×

On self-similar sets with overlaps and inverse theorems for entropy. (English) Zbl 1337.28015

Let \(\Phi :=\{\varphi_i : i\in \Lambda\}\) be a finite collection of linear contractions of the form \(\varphi_i (x) := r_i x + a_i\), where \(|r_i| < 1\) and \(a_i\in \mathbb R\). Let \(X\neq\emptyset\) be the attractor of the iterated function system \(\Phi\) and let the self-similar measure associated with \(\Phi\) and a probability vector \((p_i)_{i\in \Lambda}\) be denoted by \(\mu\). The author investigates the Hausdorff dimension \(\dim \mu\) of \(\mu\) in case the images \(\varphi_i X\) have significant overlap. The main result is the following: If \(\dim\mu < \min\{1, s\}\), where \(s\) is the similarity dimension of \(X\), then there exist superexponentially close cylinders at small enough scales. As a consequence of this result, the Furstenberg conjecture on the projections of the one-dimensional Sierpiński gasket is proven and some progress on the Bernoulli convolutions problem is achieved.

MSC:

28A80 Fractals
37L40 Invariant measures for infinite-dimensional dissipative dynamical systems
37C35 Orbit growth in dynamical systems
28D20 Entropy and other invariants
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] D. Feng and H. Hu, ”Dimension theory of iterated function systems,” Comm. Pure Appl. Math., vol. 62, iss. 11, pp. 1435-1500, 2009. · Zbl 1230.37031 · doi:10.1002/cpa.20276
[2] Y. Peres and B. Solomyak, ”Problems on self-similar sets and self-affine sets: an update,” in Fractal Geometry and Stochastics, II, Basel: Birkhäuser, 2000, vol. 46, pp. 95-106. · Zbl 0946.28003
[3] A. M. Garsia, ”Arithmetic properties of Bernoulli convolutions,” Trans. Amer. Math. Soc., vol. 102, pp. 409-432, 1962. · Zbl 0103.36502 · doi:10.2307/1993615
[4] H. Furstenberg, ”Intersections of Cantor sets and transversality of semigroups,” in Problems in Analysis, Princeton, N.J.: Princeton Univ. Press, 1970, pp. 41-59. · Zbl 0208.32203
[5] H. Furstenberg, ”Ergodic fractal measures and dimension conservation,” Ergodic Theory Dynam. Systems, vol. 28, iss. 2, pp. 405-422, 2008. · Zbl 1154.37322 · doi:10.1017/S0143385708000084
[6] R. Kenyon, ”Projecting the one-dimensional Sierpinski gasket,” Israel J. Math., vol. 97, pp. 221-238, 1997. · Zbl 0871.28006 · doi:10.1007/BF02774038
[7] G. Świcatek and J. J. P. Veerman, ”On a conjecture of Furstenberg,” Israel J. Math., vol. 130, pp. 145-155, 2002. · Zbl 1022.37016 · doi:10.1007/BF02764075
[8] Y. Peres and W. Schlag, ”Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions,” Duke Math. J., vol. 102, iss. 2, pp. 193-251, 2000. · Zbl 0961.42007 · doi:10.1215/S0012-7094-00-10222-0
[9] J. Bourgain, ”The discretized sum-product and projection theorems,” J. Anal. Math., vol. 112, pp. 193-236, 2010. · Zbl 1234.11012 · doi:10.1007/s11854-010-0028-x
[10] M. Hochman, Self-similar sets with overlaps and sumset phenomena for entropy, the multidimensional case, 2012.
[11] M. Pollicott and K. Simon, ”The Hausdorff dimension of \(\lambda\)-expansions with deleted digits,” Trans. Amer. Math. Soc., vol. 347, iss. 3, pp. 967-983, 1995. · Zbl 0831.28005 · doi:10.2307/2154881
[12] B. Solomyak, ”On the random series \(\sum\pm\lambda^n\) (an Erd\Hos problem),” Ann. of Math., vol. 142, iss. 3, pp. 611-625, 1995. · Zbl 0837.28007 · doi:10.2307/2118556
[13] P. Shmerkin and B. Solomyak, ”Zeros of \(\{-1,0,1\}\) power series and connectedness loci for self-affine sets,” Experiment. Math., vol. 15, iss. 4, pp. 499-511, 2006. · Zbl 1122.30002 · doi:10.1080/10586458.2006.10128977
[14] P. Erdös, ”On a family of symmetric Bernoulli convolutions,” Amer. J. Math., vol. 61, pp. 974-976, 1939. · Zbl 0022.35402 · doi:10.2307/2371641
[15] P. Erdös, ”On the smoothness properties of a family of Bernoulli convolutions,” Amer. J. Math., vol. 62, pp. 180-186, 1940. · Zbl 0022.35403 · doi:10.2307/2371446
