µn(f)}n=1,2,....
In addition, we give a necessary condition for f to have finite exclusion
type, i.e., for there to be finitely many order patterns that generate all order
patterns not realized by f. Using entropy we show that if f is piecewise
continuous, piecewise monotone, and either ergodic or with points of arbitrarily
high period, then f cannot have finite exclusion type. This generalizes results
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