# American Institute of Mathematical Sciences

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

## Smooth symmetries of $\times a$-invariant sets

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

Received  July 19, 2017 Revised  March 18, 2018 Published  December 2018

Fund Project: Supported by ERC grant 306494 and ISF grant 1702/17

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.

Citation: Michael Hochman. Smooth symmetries of $\times a$-invariant sets. Journal of Modern Dynamics, 2018, 13: 187-197. doi: 10.3934/jmd.2018017
##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] K. J. Falconer and D. T. Marsh, On the Lipschitz equivalence of Cantor sets, Mathematika, 39 (1992), 223-233. doi: 10.1112/S0025579300014959. Google Scholar [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. Google Scholar [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. Google Scholar [9] M. Hochman, Dynamics on fractal measures, preprint, arXiv: 1008.3731, 2010.Google Scholar [10] M. Hochman, Geometric rigidity of $× m$ invariant measures, J. Eur. Math. Soc. (JEMS), 14 (2012), 1539-1563. doi: 10.4171/JEMS/340. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [14] M. Hochman and P. Shmerkin, Equidistribution from fractal measures, Inventiones Mathematicae, 202 (2015), 427-479. doi: 10.1007/s00222-014-0573-5. Google Scholar [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. Google Scholar [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. Google Scholar [17] P. Shmerkin, On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions, preprint, arXiv: 1609.07802, 2016.Google Scholar [18] M. Wu, A proof of Furstenberg's conjecture on the intersections of $×$p and $×$q-invariant sets, preprint, arXiv: 1609.08053, 2016.Google Scholar

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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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] K. J. Falconer and D. T. Marsh, On the Lipschitz equivalence of Cantor sets, Mathematika, 39 (1992), 223-233. doi: 10.1112/S0025579300014959. Google Scholar [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. Google Scholar [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. Google Scholar [9] M. Hochman, Dynamics on fractal measures, preprint, arXiv: 1008.3731, 2010.Google Scholar [10] M. Hochman, Geometric rigidity of $× m$ invariant measures, J. Eur. Math. Soc. (JEMS), 14 (2012), 1539-1563. doi: 10.4171/JEMS/340. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [14] M. Hochman and P. Shmerkin, Equidistribution from fractal measures, Inventiones Mathematicae, 202 (2015), 427-479. doi: 10.1007/s00222-014-0573-5. Google Scholar [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. Google Scholar [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. Google Scholar [17] P. Shmerkin, On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions, preprint, arXiv: 1609.07802, 2016.Google Scholar [18] M. Wu, A proof of Furstenberg's conjecture on the intersections of $×$p and $×$q-invariant sets, preprint, arXiv: 1609.08053, 2016.Google Scholar
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