Smooth symmetries of $\times a$-invariant sets

Michael Hochman

doi:10.3934/jmd.2018017

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2018, Volume 13: 187-197. Doi: 10.3934/jmd.2018017

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Smooth symmetries of $\times a$-invariant sets

Michael Hochman

Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

Received: July 18, 2017

Revised: March 17, 2018

Published: December 2018

Supported by ERC grant 306494 and ISF grant 1702/17.

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We study the smooth self-maps $f$ of $× a$-invariant sets $X\subseteq[0,1]$. Under various assumptions we show that this forces $\log f'(x)/\log a∈\mathbb{Q}$ at many points in $X$. Our method combines scenery flow methods and equidistribution results in the positive entropy case, where we improve previous work of the author and Shmerkin, with a new topological variant of the scenery flow which applies in the zero-entropy case.

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Mathematics Subject Classification: Primary: 28A80; Secondary: 11K55, 11B30, 11P70.

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References

[1]	D. Berend, Multi-invariant sets on tori, Trans. Amer. Math. Soc., 280 (1983), 509-532. doi: 10.1090/S0002-9947-1983-0716835-6.
[2]	D. Berend, Multi-invariant sets on compact abelian groups, Trans. Amer. Math. Soc., 286 (1984), 505-535. doi: 10.1090/S0002-9947-1984-0760973-X.
[3]	M. Einsiedler, A. Katok and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Ann. of Math. (2), 164 (2006), 513-560. doi: 10.4007/annals.2006.164.513.
[4]	M. Einsiedler and E. Lindenstrauss, Rigidity properties of $\Bbb Z^d$-actions on tori and solenoids, Electron. Res. Announc. Amer. Math. Soc., 9 (2003), 99-110 (electronic). doi: 10.1090/S1079-6762-03-00117-3.
[5]	M. Elekes, T. Keleti and A. Máthé, Self-similar and self-affine sets: Measure of the intersection of two copies, Ergodic Theory and Dynamical Systems, 30 (2010), 399-440. doi: 10.1017/S0143385709000121.
[6]	K. J. Falconer and D. T. Marsh, On the Lipschitz equivalence of Cantor sets, Mathematika, 39 (1992), 223-233. doi: 10.1112/S0025579300014959.
[7]	D.-J. Feng, W. Huang and H. Rao, Affine embeddings and intersections of Cantor sets, Journal de Mathématiques Pures et Appliquées, 102 (2014), 1062-1079. doi: 10.1016/j.matpur.2014.03.003.
[8]	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.
[9]	M. Hochman, Dynamics on fractal measures, preprint, arXiv: 1008.3731, 2010.
[10]	M. Hochman, Geometric rigidity of $× m$ invariant measures, J. Eur. Math. Soc. (JEMS), 14 (2012), 1539-1563. doi: 10.4171/JEMS/340.
[11]	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.
[12]	M. Hochman, Some problems on the boundary of fractal geometry and additive combinatorics, to appear in Proceedings of FARF 3, arXiv: 1608.02711, 2016.
[13]	M. Hochman and P. Shmerkin, Local entropy averages and projections of fractal measures, Ann. of Math. (2), 175 (2012), 1001-1059. doi: 10.4007/annals.2012.175.3.1.
[14]	M. Hochman and P. Shmerkin, Equidistribution from fractal measures, Inventiones Mathematicae, 202 (2015), 427-479. doi: 10.1007/s00222-014-0573-5.
[15]	B. Kalinin, A. Katok and F. Rodriguez Hertz, New progress in nonuniform measure and cocycle rigidity, Electron. Res. Announc. Math. Sci., 15 (2008), 79-92.
[16]	A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems, 16 (1996), 751-778. doi: 10.1017/S0143385700009081.
[17]	P. Shmerkin, On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions, preprint, arXiv: 1609.07802, 2016.
[18]	M. Wu, A proof of Furstenberg's conjecture on the intersections of $×$p and $×$q-invariant sets, preprint, arXiv: 1609.08053, 2016.