{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,2,2]],"date-time":"2023-02-02T15:28:29Z","timestamp":1675351709036},"reference-count":30,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["DCDS"],"published-print":{"date-parts":[[2022]]},"abstract":"<p style='text-indent:20px;'>Consider <inline-formula><tex-math id=\"M2\">\\begin{document}$ \\beta &gt; 1 $\\end{document}<\/tex-math><\/inline-formula> and <inline-formula><tex-math id=\"M3\">\\begin{document}$ \\lfloor \\beta \\rfloor $\\end{document}<\/tex-math><\/inline-formula> its integer part. It is widely known that any real number <inline-formula><tex-math id=\"M4\">\\begin{document}$ \\alpha \\in \\Bigl[0, \\frac{\\lfloor \\beta \\rfloor}{\\beta - 1}\\Bigr] $\\end{document}<\/tex-math><\/inline-formula> can be represented in base <inline-formula><tex-math id=\"M5\">\\begin{document}$ \\beta $\\end{document}<\/tex-math><\/inline-formula> using a development in series of the form <inline-formula><tex-math id=\"M6\">\\begin{document}$ \\alpha = \\sum_{n = 1}^\\infty x_n\\beta^{-n} $\\end{document}<\/tex-math><\/inline-formula>, where <inline-formula><tex-math id=\"M7\">\\begin{document}$ x = (x_n)_{n \\geq 1} $\\end{document}<\/tex-math><\/inline-formula> is a sequence taking values into the alphabet <inline-formula><tex-math id=\"M8\">\\begin{document}$ \\{0,\\; ...\\; ,\\; \\lfloor \\beta \\rfloor\\} $\\end{document}<\/tex-math><\/inline-formula>. The so called <inline-formula><tex-math id=\"M9\">\\begin{document}$ \\beta $\\end{document}<\/tex-math><\/inline-formula>-shift, denoted by <inline-formula><tex-math id=\"M10\">\\begin{document}$ \\Sigma_\\beta $\\end{document}<\/tex-math><\/inline-formula>, is given as the set of sequences such that all their iterates by the shift map are less than or equal to the quasi-greedy <inline-formula><tex-math id=\"M11\">\\begin{document}$ \\beta $\\end{document}<\/tex-math><\/inline-formula>-expansion of <inline-formula><tex-math id=\"M12\">\\begin{document}$ 1 $\\end{document}<\/tex-math><\/inline-formula>. Fixing a H\u00f6lder continuous potential <inline-formula><tex-math id=\"M13\">\\begin{document}$ A $\\end{document}<\/tex-math><\/inline-formula>, we show an explicit expression for the main eigenfunction of the Ruelle operator <inline-formula><tex-math id=\"M14\">\\begin{document}$ \\psi_A $\\end{document}<\/tex-math><\/inline-formula>, in order to obtain a natural extension to the bilateral <inline-formula><tex-math id=\"M15\">\\begin{document}$ \\beta $\\end{document}<\/tex-math><\/inline-formula>-shift of its corresponding Gibbs state <inline-formula><tex-math id=\"M16\">\\begin{document}$ \\mu_A $\\end{document}<\/tex-math><\/inline-formula>. Our main goal here is to prove a first level large deviations principle for the family <inline-formula><tex-math id=\"M17\">\\begin{document}$ (\\mu_{tA})_{t&gt;1} $\\end{document}<\/tex-math><\/inline-formula> with a rate function <inline-formula><tex-math id=\"M18\">\\begin{document}$ I $\\end{document}<\/tex-math><\/inline-formula> attaining its maximum value on the union of the supports of all the maximizing measures of <inline-formula><tex-math id=\"M19\">\\begin{document}$ A $\\end{document}<\/tex-math><\/inline-formula>. The above is proved through a technique using the representation of <inline-formula><tex-math id=\"M20\">\\begin{document}$ \\Sigma_\\beta $\\end{document}<\/tex-math><\/inline-formula> and its bilateral extension <inline-formula><tex-math id=\"M21\">\\begin{document}$ \\widehat{\\Sigma_\\beta} $\\end{document}<\/tex-math><\/inline-formula> in terms of the quasi-greedy <inline-formula><tex-math id=\"M22\">\\begin{document}$ \\beta $\\end{document}<\/tex-math><\/inline-formula>-expansion of <inline-formula><tex-math id=\"M23\">\\begin{document}$ 1 $\\end{document}<\/tex-math><\/inline-formula> and the so called involution kernel associated to the potential <inline-formula><tex-math id=\"M24\">\\begin{document}$ A $\\end{document}<\/tex-math><\/inline-formula>.<\/p><\/jats:p>","DOI":"10.3934\/dcds.2021208","type":"journal-article","created":{"date-parts":[[2022,1,12]],"date-time":"2022-01-12T14:00:30Z","timestamp":1641996030000},"page":"2699","source":"Crossref","is-referenced-by-count":1,"title":["On involution kernels and large deviations principles on $ \\beta $-shifts"],"prefix":"10.3934","volume":"42","author":[{"given":"Victor","family":"Vargas","sequence":"first","affiliation":[{"name":"School of Mathematics, National University of Colombia, Medell\u00edn 050034, Colombia"}]}],"member":"2321","reference":[{"key":"key-10.3934\/dcds.2021208-1","unstructured":"A. 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