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Distributions of Order Patterns of Interval Maps

Published online by Cambridge University Press:  05 March 2013

AARON ABRAMS ,
ERIC BABSON ,
HENRY LANDAU ,
ZEPH LANDAU  and
JAMES POMMERSHEIM
AARON ABRAMS
Affiliation:
Washington and Lee University (e-mail: abramsa@wlu.edu)
ERIC BABSON
Affiliation:
University of California, Davis (e-mail: babson@math.ucdavis.edu)
HENRY LANDAU
Affiliation:
AT&T Research (e-mail: henryj.landau@gmail.com)
ZEPH LANDAU
Affiliation:
University of California, Berkeley (e-mail: zeph.landau@gmail.com)
JAMES POMMERSHEIM
Affiliation:
Reed College (e-mail: jamie@reed.edu)
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Abstract

A permutation σ describing the relative orders of the first n iterates 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 σ 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, that is, 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 of S. Elizalde.

Keywords

Primary 37E05Secondary 37A0537B40

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 3 , May 2013 , pp. 319 - 341
Copyright
Copyright © Cambridge University Press 2013

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