{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T16:09:34Z","timestamp":1776096574481,"version":"3.50.1"},"reference-count":12,"publisher":"American Mathematical Society (AMS)","issue":"4","license":[{"start":{"date-parts":[[2005,10,15]],"date-time":"2005-10-15T00:00:00Z","timestamp":1129334400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Proc. Amer. Math. Soc."],"abstract":"<p>\n                    The famous\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3 x plus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">3x+1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    problem involves applying two maps:\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T 0 left-parenthesis x right-parenthesis equals x slash 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">T_0(x)=x\/2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T 1 left-parenthesis x right-parenthesis equals left-parenthesis 3 x plus 1 right-parenthesis slash 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">T_1(x)=(3x+1)\/2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to positive integers. If\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n                        <mml:semantics>\n                          <mml:mi>x<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is even, one applies\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">T_0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , if it is odd, one applies\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T 1\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">T_1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The conjecture states that each trajectory of the system arrives to the periodic orbit\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartSet 1 comma 2 EndSet\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo fence=\"false\" stretchy=\"false\">{<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo fence=\"false\" stretchy=\"false\">}<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\{1,2\\}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In this paper, instead of choosing each time which map to apply, we allow ourselves more freedom and apply both\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">T_0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T 1\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">T_1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    independently of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n                        <mml:semantics>\n                          <mml:mi>x<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . That is, we consider the action of the free semigroup with generators\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">T_0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T 1\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">T_1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    on the space of positive real numbers. We prove that this action is minimal (each trajectory is dense) and that the periodic points are dense. Moreover, we give a full characterization of the group of transformations of the real line generated by\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T 0\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">T_0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T 1\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">T_1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0002-9939-04-07696-8","type":"journal-article","created":{"date-parts":[[2004,12,20]],"date-time":"2004-12-20T08:10:20Z","timestamp":1103530220000},"page":"1109-1118","source":"Crossref","is-referenced-by-count":3,"special_numbering":"550","title":["Real 3\ud835\udc65+1"],"prefix":"10.1090","volume":"133","author":[{"given":"Micha\u0142","family":"Misiurewicz","sequence":"first","affiliation":[]},{"given":"Ana","family":"Rodrigues","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2004,10,15]]},"reference":[{"issue":"3","key":"1","first-page":"260","article-title":"On the existence of cycles of given length in integer sequences like \ud835\udc65_{\ud835\udc5b+1}=\ud835\udc65_{\ud835\udc5b}\/2 if \ud835\udc65_{\ud835\udc5b} even, and \ud835\udc65_{\ud835\udc5b+1}=3\ud835\udc65_{\ud835\udc5b}+1 otherwise","volume":"64","author":"B\u00f6hm, Corrado","year":"1978","journal-title":"Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8)","ISSN":"https:\/\/id.crossref.org\/issn\/0392-7881","issn-type":"print"},{"issue":"181","key":"2","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1093\/qmath\/46.1.21","article-title":"The group of translations and positive rational powers is free","volume":"46","author":"Cohen, Stephen D.","year":"1995","journal-title":"Quart. J. Math. Oxford Ser. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0033-5606","issn-type":"print"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"1279","DOI":"10.1090\/S0002-9947-00-02704-5","article-title":"The topological dynamics of semigroup actions","volume":"353","author":"Ellis, David B.","year":"2001","journal-title":"Trans. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"},{"key":"4","first-page":"119","article-title":"An ergodic theorem for actions of a free semigroup","volume":"231","author":"Grigorchuk, R. I.","year":"2000","journal-title":"Tr. Mat. Inst. Steklova","ISSN":"https:\/\/id.crossref.org\/issn\/0371-9685","issn-type":"print"},{"key":"5","unstructured":"C. Gurwood, On periodicity in Collatz\u2019s Conjecture, preprint."},{"key":"6","unstructured":"J. C. Lagarias, 3\ud835\udc65+1 Problem annotated bibliography, \\url{http:\/\/www.research.att.com\/ jcl\/doc\/3x+1bib.ps}."},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"3","DOI":"10.2307\/2322189","article-title":"The 3\ud835\udc65+1 problem and its generalizations","volume":"92","author":"Lagarias, Jeffrey C.","year":"1985","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"33","DOI":"10.4064\/aa-56-1-33-53","article-title":"The set of rational cycles for the 3\ud835\udc65+1 problem","volume":"56","author":"Lagarias, Jeffrey C.","year":"1990","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1017\/S0143385700005629","article-title":"\u00d72 and \u00d73 invariant measures and entropy","volume":"10","author":"Rudolph, Daniel J.","year":"1990","journal-title":"Ergodic Theory Dynam. Systems","ISSN":"https:\/\/id.crossref.org\/issn\/0143-3857","issn-type":"print"},{"key":"10","first-page":"64","article-title":"On the uniform distribution of the orbits of actions of free groups and semigroups on the plane","volume":"231","author":"Vorobets, Ya. B.","year":"2000","journal-title":"Tr. Mat. Inst. Steklova","ISSN":"https:\/\/id.crossref.org\/issn\/0371-9685","issn-type":"print"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"408","DOI":"10.1016\/0021-8693(88)90030-0","article-title":"The group generated by \ud835\udc65\\mapsto\ud835\udc65+1 and \ud835\udc65\\mapsto\ud835\udc65^{\ud835\udc5d} is free","volume":"118","author":"White, Samuel","year":"1988","journal-title":"J. Algebra","ISSN":"https:\/\/id.crossref.org\/issn\/0021-8693","issn-type":"print"},{"key":"12","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0095985","volume-title":"The dynamical system generated by the $3n+1$ function","volume":"1681","author":"Wirsching, G\u00fcnther J.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/3540639705"}],"container-title":["Proceedings of the American Mathematical Society"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/proc\/2005-133-04\/S0002-9939-04-07696-8\/S0002-9939-04-07696-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/proc\/2005-133-04\/S0002-9939-04-07696-8\/S0002-9939-04-07696-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T15:29:42Z","timestamp":1776094182000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/proc\/2005-133-04\/S0002-9939-04-07696-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,10,15]]},"references-count":12,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2005,4]]}},"alternative-id":["S0002-9939-04-07696-8"],"URL":"https:\/\/doi.org\/10.1090\/s0002-9939-04-07696-8","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6826","0002-9939"],"issn-type":[{"value":"1088-6826","type":"electronic"},{"value":"0002-9939","type":"print"}],"subject":[],"published":{"date-parts":[[2004,10,15]]}}}