{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T11:04:57Z","timestamp":1648897497022},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2008,9,19]],"date-time":"2008-09-19T00:00:00Z","timestamp":1221782400000},"content-version":"unspecified","delay-in-days":4798,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ergod. Th. Dynam. Sys."],"published-print":{"date-parts":[[1995,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Sullivan's scaling function provides a complete description of the smooth conjugacy classes of cookie-cutters. However, for smooth conjugacy classes of Markov maps on a train track, such as expanding circle maps and train track mappings induced by pseudo-Anosov systems, the generalisation of the scaling function suffers from a deficiency. It is difficult to characterise the structure of the set of those scaling functions which correspond to smooth mappings. We introduce a new invariant for Markov maps called the <jats:italic>solenoid function<\/jats:italic>. We prove that for any prescribed topological structure, there is a one-to-one correspondence between smooth conjugacy classes of smooth Markov maps and pseudo-H\u00f6lder solenoid functions. This gives a characterisation of the moduli space for smooth conjugacy classes of smooth Markov maps. For smooth expanding maps of the circle with degree <jats:italic>d<\/jats:italic> this moduli space is the space of H\u00f6lder continuous functions on the space {0,\u2026, <jats:italic>d<\/jats:italic> \u2212 1}<jats:sup>\u2115<\/jats:sup> satisfying the matching condition.<\/jats:p>","DOI":"10.1017\/s0143385700008622","type":"journal-article","created":{"date-parts":[[2008,9,19]],"date-time":"2008-09-19T11:10:36Z","timestamp":1221822636000},"page":"697-734","source":"Crossref","is-referenced-by-count":6,"title":["Classifying <i>C<\/i><sup>1+<\/sup> structures on dynamical fractals: 1. The moduli space of solenoid functions for Markov maps on train tracks"],"prefix":"10.1017","volume":"15","author":[{"given":"A. A.","family":"Pinto","sequence":"first","affiliation":[]},{"given":"D. A.","family":"Rand","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2008,9,19]]},"reference":[{"key":"S0143385700008622_ref014","first-page":"1216","volume-title":"Proc. Int. Congress of Mathematicians","author":"Sullivan","year":"1988"},{"key":"S0143385700008622_ref017","first-page":"169","article-title":"Expanding attractors","volume":"43","author":"Williams","year":"1974","journal-title":"Publ."},{"key":"S0143385700008622_ref003","unstructured":"[3] Pinto A. A. and Sullivan D. . The solenoid and the circle. In preparation."},{"key":"S0143385700008622_ref010","first-page":"1","volume-title":"In New Directions in Dynamical Systems","author":"Rand","year":"1988"},{"key":"S0143385700008622_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF01232439"},{"key":"S0143385700008622_ref002","unstructured":"[2] Fisher A. . Private communication."},{"key":"S0143385700008622_ref015","volume-title":"Mathematics into the Twenty-first Century","volume":"2","author":"Sullivan","year":"1991"},{"key":"S0143385700008622_ref013","volume-title":"In Nonlinear Evolution and Chaotic Phenomena","author":"Sullivan","year":"1988"},{"key":"S0143385700008622_ref005","unstructured":"[5] Pinto A. A. and Rand D. A. . Characterising rigidity and flexibility of pseudo-Anosov and other transversally laminated dynamical systems on surfaces. In preparation."},{"key":"S0143385700008622_ref016","article-title":"Linking the universalities of Milnor-Thurston, Feigenbaum and Alfors-Bers","author":"Sullivan","year":"1992","journal-title":"Topological Methods in Modern Mathematics"},{"key":"S0143385700008622_ref004","unstructured":"[4] Pinto A. A. and Rand D. A. . The renormalisation dynamics and Teichm\u00fcller spaces of less-smooth conjugacy classes of golden rotations. In preparation."},{"key":"S0143385700008622_ref012","doi-asserted-by":"publisher","DOI":"10.2307\/2373276"},{"key":"S0143385700008622_ref007","doi-asserted-by":"publisher","DOI":"10.1017\/S014338570000972X"},{"key":"S0143385700008622_ref006","first-page":"1","article-title":"Global phase space universality, smooth conjugacies and renormalisation: 2","volume":"4","author":"Pinto","year":"1991","journal-title":"Nonlinearity"},{"key":"S0143385700008622_ref009","unstructured":"[9] Pinto A. A. . Convergence of renormalisation and rigidity of dynamical systems. Warwick PhD thesis, 1991."},{"key":"S0143385700008622_ref011","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/1\/1\/007"},{"key":"S0143385700008622_ref008","unstructured":"[8] Pinto A. A. and Rand D. A. . Families of solenoid functions for families of smooth Markov maps. In preparation."}],"container-title":["Ergodic Theory and Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0143385700008622","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T17:19:15Z","timestamp":1557767955000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0143385700008622\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,8]]},"references-count":17,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1995,8]]}},"alternative-id":["S0143385700008622"],"URL":"https:\/\/doi.org\/10.1017\/s0143385700008622","relation":{},"ISSN":["0143-3857","1469-4417"],"issn-type":[{"value":"0143-3857","type":"print"},{"value":"1469-4417","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,8]]}}}