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  2019, 15: 437-449. doi: 10.3934/jmd.2019027

From invariance to self-similarity: The work of Michael Hochman on fractal dimension and its aftermath

Hillel Furstenberg

Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Received  November 20, 2019 Published  December 2019

M. Hochman's work on the dimension of self-similar sets has given impetus to resolving other questions regarding fractal dimension. We describe Hochman's approach and its influence on the subsequent resolution by P. Shmerkin of the conjecture on the dimension of the intersection of $ \times p $- and $ \times q $-Cantor sets.

Keywords: Hausdorff dimension, probability measures, partitions, entropy, self-similarity, convolution, Cantor sets, $ L^q $-norm.
Mathematics Subject Classification: Primary: 28A80, 37C45; Secondary: 11K55, 11B30.
Citation: Hillel Furstenberg. From invariance to self-similarity: The work of Michael Hochman on fractal dimension and its aftermath. Journal of Modern Dynamics, 2019, 15: 437-449. doi: 10.3934/jmd.2019027
References:
[1]

H. Furstenberg, Intersections of Cantor sets and transversality of semigroups. In Problems in Analysis (Symposium in honor of Salomon Bochner, Princeton University Press, Princeton, N.J. 1969), 41–59, 1970.  Google Scholar

[2]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.  Google Scholar

[3]

M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2), 180 (2014), 773-822.  doi: 10.4007/annals.2014.180.2.7.  Google Scholar

[4]

P. Shmerkin, On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$-norms of convolutions, Ann. of Math. (2), 189 (2019), 319–391. doi: 10.4007/annals.2019.189.2.1.  Google Scholar

[5]

M. Wu, A proof of Furstenberg's conjecture on the intersection of $\times p$ and $\times q$-invariant sets, arXiv: 1609.08053v3, February 2019. Google Scholar

show all references

References:
[1]

H. Furstenberg, Intersections of Cantor sets and transversality of semigroups. In Problems in Analysis (Symposium in honor of Salomon Bochner, Princeton University Press, Princeton, N.J. 1969), 41–59, 1970.  Google Scholar

[2]

H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, 1 (1967), 1-49.  doi: 10.1007/BF01692494.  Google Scholar

[3]

M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2), 180 (2014), 773-822.  doi: 10.4007/annals.2014.180.2.7.  Google Scholar

[4]

P. Shmerkin, On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$-norms of convolutions, Ann. of Math. (2), 189 (2019), 319–391. doi: 10.4007/annals.2019.189.2.1.  Google Scholar

[5]

M. Wu, A proof of Furstenberg's conjecture on the intersection of $\times p$ and $\times q$-invariant sets, arXiv: 1609.08053v3, February 2019. Google Scholar

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