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Symmetry of entropy in higher rank diagonalizable actions and measure classification
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 |
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.
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. |
show all references
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. |
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