Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-15T20:23:26.879Z Has data issue: false hasContentIssue false
  • English
  • Français

SHRINKING TARGETS FOR NONAUTONOMOUS DYNAMICAL SYSTEMS CORRESPONDING TO CANTOR SERIES EXPANSIONS

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Complex dynamical systems Ergodic theory

Published online by Cambridge University Press:  02 June 2015

LIOR FISHMAN ,
BILL MANCE ,
DAVID SIMMONS  and
MARIUSZ URBAŃSKI
LIOR FISHMAN
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA email lior.fishman@unt.edu
BILL MANCE
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA email mance@unt.edu
DAVID SIMMONS*
Affiliation:
Ohio State University, Department of Mathematics, 231 W. 18th Avenue, Columbus, OH 43210-1174, USA email simmons.465@osu.edu
MARIUSZ URBAŃSKI
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX 76203-5017, USA email urbanski@unt.edu
*
✉simmons.465@osu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the interval $[0,1)$, viewed as $\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilising our theorem. In particular, we give a Diophantine approximation interpretation of our scheme.

Keywords

nonautonomous dynamical systemCantor series expansionHausdorff dimensionshrinking target schemeDiophantine approximation

MSC classification

Primary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Secondary: 37A50: Relations with probability theory and stochastic processes 37F35: Conformal densities and Hausdorff dimension
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 2 , October 2015 , pp. 205 - 213
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Airey, D. and Mance, B., ‘Unexpected distribution phenomenon resulting from Cantor series expansions’, Preprint, 2014.CrossRefGoogle Scholar
Airey, D., Mance, B. and Vandehey, J., ‘Normality preserving operations for Cantor series expansions and associated fractals, II’, Preprint, 2014.CrossRefGoogle Scholar
Besicovic̆, A., ‘Sets of fractional dimension (IV): on rational approximation to real numbers’, J. Lond. Math. Soc. (2) 9 (1934), 126–131.CrossRefGoogle Scholar
Bugeaud, Y. and Liao, L., ‘Uniform Diophantine approximation related to $b$-ary and ${\it\beta}$-expansions’, Preprint, 2014.CrossRefGoogle Scholar
Bugeaud, Y. and Wang, B., ‘Distribution of full cylinders and the Diophantine properties of the orbits in 𝛽-expansions’, J. Fractal Geom. 1 (2014), 221–241.CrossRefGoogle Scholar
Cantor, G., ‘Uber die einfachen Zahlensysteme’, Z. Math. Phys. 14 (1869), 121–128.Google Scholar
Erdős, P. and Rényi, A., ‘On Cantor’s series with convergent ∑1∕q n’, Ann. Univ. Sci. Eötvös Budapest. Sect. Math. 2 (1959), 93–109.Google Scholar
Erdős, P. and Rényi, A., ‘Some further statistical properties of the digits in Cantor’s series’, Acta Math. Acad. Sci. Hungar. 10 (1959), 21–29.CrossRefGoogle Scholar
Fishman, L., Simmons, D. and Urbański, M., Diophantine approximation in Banach spaces’, Preprint, 2013, J. Théor. Nombres Bordeaux, to appear.CrossRefGoogle Scholar
Galambos, J., ‘Uniformly distributed sequences mod 1 and Cantor’s series representation’, Czechoslovak Math. J. 26 (1976), 636–641.CrossRefGoogle Scholar
Hill, R. and Velani, S., ‘Metric Diophantine approximation in Julia sets of expanding rational maps’, Publ. Math. Inst. Hautes Études Sci. 85 (1997), 193–216.CrossRefGoogle Scholar
Hill, R. and Velani, S., ‘The Jarník–Besicovitch theorem for geometrically finite Kleinian groups’, Proc. Lond. Math. Soc. 77 (1998), 524–550.CrossRefGoogle Scholar
Jarník, V., ‘Diophantische Approximationen und Hausdorffsches Maß’, Mat. Sb. 36 (1929), 371–382.Google Scholar
Mance, B., ‘Number theoretic applications of a class of Cantor series fractal functions, I’, Acta Math. Hungar. 144(2) (2014), 449–493.CrossRefGoogle Scholar
Reeve, H., ‘Shrinking targets for countable Markov maps’, Preprint, 2011.Google Scholar
Rényi, A., ‘On a new axiomatic theory of probability’, Acta Math. Acad. Sci. Hungar. 6 (1955), 329–332.CrossRefGoogle Scholar
Rényi, A., ‘On the distribution of the digits in Cantor’s series’, Mat. Lapok 7 (1956), 77–100.Google Scholar
Rényi, A., ‘Probabilistic methods in number theory’, Shuxue Jinzhan 4 (1958), 465–510.Google Scholar
S̆alát, T., ‘Eine metrische Eigenschaft der Cantorschen Entwicklungen der reelen Zahlen und Irrationalitätskriterien’, Czechoslovak Math. J. 14 (1964), 254–266.Google Scholar
Schweiger, F., ‘Über den Satz von Borel–Rényi in der Theorie der Cantorschen Reihen’, Monatsh. Math. 74 (1969), 150–153.CrossRefGoogle Scholar
Stratmann, B., ‘Fractal dimensions for Jarník limit sets, the semi-classical approach’, Ark. Mat. 33 (1995), 385–403.CrossRefGoogle Scholar
Stratmann, B. and Urbański, M., ‘Jarník and Julia; a Diophantine analysis for parabolic rational maps for geometrically finite Kleinian groups with parabolic elements’, Math. Scand. 91 (2002), 27–54.CrossRefGoogle Scholar
Turán, P., ‘On the distribution of ‘digits’ in Cantor systems’, Mat. Lapok 7 (1956), 71–76.Google Scholar
Urbański, M., ‘The Diophantine analysis of conformal iterated function systems’, Monatsh. Math. 137 (2002), 325–340.CrossRefGoogle Scholar