-
Previous Article
Countable Markov partitions suitable for thermodynamic formalism
- JMD Home
- This Volume
-
Next Article
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. 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. |
[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. 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. |
[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. 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 |
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. 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. |
[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. 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. |
[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. 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 |
[1] |
Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055 |
[2] |
Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026 |
[3] |
Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145. |
[4] |
Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281 |
[5] |
Johannes Kellendonk, Lorenzo Sadun. Conjugacies of model sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3805-3830. doi: 10.3934/dcds.2017161 |
[6] |
Jonathan DeWitt. Local Lyapunov spectrum rigidity of nilmanifold automorphisms. Journal of Modern Dynamics, 2021, 17: 65-109. doi: 10.3934/jmd.2021003 |
[7] |
Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 |
[8] |
Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675 |
[9] |
M. R. S. Kulenović, J. Marcotte, O. Merino. Properties of basins of attraction for planar discrete cooperative maps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2721-2737. doi: 10.3934/dcdsb.2020202 |
[10] |
Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597 |
[11] |
Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397 |
[12] |
Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119 |
[13] |
Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004 |
[14] |
Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166 |
[15] |
Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83 |
[16] |
Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 |
[17] |
Elena K. Kostousova. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021015 |
2019 Impact Factor: 0.465
Tools
Metrics
Other articles
by authors
[Back to Top]