{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,8,24]],"date-time":"2023-08-24T11:20:25Z","timestamp":1692876025396},"reference-count":10,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2013,3,5]],"date-time":"2013-03-05T00:00:00Z","timestamp":1362441600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2013,5]]},"abstract":"<jats:p>A permutation \u03c3 describing the relative orders of the first <jats:italic>n<\/jats:italic> iterates of a point <jats:italic>x<\/jats:italic> under a self-map <jats:italic>f<\/jats:italic> of the interval <jats:italic>I<\/jats:italic>=[0,1] is called an <jats:italic>order pattern<\/jats:italic>. For fixed <jats:italic>f<\/jats:italic> and <jats:italic>n<\/jats:italic>, measuring the points <jats:italic>x<\/jats:italic> \u2208 <jats:italic>I<\/jats:italic> (according to Lebesgue measure) that generate the order pattern \u03c3 gives a probability distribution \u03bc<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>f<\/jats:italic>) on the set of length <jats:italic>n<\/jats:italic> permutations. We study the distributions that arise this way for various classes of functions <jats:italic>f<\/jats:italic>.<\/jats:p><jats:p>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 <jats:italic>n<\/jats:italic> 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 <jats:italic>f<\/jats:italic>, apart from an obvious compatibility condition, there is no restriction on the sequence {\u03bc<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>(<jats:italic>f<\/jats:italic>)}<jats:sub>n=1,2,.\u00a0.\u00a0.<\/jats:sub>.<\/jats:p><jats:p>In addition, we give a necessary condition for <jats:italic>f<\/jats:italic> to have <jats:italic>finite exclusion type<\/jats:italic>, that is, for there to be finitely many order patterns that generate all order patterns not realized by <jats:italic>f<\/jats:italic>. Using entropy we show that if <jats:italic>f<\/jats:italic> is piecewise continuous, piecewise monotone, and either ergodic or with points of arbitrarily high period, then <jats:italic>f<\/jats:italic> cannot have finite exclusion type. This generalizes results of S. Elizalde.<\/jats:p>","DOI":"10.1017\/s0963548313000035","type":"journal-article","created":{"date-parts":[[2013,3,5]],"date-time":"2013-03-05T12:45:29Z","timestamp":1362487529000},"page":"319-341","source":"Crossref","is-referenced-by-count":2,"title":["Distributions of Order Patterns of Interval Maps"],"prefix":"10.1017","volume":"22","author":[{"given":"AARON","family":"ABRAMS","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ERIC","family":"BABSON","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"HENRY","family":"LANDAU","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ZEPH","family":"LANDAU","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"JAMES","family":"POMMERSHEIM","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2013,3,5]]},"reference":[{"key":"S0963548313000035_ref7","unstructured":"Hesterberg A. (2010) Iterated iteratedly piecewise continuous function order pattern probability distributions. Preprint."},{"key":"S0963548313000035_ref9","first-page":"61","article-title":"Co-existence of cycles of a continuous mapping of a line into itself.","volume":"16","author":"Sarkovskii","year":"1964","journal-title":"Ukrain. Mat. Z."},{"key":"S0963548313000035_ref4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.88.174102"},{"key":"S0963548313000035_ref5","doi-asserted-by":"publisher","DOI":"10.1137\/080726689"},{"key":"S0963548313000035_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2007.07.004"},{"key":"S0963548313000035_ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2007.04.010"},{"key":"S0963548313000035_ref3","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/15\/5\/312"},{"key":"S0963548313000035_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2011.04.012"},{"key":"S0963548313000035_ref8","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/16\/3\/310"},{"key":"S0963548313000035_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5775-2"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548313000035","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,23]],"date-time":"2019-04-23T22:00:44Z","timestamp":1556056844000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548313000035\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,3,5]]},"references-count":10,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2013,5]]}},"alternative-id":["S0963548313000035"],"URL":"https:\/\/doi.org\/10.1017\/s0963548313000035","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,3,5]]}}}