@article{article_1089480, title={Farey graph and rational fixed points of the extended modular group}, journal={Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics}, volume={71}, pages={1029–1043}, year={2022}, DOI={10.31801/cfsuasmas.1089480}, author={Demir, Bilal and Karataş, Mustafa}, keywords={Extended modular group, fixed points, Farey sequence, Farey graph}, abstract={Fixed points of matrices have many applications in various areas of science and mathematics. Extended modular group <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-munderover"> <span class="mjx-stack"> <span class="mjx-over" style="font-size:70.7%;height:.096em;padding-bottom:.283em;padding-top:.141em;"> <span class="mjx-mo" style="vertical-align:top;"> <span class="mjx-delim-h"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.301em;padding-bottom:.833em;margin:0px -.07em 0px -.069em;">¯ </span> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.301em;padding-bottom:.833em;margin:0px .095em 0px -.234em;letter-spacing:-.369em;">¯¯ </span> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.301em;padding-bottom:.833em;margin-right:-.07em;margin-bottom:0px;margin-top:0px;">¯ </span> </span> </span> </span> <span class="mjx-op"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.316em;">Γ </span> </span> </span> </span> </span> </span> </span> <span class="MJX_Assistive_MathML">Γ¯ </span> </span> is the group of <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-mn"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.316em;">2 </span> </span> <span class="mjx-mo MJXc-space2"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.191em;padding-bottom:.316em;">× </span> </span> <span class="mjx-mn MJXc-space2"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.316em;">2 </span> </span> </span> </span> <span class="MJX_Assistive_MathML">2×2 </span> </span> matrices with integer entries and determinant <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-mo"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.316em;">± </span> </span> <span class="mjx-mn"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.316em;">1 </span> </span> </span> </span> <span class="MJX_Assistive_MathML">±1 </span> </span>. There are strong connections between extended modular group, continued fractions and Farey graph. Farey graph is a graph with vertex set <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-munderover"> <span class="mjx-stack"> <span class="mjx-over" style="height:.213em;padding-bottom:.06em;padding-left:.139em;"> <span class="mjx-mo" style="vertical-align:top;"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.378em;padding-bottom:.316em;">^ </span> </span> </span> <span class="mjx-op"> <span class="mjx-texatom"> <span class="mjx-mrow"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-ams-R" style="padding-top:.441em;padding-bottom:.503em;">Q </span> </span> </span> </span> </span> </span> </span> </span> </span> <span class="mjx-mo MJXc-space3"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.066em;padding-bottom:.316em;">= </span> </span> <span class="mjx-texatom MJXc-space3"> <span class="mjx-mrow"> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-ams-R" style="padding-top:.441em;padding-bottom:.503em;">Q </span> </span> </span> </span> <span class="mjx-mo MJXc-space2"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.316em;padding-bottom:.378em;">∪ </span> </span> <span class="mjx-mo MJXc-space2"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.441em;padding-bottom:.566em;">{ </span> </span> <span class="mjx-mi"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.128em;padding-bottom:.378em;">∞ </span> </span> <span class="mjx-mo"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.441em;padding-bottom:.566em;">} </span> </span> </span> </span> <span class="MJX_Assistive_MathML">Q^=Q∪{∞} </span> </span>. In this study, we consider the elements in <span class="MathJax_Preview" style="color:inherit;"> </span> <span class="mjx-chtml MathJax_CHTML" style="font-size:123%;"> <span class="mjx-math"> <span class="mjx-mrow"> <span class="mjx-munderover"> <span class="mjx-stack"> <span class="mjx-over" style="font-size:70.7%;height:.096em;padding-bottom:.283em;padding-top:.141em;"> <span class="mjx-mo" style="vertical-align:top;"> <span class="mjx-delim-h"> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.301em;padding-bottom:.833em;margin:0px -.07em 0px -.069em;">¯ </span> <span class="mjx-char MJXc-TeX-main-R" style="padding-top:.301em;padding-bottom:.833em;margin:0px .095em 0px -.234em;letter-spacing:-.369em;">¯¯ </span> <span class="mjx-char MJXc-TeX-main-R" style="padd}, number={4}, publisher={Ankara University}