Parry’s topological transitivity and $f$-expansions
HTML articles powered by AMS MathViewer
- by Jr. E. Arthur Robinson PDF
- Proc. Amer. Math. Soc. 144 (2016), 2093-2107 Request permission
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.References
- Ethan Akin, Dynamics of discontinuous maps via closed relations, Topology Proc. 41 (2013), 271–310. MR 2988034
- Ethan Akin, The general topology of dynamical systems, Graduate Studies in Mathematics, vol. 1, American Mathematical Society, Providence, RI, 1993. MR 1219737, DOI 10.1090/gsm/001
- Ethan Akin and Jeffrey D. Carlson, Conceptions of topological transitivity, Topology Appl. 159 (2012), no. 12, 2815–2830. MR 2942654, DOI 10.1016/j.topol.2012.04.016
- Pierre Arnoux, Donald S. Ornstein, and Benjamin Weiss, Cutting and stacking, interval exchanges and geometric models, Israel J. Math. 50 (1985), no. 1-2, 160–168. MR 788073, DOI 10.1007/BF02761122
- B. H. Bissinger, A generalization of continued fractions, Bull. Amer. Math. Soc. 50 (1944), 868–876. MR 11338, DOI 10.1090/S0002-9904-1944-08254-2
- F. Blanchard, $\beta$-expansions and symbolic dynamics, Theoret. Comput. Sci. 65 (1989), no. 2, 131–141. MR 1020481, DOI 10.1016/0304-3975(89)90038-8
- Abraham Boyarsky and PawełGóra, Laws of chaos, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1997. Invariant measures and dynamical systems in one dimension. MR 1461536, DOI 10.1007/978-1-4612-2024-4
- Karma Dajani and Cor Kraaikamp, Ergodic theory of numbers, Carus Mathematical Monographs, vol. 29, Mathematical Association of America, Washington, DC, 2002. MR 1917322
- Karma Dajani, Cor Kraaikamp, and Niels van der Wekken, Ergodicity of $N$-continued fraction expansions, J. Number Theory 133 (2013), no. 9, 3183–3204. MR 3057071, DOI 10.1016/j.jnt.2013.02.017
- Robert L. Devaney, An introduction to chaotic dynamical systems, Studies in Nonlinearity, Westview Press, Boulder, CO, 2003. Reprint of the second (1989) edition. MR 1979140
- C. J. Everett, Representations for real numbers, Bull. Amer. Math. Soc. 52 (1946), 861–869. MR 18221, DOI 10.1090/S0002-9904-1946-08659-0
- PawełGóra, Invariant densities for piecewise linear maps of the unit interval, Ergodic Theory Dynam. Systems 29 (2009), no. 5, 1549–1583. MR 2545017, DOI 10.1017/S0143385708000801
- S. Kakeya, On the generalized scale of notation, Japan J. Math. 1 (1924), 95–108.
- Michael Keane, Interval exchange transformations, Math. Z. 141 (1975), 25–31. MR 357739, DOI 10.1007/BF01236981
- Sergiĭ Kolyada and Ľubomír Snoha, Some aspects of topological transitivity—a survey, Iteration theory (ECIT 94) (Opava), Grazer Math. Ber., vol. 334, Karl-Franzens-Univ. Graz, Graz, 1997, pp. 3–35. MR 1644768
- Howard Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2) 115 (1982), no. 1, 169–200. MR 644018, DOI 10.2307/1971341
- Erblin Mehmetaj. Properties of $r$-continued fractions. PhD thesis, George Washington University (2014).
- John Milnor and William Thurston, On iterated maps of the interval, Dynamical systems (College Park, MD, 1986–87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 465–563. MR 970571, DOI 10.1007/BFb0082847
- Anima Nagar, V. Kannan, and S. P. Sesha Sai, Properties of topologically transitive maps on the real line, Real Anal. Exchange 27 (2001/02), no. 1, 325–334. MR 1887863
- Arnaldo Nogueira, Almost all interval exchange transformations with flips are nonergodic, Ergodic Theory Dynam. Systems 9 (1989), no. 3, 515–525. MR 1016669, DOI 10.1017/S0143385700005150
- V. I. Oseledec, The spectrum of ergodic automorphisms, Dokl. Akad. Nauk SSSR 168 (1966), 1009–1011 (Russian). MR 0199347
- W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416 (English, with Russian summary). MR 142719, DOI 10.1007/BF02020954
- W. Parry, Representations for real numbers, Acta Math. Acad. Sci. Hungar. 15 (1964), 95–105. MR 166332, DOI 10.1007/BF01897025
- A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. MR 97374, DOI 10.1007/BF02020331
- E. Arthur Robinson Jr., Sturmian expansions and entropy, Integers 11B (2011), Paper No. A13, 16. MR 3054432
- Sebastian van Strien, Smooth dynamics on the interval (with an emphasis on quadratic-like maps), New directions in dynamical systems, London Math. Soc. Lecture Note Ser., vol. 127, Cambridge Univ. Press, Cambridge, 1988, pp. 57–119. MR 953970
- William A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 (1982), no. 1, 201–242. MR 644019, DOI 10.2307/1971391
Additional Information
- Jr. E. Arthur Robinson
- Affiliation: Department of Mathematics, George Washington University, 2115 G Street NW, Washington, DC 20052
- Email: robinson@gwu.edu
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2093-2107
- MSC (2010): Primary 37E05, 37B20, 11K55
- DOI: https://doi.org/10.1090/proc/12857
- MathSciNet review: 3460170