[16] Y. Peres, W. Schlag, and B. Solomyak, ”Sixty years of Bernoulli convolutions,” in Fractal Geometry and Stochastics, II, Basel: Birkhäuser, 2000, vol. 46, pp. 39-65. · Zbl 0961.42006
[17] J. C. Alexander and J. A. Yorke, ”Fat baker’s transformations,” Ergodic Theory Dynam. Systems, vol. 4, iss. 1, pp. 1-23, 1984. · Zbl 0553.58020 · doi:10.1017/S0143385700002236
[18] F. Przytycki and M. Urbański, ”On the Hausdorff dimension of some fractal sets,” Studia Math., vol. 93, iss. 2, pp. 155-186, 1989. · Zbl 0691.58029
[19] M. Keane, M. Smorodinsky, and B. Solomyak, ”On the morphology of \(\gamma\)-expansions with deleted digits,” Trans. Amer. Math. Soc., vol. 347, iss. 3, pp. 955-966, 1995. · Zbl 0834.11033 · doi:10.2307/2154880
[20] M. Nicol, N. Sidorov, and D. Broomhead, ”On the fine structure of stationary measures in systems which contract-on-average,” J. Theoret. Probab., vol. 15, iss. 3, pp. 715-730, 2002. · Zbl 1026.28006 · doi:10.1023/A:1016224000145
[21] Y. Peres, K. Simon, and B. Solomyak, ”Absolute continuity for random iterated function systems with overlaps,” J. London Math. Soc., vol. 74, iss. 3, pp. 739-756, 2006. · Zbl 1122.37018 · doi:10.1112/S0024610706023258
[22] A. M. Garsia, ”Entropy and singularity of infinite convolutions,” Pacific J. Math., vol. 13, pp. 1159-1169, 1963. · Zbl 0126.14901 · doi:10.2140/pjm.1963.13.1159
[23] P. Shmerkin, ”On the exceptional set for absolute continuity of Bernoulli convolutions,” Geom. Funct. Anal., vol. 24, pp. 946-958. · Zbl 1305.28012 · doi:10.1007/s00039-014-0285-4
[24] T. Tao and V. Vu, Additive Combinatorics, Cambridge: Cambridge Univ. Press, 2006, vol. 105. · Zbl 1127.11002 · doi:10.1017/CBO9780511755149
[25] T. Tao, ”Sumset and inverse sumset theory for Shannon entropy,” Combin. Probab. Comput., vol. 19, iss. 4, pp. 603-639, 2010. · Zbl 1239.11015 · doi:10.1017/S0963548309990642
[26] J. Bourgain, N. Katz, and T. Tao, ”A sum-product estimate in finite fields, and applications,” Geom. Funct. Anal., vol. 14, iss. 1, pp. 27-57, 2004. · Zbl 1145.11306 · doi:10.1007/s00039-004-0451-1
[27] N. H. Katz and T. Tao, ”Some connections between Falconer’s distance set conjecture and sets of Furstenburg type,” New York J. Math., vol. 7, pp. 149-187, 2001. · Zbl 0991.28006
[28] J. Bourgain, ”On the Erd\Hos-Volkmann and Katz-Tao ring conjectures,” Geom. Funct. Anal., vol. 13, iss. 2, pp. 334-365, 2003. · Zbl 1115.11049 · doi:10.1007/s000390300008
[29] M. Hochman and P. Shmerkin, Equidistribution from fractals, 2011. · Zbl 1409.11054
[30] P. ErdHos and B. Volkmann, ”Additive Gruppen mit vorgegebener Hausdorffscher Dimension,” J. Reine Angew. Math., vol. 221, pp. 203-208, 1966. · Zbl 0135.10202
[31] T. W. Körner, ”Hausdorff dimension of sums of sets with themselves,” Studia Math., vol. 188, iss. 3, pp. 287-295, 2008. · Zbl 1200.28007 · doi:10.4064/sm188-3-4
[32] J. Schmeling and P. Shmerkin, ”On the dimension of iterated sumsets,” in Recent Developments in Fractals and Related Fields, Boston, MA: Birkhäuser, 2010, pp. 55-72. · Zbl 1216.28011 · doi:10.1007/978-0-8176-4888-6_5
[33] C. Esseen, ”On the Liapounoff limit of error in the theory of probability,” Ark. Mat. Astr. Fys., vol. 28A, iss. 9, p. 19, 1942. · Zbl 0027.33902
[34] V. A. Kauimanovich and A. M. Vershik, ”Random walks on discrete groups: boundary and entropy,” Ann. Probab., vol. 11, iss. 3, pp. 457-490, 1983. · Zbl 0641.60009 · doi:10.1214/aop/1176993497
[35] M. Madiman, ”On the entropy of sums,” in Information Theory Workshop, 2008. ITW \('\)08, , 2008, pp. 303-307. · doi:10.1109/ITW.2008.4578674
[36] M. Madiman, A. W. Marcus, and P. Tetali, ”Entropy and set cardinality inequalities for partition-determined functions,” Random Structures Algorithms, vol. 40, iss. 4, pp. 399-424, 2012. · Zbl 1244.05024 · doi:10.1002/rsa.20385
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.