{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T01:08:49Z","timestamp":1649034529921},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"We introduce a square root map on Sturmian words and study its properties. Given a Sturmian word of slope $\\alpha$,\u00a0there exists exactly six minimal squares in its language (a minimal square does not have a square as a proper\u00a0prefix). A Sturmian word $s$ of slope $\\alpha$ can be written as a product of these six minimal squares:\u00a0$s = X_1^2 X_2^2 X_3^2 \\cdots$. The square root of $s$ is defined to be the word $\\sqrt{s} = X_1 X_2 X_3 \\cdots$.\u00a0The main result of this paper is that $\\sqrt{s}$ is also a Sturmian word of slope $\\alpha$. Further, we\u00a0characterize the Sturmian fixed points of the square root map, and we describe how to find the intercept of\u00a0$\\sqrt{s}$ and an occurrence of any prefix of $\\sqrt{s}$ in $s$. Related to the square root map, we characterize\u00a0the solutions of the word equation $X_1^2 X_2^2 \\cdots X_n^2 = (X_1 X_2 \\cdots X_n)^2$ in the language of Sturmian\u00a0words of slope $\\alpha$ where the words $X_i^2$ are minimal squares of slope $\\alpha$.We also study the square root map in a more general setting. We explicitly construct an infinite set of\u00a0non-Sturmian fixed points of the square root map. We show that the subshifts $\\Omega$ generated by these words have\u00a0a curious property: for all $w \\in \\Omega$ either $\\sqrt{w} \\in \\Omega$ or $\\sqrt{w}$ is periodic. In particular,\u00a0the square root map can map an aperiodic word to a periodic word.<\/jats:p>","DOI":"10.37236\/6074","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:59:06Z","timestamp":1578671946000},"source":"Crossref","is-referenced-by-count":1,"title":["A Square Root Map on Sturmian Words"],"prefix":"10.37236","volume":"24","author":[{"given":"Jarkko","family":"Peltom\u00e4ki","sequence":"first","affiliation":[]},{"given":"Markus A.","family":"Whiteland","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,3,31]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p54\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p54\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:03:12Z","timestamp":1579237392000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i1p54"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,3,31]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1,20]]}},"URL":"http:\/\/dx.doi.org\/10.37236\/6074","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":["Computational Theory and Mathematics","Geometry and Topology","Theoretical Computer Science","Applied Mathematics","Discrete Mathematics and Combinatorics"],"published":{"date-parts":[[2017,3,31]]}}}