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Parry's topological transitivity and $ f$-expansions


Author: E. Arthur Robinson, Jr.
Journal: Proc. Amer. Math. Soc. 144 (2016), 2093-2107
MSC (2010): Primary 37E05, 37B20, 11K55
DOI: https://doi.org/10.1090/proc/12857
Published electronically: August 12, 2015
MathSciNet review: 3460170
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Abstract: In his 1964 paper on $ f$-expansions, Parry studied piecewise-
continuous, piecewise-monotonic maps $ F$ of the interval $ [0,1]$, and introduced a notion of topological transitivity different from any of the modern definitions. This notion, which we call Parry topological transitivity (PTT), is that the backward orbit $ O^-(x)=\{y:x=F^ny$$ \text {\ for\ some\ }n\ge 0\}$ of some $ x\in [0,1]$ is dense. We take topological transitivity (TT) to mean that some $ x$ has a dense forward orbit. Parry's application of PTT to $ f$-expansions is that PTT implies the partition of $ [0,1]$ into the ``fibers'' of $ F$ is a generating partition (i.e., $ f$-expansions are ``valid''). We prove the same result for TT, and use this to show that for interval maps $ F$, TT implies PTT. A separate proof is provided for continuous maps $ F$ of perfect Polish spaces. The converse is false.


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Additional Information

E. Arthur Robinson, Jr.
Affiliation: Department of Mathematics, George Washington University, 2115 G Street NW, Washington, DC 20052
Email: robinson@gwu.edu

DOI: https://doi.org/10.1090/proc/12857
Received by editor(s): May 22, 2014
Received by editor(s) in revised form: May 29, 2015, and June 8, 2015
Published electronically: August 12, 2015
Additional Notes: This work partially supported by a grant from the Simons Foundation (award number 244739 to E. Arthur Robinson, Jr.)
Communicated by: Yingfei Yi
Article copyright: © Copyright 2015 American Mathematical Society

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