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

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

Published electronically:
July 23, 2007

MathSciNet review:
2336321

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.**E. Akin, Good measures on Cantor space, Trans. Amer. Math. Soc. 357 (2005), 2681-2722. MR**2139523 (2006e:37003)****2.**S. Alpern and V. S. Prasad, Typical dynamics of volume preserving homeomorphisms, Cambridge Tracts in Mathematics, 139, Cambridge University Press, Cambridge, 2000. MR**1826331 (2002i:37006)****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), 560-562. MR**826481 (87b:11100)****5.**R. D. MAULDIN, Problems in topology arising from analysis, in Open problems in topology (J. van Mill and G. M. Reed, eds.), North-Holland, Amsterdam, 1990, pp. 617-629. MR**1078668****6.**F. J. Navarro-Bermudez, Topologically equivalent measures in the Cantor space, Proc. Amer. Math. Soc. 77 (1979), 229-236. MR**542090 (80k:28017)****7.**F. J. Navarro-Bermudez and J. C. Oxtoby, Four topologically equivalent measures in the Cantor space, Proc. Amer. Math. Soc. 104 (1988), 859-860. MR**939966 (90c:28020)****8.**J. C. Oxtoby, Homeomorphic measures in metric spaces, Proc. Amer. Math. Soc. 24 (1970), 419-423. MR**0260961 (41:5581)****9.**J. C. Oxtoby and V. S. Prasad, Homeomorphic measures in the Hilbert cube, Pac. J. Math. 77 (1978), 483-497. MR**510936 (80h:28006)****10.**J. C. OXTOBY AND S. M. ULAM, Measure preserving homeomorphisms and metrical transitivity, Ann. Math. 42 (1941), 874-920. MR**0005803 (3:211b)****11.**R. G. E. Pinch, Binomial equivalence of algebraic numbers, J. Indian Math. Soc. (N.S.) 58 (1992), 33-37. MR**1207024 (94a:11160)****12.**E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand. 4 (1956), 287-302. MR**0085223 (19:7f)****13.**F. Hausdorff, Summationsmethoden und Momentfolgen I, Math. Z. 9 (1921), 74-109. MR**1544453****14.**G. Polya and G Szego, Problems and Theorems in Analysis II, Springer, 1972. MR**1492448****15.**G. G. Lorentz, Bernstein Polynomials, Toronto: University of Toronto Press, 1953. MR**0057370 (15:217a)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
37B05,
28D05,
28C15

Retrieve articles in all journals with MSC (2000): 37B05, 28D05, 28C15

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.