Geodesic
A certain family of self-similar sets
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2011), pp. 31-45.

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In this paper we consider an one-parameter family of iterated function systems. For every value of the parameter we find the set of top addresses. We prove that this set is a countable disjoint union of self-similar sets and calculate its Hausdorff dimension.
Keywords: iterated function systems, attractor, top address, self-similarity
Mots-clés : Hausdorff dimension. 
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K. B. Igudesman. A certain family of self-similar sets. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2011), pp. 31-45. http://geodesic.mathdoc.fr/item/IVM_2011_2_a3/

[1] Hutchinson J., “Fractals and self similarity”, Indiana Univ. Math J., 30 (1981), 713–747 | DOI | MR | Zbl

[2] Barnsley M. F., “Theory and applications of fractal tops”, Fractals in Engineering: New Trends in Theory and Applications, Springer-Verlag, London, 2005, 3–20 | Zbl

[3] Rényi A., “Representations for real numbers and their ergodic properties”, Acta Math. Acad. Sci. Hung., 8 (1957), 477–493 | DOI | MR | Zbl

[4] Parry W., “On the $\beta$-expansions of real numbers”, Acta Math. Acad. Sci. Hung., 11 (1960), 401–416 | DOI | MR | Zbl

[5] Erdős P., “On a family of symmetric Bernoulli convolutions”, Amer. J. Math., 61 (1939), 974–976 | DOI | MR | Zbl

[6] Peres Y., Schlag W., Solomyak B., “Sixty years of Bernoulli convolutions”, Fractal geom. and stochastics, Proc. of the 2nd conf. (Greifswald, Germany), v. II, Birkhäuser, Basel, 2000, 39–65 | MR | Zbl

[7] Moran M., “Hausdorff measure of infinitely generated self-similar sets”, Monatsh. Math., 122:4 (1996), 387–399 | DOI | MR | Zbl

[8] Sidorov N., “Almost every number has a continuum of $\beta$-expansions”, Amer. Math. Mon., 110:9 (2003), 838–842 | DOI | MR | Zbl

[9] Barnsley M., Superfractals, Cambridge University Press, Cambridge, 2006 | MR | Zbl

[10] Baiocchi C., Komornik V., Greedy and quasi-greedy expansions in non-integer bases, 16 Oct. 2007, arXiv: 0710.3001v1[math.NT]

[11] Igudesman K. B., “Verkhnie adresa dlya odnogo semeistva sistem iterirovannykh funktsii na otrezke”, Izv. vuzov. Matematika, 2009, no. 9, 75–81 | MR | Zbl

[12] Sidorov N., Vershik A., “Ergodic properties of the Erdős measure the entropy of the goldenshift, and related problems”, Monatsh. Math., 126:3 (1998), 215–261 | DOI | MR | Zbl

[13] Falconer K., The geometry of fractal sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge etc., 1985 | MR | Zbl