Real
Authors:
Michal Misiurewicz and Ana Rodrigues
Journal:
Proc. Amer. Math. Soc. 133 (2005), 11091118
MSC (2000):
Primary 37B05; Secondary 20M20, 37C25, 11B83
Published electronically:
October 15, 2004
MathSciNet review:
2117212
Fulltext PDF Free Access
Abstract 
References 
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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 .
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Additional Information
Michal Misiurewicz
Affiliation:
Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 462023216
Email:
mmisiure@math.iupui.edu
Ana Rodrigues
Affiliation:
Universidade do Minho, Escola de Ciencias, Departamento de Matematica, Campus de Gualtar, 4710057 Braga, Portugal
Email:
anarodrigues@math.uminho.pt
DOI:
http://dx.doi.org/10.1090/S0002993904076968
PII:
S 00029939(04)076968
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.
