On homeomorphic Bernoulli measures on the Cantor space

Authors:
Randall Dougherty, R. Daniel Mauldin and Andrew Yingst

Journal:
Trans. Amer. Math. Soc. **359** (2007), 6155-6166

MSC (2000):
Primary 37B05; Secondary 28D05, 28C15

Published electronically:
July 23, 2007

MathSciNet review:
2336321

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Abstract: Let be the Bernoulli measure on the Cantor space given as the infinite product of two-point measures with weights and . It is a long-standing open problem to characterize those and such that and are topologically equivalent (i.e., there is a homeomorphism from the Cantor space to itself sending to ). The (possibly) weaker property of and being continuously reducible to each other is equivalent to a property of and called binomial equivalence. In this paper we define an algebraic property called ``refinability'' and show that, if and are refinable and binomially equivalent, then and 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 and such that and are refinable, so the corresponding measures are topologically equivalent.

**1.**Ethan Akin,*Good measures on Cantor space*, Trans. Amer. Math. Soc.**357**(2005), no. 7, 2681–2722 (electronic). MR**2139523**, 10.1090/S0002-9947-04-03524-X**2.**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****3.**T. D. Austin, A pair of non-homeomorphic measures on the Cantor set, Math. Proc. Cam. Phil. Soc., to appear.**4.**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**, 10.1090/S0002-9939-1986-0826481-X**5.**R. Daniel Mauldin,*Problems in topology arising from analysis*, Open problems in topology, North-Holland, Amsterdam, 1990, pp. 617–629. MR**1078668****6.**Francisco J. Navarro-Bermúdez,*Topologically equivalent measures in the Cantor space*, Proc. Amer. Math. Soc.**77**(1979), no. 2, 229–236. MR**542090**, 10.1090/S0002-9939-1979-0542090-0**7.**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**, 10.1090/S0002-9939-1988-0939966-4**8.**John C. Oxtoby,*Homeomorphic measures in metric spaces*, Proc. Amer. Math. Soc.**24**(1970), 419–423. MR**0260961**, 10.1090/S0002-9939-1970-0260961-1**9.**John C. Oxtoby and Vidhu S. Prasad,*Homeomorphic measures in the Hilbert cube*, Pacific J. Math.**77**(1978), no. 2, 483–497. MR**510936****10.**J. C. Oxtoby and S. M. Ulam,*Measure-preserving homeomorphisms and metrical transitivity*, Ann. of Math. (2)**42**(1941), 874–920. MR**0005803****11.**R. G. E. Pinch,*Binomial equivalence of algebraic integers*, J. Indian Math. Soc. (N.S.)**58**(1992), no. 1-4, 33–37. MR**1207024****12.**Ernst S. Selmer,*On the irreducibility of certain trinomials*, Math. Scand.**4**(1956), 287–302. MR**0085223****13.**Felix Hausdorff,*Summationsmethoden und Momentfolgen. I*, Math. Z.**9**(1921), no. 1-2, 74–109 (German). MR**1544453**, 10.1007/BF01378337**14.**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****15.**G. G. Lorentz,*Bernstein polynomials*, Mathematical Expositions, no. 8, University of Toronto Press, Toronto, 1953. MR**0057370**

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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

DOI:
https://doi.org/10.1090/S0002-9947-07-04352-8

Keywords:
Homeomorphic measures,
Cantor space,
binomially reducible

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

Article copyright:
© Copyright 2007
American Mathematical Society

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