 Previous Article
 JMD Home
 This Volume

Next Article
The work of Mike Hochman on multidimensional symbolic dynamics and Borel dynamics
From invariance to selfsimilarity: 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 
M. Hochman's work on the dimension of selfsimilar 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.
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), 149. doi: 10.1007/BF01692494. 
[3] 
M. Hochman, On selfsimilar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2), 180 (2014), 773822. doi: 10.4007/annals.2014.180.2.7. 
[4] 
P. Shmerkin, On Furstenberg's intersection conjecture, selfsimilar 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), 149. doi: 10.1007/BF01692494. 
[3] 
M. Hochman, On selfsimilar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2), 180 (2014), 773822. doi: 10.4007/annals.2014.180.2.7. 
[4] 
P. Shmerkin, On Furstenberg's intersection conjecture, selfsimilar 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. 
[1] 
Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 24172436. doi: 10.3934/dcds.2012.32.2417 
[2] 
S. Astels. Thickness measures for Cantor sets. Electronic Research Announcements, 1999, 5: 108111. 
[3] 
Krzysztof Barański. Hausdorff dimension of selfaffine limit sets with an invariant direction. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 10151023. doi: 10.3934/dcds.2008.21.1015 
[4] 
Meng Ding, TingZhu Huang, XiLe Zhao, Michael K. Ng, TianHui Ma. Tensor train rank minimization with nonlocal selfsimilarity for tensor completion. Inverse Problems and Imaging, 2021, 15 (3) : 475498. doi: 10.3934/ipi.2021001 
[5] 
Rogelio Valdez. Selfsimilarity of the Mandelbrot set for real essentially bounded combinatorics. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 897922. doi: 10.3934/dcds.2006.16.897 
[6] 
Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457464. doi: 10.3934/jmd.2008.2.457 
[7] 
Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114118. 
[8] 
Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 32933313. doi: 10.3934/dcds.2015.35.3293 
[9] 
José Ignacio AlvarezHamelin, Luca Dall'Asta, Alain Barrat, Alessandro Vespignani. Kcore decomposition of Internet graphs: hierarchies, selfsimilarity and measurement biases. Networks and Heterogeneous Media, 2008, 3 (2) : 371393. doi: 10.3934/nhm.2008.3.371 
[10] 
Peter V. Gordon, Cyrill B. Muratov. Selfsimilarity and longtime behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks and Heterogeneous Media, 2012, 7 (4) : 767780. doi: 10.3934/nhm.2012.7.767 
[11] 
Changming Song, Yun Wang. Nonlocal latent low rank sparse representation for single image super resolution via selfsimilarity learning. Inverse Problems and Imaging, 2021, 15 (6) : 13471362. doi: 10.3934/ipi.2021017 
[12] 
Silvére Gangloff, Alonso Herrera, Cristobal Rojas, Mathieu Sablik. Computability of topological entropy: From general systems to transformations on Cantor sets and the interval. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 42594286. doi: 10.3934/dcds.2020180 
[13] 
Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 235246. doi: 10.3934/dcds.2008.22.235 
[14] 
Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 23752393. doi: 10.3934/dcds.2018098 
[15] 
Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of nonuniformly hyperbolic horseshoes. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 431448. doi: 10.3934/dcds.2018020 
[16] 
Fritz Colonius. Invariance entropy, quasistationary measures and control sets. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 20932123. doi: 10.3934/dcds.2018086 
[17] 
Maik Gröger, Olga Lukina. Measures and stabilizers of group Cantor actions. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 20012029. doi: 10.3934/dcds.2020350 
[18] 
Mehdi Pourbarat. On the arithmetic difference of middle Cantor sets. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 42594278. doi: 10.3934/dcds.2018186 
[19] 
Bernard Helffer, Thomas HoffmannOstenhof, Susanna Terracini. Nodal minimal partitions in dimension $3$. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 617635. doi: 10.3934/dcds.2010.28.617 
[20] 
Guozhen Lu, Yunyan Yang. Sharp constant and extremal function for the improved MoserTrudinger inequality involving $L^p$ norm in two dimension. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 963979. doi: 10.3934/dcds.2009.25.963 
2020 Impact Factor: 0.848
Tools
Article outline
[Back to Top]