Conformal iterated function systems

with applications to the geometry

of continued fractions

Authors:
R. Daniel Mauldin and Mariusz Urbanski

Journal:
Trans. Amer. Math. Soc. **351** (1999), 4995-5025

MSC (1991):
Primary 28A80; Secondary 58F08, 58F11, 28A78

DOI:
https://doi.org/10.1090/S0002-9947-99-02268-0

Published electronically:
July 21, 1999

MathSciNet review:
1487636

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior. We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmetic density properties of the set of allowed entries.

**[Cu]**T.W. Cusick,*Hausdorff dimension of sets of continued fractions*, Quart. J. Math. Oxford (2), 41 (1990), 277-286. MR**91m:11057****[E]**P. Erdös,*On the difference of consecutive primes*, Quart. J. Oxford 6 (1935), 124-128.**[Go]**I.J. Good,*The fractional dimension theory of continued fractions*, Proc. Camb. Philos. Soc. 37(1941), 199-228; Corrigenda, Math. Proc. Camb. Phil. Soc. 105(1989), 607; Addenda, J. Statist. Comput. and Simul. 32, Nos. 1-2 (1989), 64-68. MR**3:7b**; MR**90i:28013****[He]**D. Hensley,*Continued fraction Cantor sets, Hausdorff dimension, and functional analysis*, J. Number Th. 40(1992), 336-358. MR**93c:11058****[M]**H. Maier,*Chains of large gaps between consecutive primes*, Advances Math. 39 (1981), 257-269. MR**82g:10061****[Ma]**P. Mattila,*Geometry of sets and measures in euclidean spaces*, Cambridge Studies in Advanced Mathematics 44(Cambridge University Press, 1995). MR**96h:28006****[MU]**R.D. Mauldin, M. Urbanski,*Dimensions and measures in infinite iterated function systems*, Proceedings London Math. Soc., (3) 73(1996), 105-154. MR**97c:28020****[Ra]**G. Ramharter,*Some metrical properties of continued fractions*, Mathematika 30 (1983), 117-132. MR**85c:11058**

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Additional Information

**R. Daniel Mauldin**

Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203

Email:
mauldin@unt.edu

**Mariusz Urbanski**

Affiliation:
Department of Mathematics, University of North Texas, Denton, Texas 76203

Email:
urbanski@unt.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02268-0

Keywords:
Iterated function systems,
continued fractions,
Hausdorff dimension,
Hausdorff and packing measures,
arithmetic densities

Received by editor(s):
April 4, 1997

Published electronically:
July 21, 1999

Additional Notes:
Research supported by NSF Grant DMS-9502952

Article copyright:
© Copyright 1999
American Mathematical Society