Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the morphology of $ \gamma$-expansions with deleted digits


Authors: Mike Keane, Meir Smorodinsky and Boris Solomyak
Journal: Trans. Amer. Math. Soc. 347 (1995), 955-966
MSC: Primary 11K55; Secondary 28A12
DOI: https://doi.org/10.1090/S0002-9947-1995-1290723-X
MathSciNet review: 1290723
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the size of the set of reals which can be represented in base $ \gamma $ using only the digits $ 0,1,3$. It is shown that this set has Lebesgue measure zero for $ \gamma \leqslant 1/3$ and equals an interval for $ \gamma \geqslant 2/5$. Our main goal is to prove that it has Lebesgue measure zero for a certain countable subset of $ (1/3,2/5)$.


References [Enhancements On Off] (What's this?)

  • [P] W. Parry, On the $ \beta $-expansion of real numbers., Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416. MR 0142719 (26:288)
  • [PS] M. Pollicott and K. Simon, The Hausdorff dimension of $ \lambda $-expansions with deleted digits, Trans. Amer. Math. Soc. 347 (1995), 967-983. MR 1290729 (95h:11080)
  • [PT] J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Univ. Press, Cambridge, 1993. MR 1237641 (94h:58129)
  • [R] A. Renyi, Representations for real numbers and their ergodic properties., Acta Math. Acad. Sci. Hungar. 8 (1957), 472-493. MR 0097374 (20:3843)
  • [S] M. Smorodinsky, $ \beta $-automorphisms are Bernoulli shifts, Acta Math. Acad. Sci. Hungar. 24 (1973), 273-278. MR 0346133 (49:10859)
  • [W] P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1982. MR 648108 (84e:28017)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11K55, 28A12

Retrieve articles in all journals with MSC: 11K55, 28A12


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1290723-X
Keywords: $ \beta $-expansions, Cantor sets, Hausdorff dimension
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society