On homeomorphic Bernoulli measures on the Cantor space
HTML articles powered by AMS MathViewer
- by Randall Dougherty, R. Daniel Mauldin and Andrew Yingst PDF
- Trans. Amer. Math. Soc. 359 (2007), 6155-6166 Request permission
Abstract:
Let $\mu (r)$ be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights $r$ and $1-r$. It is a long-standing open problem to characterize those $r$ and $s$ such that $\mu (r)$ and $\mu (s)$ are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending $\mu (r)$ to $\mu (s)$). The (possibly) weaker property of $\mu (r)$ and $\mu (s)$ being continuously reducible to each other is equivalent to a property of $r$ and $s$ called binomial equivalence. In this paper we define an algebraic property called “refinability” and show that, if $r$ and $s$ are refinable and binomially equivalent, then $\mu (r)$ and $\mu (s)$ are topologically equivalent. Next we show that refinability is equivalent to a fairly simple algebraic property. Finally, we give a class of examples of binomially equivalent and refinable numbers; in particular, the positive numbers $r$ and $s$ such that $s = r^2$ and $r = 1-s^2$ are refinable, so the corresponding measures are topologically equivalent.References
- Ethan Akin, Good measures on Cantor space, Trans. Amer. Math. Soc. 357 (2005), no. 7, 2681–2722. MR 2139523, DOI 10.1090/S0002-9947-04-03524-X
- Steve Alpern and V. S. Prasad, Typical dynamics of volume preserving homeomorphisms, Cambridge Tracts in Mathematics, vol. 139, Cambridge University Press, Cambridge, 2000. MR 1826331
- T. D. Austin, A pair of non-homeomorphic measures on the Cantor set, Math. Proc. Cam. Phil. Soc., to appear.
- K. J. Huang, Algebraic numbers and topologically equivalent measures in the Cantor set, Proc. Amer. Math. Soc. 96 (1986), no. 4, 560–562. MR 826481, DOI 10.1090/S0002-9939-1986-0826481-X
- R. Daniel Mauldin, Problems in topology arising from analysis, Open problems in topology, North-Holland, Amsterdam, 1990, pp. 617–629. MR 1078668
- Francisco J. Navarro-Bermúdez, Topologically equivalent measures in the Cantor space, Proc. Amer. Math. Soc. 77 (1979), no. 2, 229–236. MR 542090, DOI 10.1090/S0002-9939-1979-0542090-0
- Francisco J. Navarro-Bermúdez and John C. Oxtoby, Four topologically equivalent measures in the Cantor space, Proc. Amer. Math. Soc. 104 (1988), no. 3, 859–860. MR 939966, DOI 10.1090/S0002-9939-1988-0939966-4
- John C. Oxtoby, Homeomorphic measures in metric spaces, Proc. Amer. Math. Soc. 24 (1970), 419–423. MR 260961, DOI 10.1090/S0002-9939-1970-0260961-1
- John C. Oxtoby and Vidhu S. Prasad, Homeomorphic measures in the Hilbert cube, Pacific J. Math. 77 (1978), no. 2, 483–497. MR 510936, DOI 10.2140/pjm.1978.77.483
- J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2) 42 (1941), 874–920. MR 5803, DOI 10.2307/1968772
- R. G. E. Pinch, Binomial equivalence of algebraic integers, J. Indian Math. Soc. (N.S.) 58 (1992), no. 1-4, 33–37. MR 1207024
- Ernst S. Selmer, On the irreducibility of certain trinomials, Math. Scand. 4 (1956), 287–302. MR 85223, DOI 10.7146/math.scand.a-10478
- Felix Hausdorff, Summationsmethoden und Momentfolgen. I, Math. Z. 9 (1921), no. 1-2, 74–109 (German). MR 1544453, DOI 10.1007/BF01378337
- George Pólya and Gabor Szegő, Problems and theorems in analysis. II, Classics in Mathematics, Springer-Verlag, Berlin, 1998. Theory of functions, zeros, polynomials, determinants, number theory, geometry; Translated from the German by C. E. Billigheimer; Reprint of the 1976 English translation. MR 1492448, DOI 10.1007/978-3-642-61905-2_{7}
- G. G. Lorentz, Bernstein polynomials, Mathematical Expositions, No. 8, University of Toronto Press, Toronto, 1953. MR 0057370
Additional Information
- Randall Dougherty
- Affiliation: IDA Center for Communications Research, 4320 Westerra Ct., San Diego, California 92121
- Email: rdough@ccrwest.org
- R. Daniel Mauldin
- Affiliation: Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76203
- Email: mauldin@unt.edu
- Andrew Yingst
- Affiliation: Department of Mathematics, P.O. Box 311430, University of North Texas, Denton, Texas 76203
- Address at time of publication: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
- Email: andyq@unt.edu, yingst@math.sc.edu
- Received by editor(s): January 18, 2006
- Received by editor(s) in revised form: May 1, 2006
- Published electronically: July 23, 2007
- Additional Notes: The second author was supported in part by NSF grant DMS 0400481
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 6155-6166
- MSC (2000): Primary 37B05; Secondary 28D05, 28C15
- DOI: https://doi.org/10.1090/S0002-9947-07-04352-8
- MathSciNet review: 2336321