Jordan, Thomas and Pollicott, Mark. (2006) Properties of measures supported on fat Sierpinski carpets. Ergodic Theory and Dynamical Systems, Vol.26 (No.3). pp. 739-754. ISSN 0143-3857

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Abstract

In this paper we study certain conformal iterated function schemes in two dimensions that are natural generalizations of the Sierpinski carpet construction. In particular, we consider scaling factors for which the open set condition fails. For such ‘fat Sierpinski carpets’ we study the range of parameters for which the dimension of the set is exactly known, or for which the set has positive measure.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Fractals, Dimension theory (Topology), Dimensional analysis
Journal or Publication Title:	Ergodic Theory and Dynamical Systems
Publisher:	Cambridge University Press
ISSN:	0143-3857
Date:	June 2006
Volume:	Vol.26
Number:	No.3
Page Range:	pp. 739-754
Identification Number:	10.1017/S0143385705000696
Status:	Peer Reviewed
Access rights to Published version:	Open Access
References:	[1] L. M. Abramov and V. A. Rohlin. The entropy of a skew product of measure preserving transformations. Trans. Amer. Math. Soc. (2) 48 (1966), 255–265. [2] C. Bishop. Topics in real analysis (unpublished lecture notes) http://www.math.sunysb.edu/∼bishop/classes/math639.S01/math639.html [3] T. Bogensch¨utz and H. Crauel. The Abramov–Rokhlin Formula (Lecture Notes in Mathematics, 1514). Springer, Berlin, 1992, 32–35. [4] D. Broomhead, J. Montaldi and N. Sidorov. Golden gaskets: variations on the Sierpinski sieve. Preprint. Available at: http://www.ma.umist.edu/∼nikita/gold-final.pdf [5] K. Falconer. Fractal Geometry. Wiley, London, 1990. [6] L. Jonker and J. Veerman. Semi-continuity of dimension and measure of locally scaled fractals. Fund. Math. 173 (2002), 113–131. [7] T. Jordan. Dimension of Fat Sierpinski gaskets. Preprint. Available at: http://www.maths.warwick.ac.uk/∼tjordan/overlapgasket.pdf. [8] F. Ledrappier and L.-S. Young. The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension. Ann. of Math. (2) 112 (1985), 540–574. [9] F. Ledrappier and P.Walters. A relativised variational principle for continuous transformations. J. London. Math. Soc. 16 (1977), 568–576. [10] C. McMullen. The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96 (1984), 1–9. [11] Y. Peres and B. Solomyak. Self-similar measures and the intersection of Cantor sets. Trans. Amer. Math. Soc. 350 (1998), 4065–4087. [12] Y. Peres and B. Solomyak. Problems on self-similar and self-affine sets; an update. Progr. Probab. 46 (2000), 95–106. [13] K. Petersen. Ergodic Theory. Cambridge University Press, Cambridge, 1983. [14] M. Pollicott and K. Simon. The Hausdorff dimension of λ-expansions with deleted digits. Trans. Amer. Math. Soc. 347 (1995), 967–983. [15] V. Rohlin. Lectures on the entropy theory of measure preserving transformations. Russian Math. Surveys 22(5) (1967), 1–52. [16] V. Rohlin. On the fundamental ideas of measure theory. Trans. Amer. Math. Soc. 71 (1952), 1–54. [17] K. Simon and B. Solomyak. On the dimension of self-similar sets. Fractals 10 (2003), 59–65. [18] K. Simon, B. Solomyak and M. Urbanski. Invariant measures for parabolic IFS with overlaps and random continued fractions. Trans. Amer. Math. Soc. 353 (2001), 5145–5164. [19] B. Solomyak. On the random series[SUM]±λn (an Erdös problem). Ann. of Math. (2) 142 (1995), 611–625. [20] P. Walters. Ergodic Theory. Springer, Berlin, 1982. [21] M. Csörnyei, T. Jordan, M. Pollicott, D. Preiss and B. Solomyak. Positive-measure self-similar sets without interior, this issue.
URI:	http://wrap.warwick.ac.uk/id/eprint/691

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