Real

Authors:
Michal Misiurewicz and Ana Rodrigues

Journal:
Proc. Amer. Math. Soc. **133** (2005), 1109-1118

MSC (2000):
Primary 37B05; Secondary 20M20, 37C25, 11B83

Published electronically:
October 15, 2004

MathSciNet review:
2117212

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The famous problem involves applying two maps: and to positive integers. If is even, one applies , if it is odd, one applies . The conjecture states that each trajectory of the system arrives to the periodic orbit . In this paper, instead of choosing each time which map to apply, we allow ourselves more freedom and apply both and independently of . That is, we consider the action of the free semigroup with generators and 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 and .

**1.**Corrado Böhm and Giovanna Sontacchi,*On the existence of cycles of given length in integer sequences like 𝑥_{𝑛+1}=𝑥_{𝑛}/2 if 𝑥_{𝑛} even, and 𝑥_{𝑛+1}=3𝑥_{𝑛}+1 otherwise*, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)**64**(1978), no. 3, 260–264 (English, with Italian summary). MR**551509****2.**Stephen D. Cohen,*The group of translations and positive rational powers is free*, Quart. J. Math. Oxford Ser. (2)**46**(1995), no. 181, 21–93. MR**1326133**, 10.1093/qmath/46.1.21**3.**David B. Ellis, Robert Ellis, and Mahesh Nerurkar,*The topological dynamics of semigroup actions*, Trans. Amer. Math. Soc.**353**(2001), no. 4, 1279–1320 (electronic). MR**1806740**, 10.1090/S0002-9947-00-02704-5**4.**R. I. Grigorchuk,*An ergodic theorem for actions of a free semigroup*, Tr. Mat. Inst. Steklova**231**(2000), no. Din. Sist., Avtom. i Beskon. Gruppy, 119–133 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math.**4 (231)**(2000), 113–127. MR**1841754****5.**C. Gurwood,*On periodicity in Collatz's Conjecture*, preprint.**6.**J. C. Lagarias,*Problem annotated bibliography*,`http://www.research.att.com/ ~jcl/doc/3x+1bib.ps`.**7.**Jeffrey C. Lagarias,*The 3𝑥+1 problem and its generalizations*, Amer. Math. Monthly**92**(1985), no. 1, 3–23. MR**777565**, 10.2307/2322189**8.**Jeffrey C. Lagarias,*The set of rational cycles for the 3𝑥+1 problem*, Acta Arith.**56**(1990), no. 1, 33–53. MR**1067980****9.**Daniel J. Rudolph,*×2 and ×3 invariant measures and entropy*, Ergodic Theory Dynam. Systems**10**(1990), no. 2, 395–406. MR**1062766**, 10.1017/S0143385700005629**10.**Ya. B. Vorobets,*On the uniform distribution of the orbits of actions of free groups and semigroups on the plane*, Tr. Mat. Inst. Steklova**231**(2000), no. Din. Sist., Avtom. i Beskon. Gruppy, 64–95 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math.**4 (231)**(2000), 59–89. MR**1841752****11.**Samuel White,*The group generated by 𝑥\mapsto𝑥+1 and 𝑥\mapsto𝑥^{𝑝} is free*, J. Algebra**118**(1988), no. 2, 408–422. MR**969681**, 10.1016/0021-8693(88)90030-0**12.**Günther J. Wirsching,*The dynamical system generated by the 3𝑛+1 function*, Lecture Notes in Mathematics, vol. 1681, Springer-Verlag, Berlin, 1998. MR**1612686**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
37B05,
20M20,
37C25,
11B83

Retrieve articles in all journals with MSC (2000): 37B05, 20M20, 37C25, 11B83

Additional Information

**Michal Misiurewicz**

Affiliation:
Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216

Email:
mmisiure@math.iupui.edu

**Ana Rodrigues**

Affiliation:
Universidade do Minho, Escola de Ciencias, Departamento de Matematica, Campus de Gualtar, 4710-057 Braga, Portugal

Email:
anarodrigues@math.uminho.pt

DOI:
https://doi.org/10.1090/S0002-9939-04-07696-8

Received by editor(s):
November 26, 2003

Published electronically:
October 15, 2004

Additional Notes:
The authors were partially supported by NSF grant DMS 0139916. The second author thanks the hospitality of the Department of Mathematical Sciences of IUPUI

Communicated by:
Michael Handel

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.