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 (Brin Prize article)

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.

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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

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