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Fourier coefficients of $\times p$-invariant measures
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China |
We consider densities $D_\Sigma(A)$, $\overline{D}_\Sigma(A)$ and $\underline{D}_\Sigma(A)$ for a subset $A$ of $\mathbb{N}$ with respect to a sequence $\Sigma$ of finite subsets of $\mathbb{N}$ and study Fourier coefficients of ergodic, weakly mixing and strongly mixing $\times p$-invariant measures on the unit circle $\mathbb{T}$. Combining these, we prove the following measure rigidity results: on $\mathbb{T}$, the Lebesgue measure is the only non-atomic $\times p$-invariant measure satisfying one of the following: (1) $\mu$ is ergodic and there exist a Følner sequence $\Sigma$ in $\mathbb{N}$ and a nonzero integer $l$ such that $\mu$ is $\times (p^j+l)$-invariant for all $j$ in a subset $A$ of $\mathbb{N}$ with $D_\Sigma(A)=1$; (2) $\mu$ is weakly mixing and there exist a Følner sequence $\Sigma$ in $\mathbb{N}$ and a nonzero integer $l$ such that $\mu$ is $\times (p^j+l)$-invariant for all $j$ in a subset $A$ of $\mathbb{N}$ with $\overline{D}_\Sigma(A)>0$; (3) $\mu$ is strongly mixing and there exists a nonzero integer $l$ such that $\mu$ is $\times (p^j+l)$-invariant for infinitely many $j$. Moreover, a $\times p$-invariant measure satisfying (2) or (3) is either a Dirac measure or the Lebesgue measure.
As an application we prove that for every increasing function $\tau$ defined on positive integers with $\lim_{n\to\infty}\tau(n)=\infty$, there exists a multiplicative semigroup $S_\tau$ of $\mathbb{Z}^+$ containing $p$ such that $|S_\tau\cap[1,n]|\leq (\log_p n)^{\tau(n)}$ and the Lebesgue measure is the only non-atomic ergodic $\!\times \!p$-invariant measure which is $\times q$-invariant for all $q$ in $S_\tau$.
References:
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T. Bowley,
Extension of the Birkhoff and von Neumannn ergodic theorems to semigroup actions, Ann. Inst. Poincaré H., Sect. B (N.S.), 7 (1971), 283-291.
|
| [2] |
M. Einsiedler and A. Fish,
Rigidity of measures invariant under the action of a multiplicative semigroup of polynomial growth on T, Ergodic Theory Dynam. Systems, 30 (2010), 151-157.
doi: 10.1017/S014338570800103X. |
| [3] |
M. Einsiedler and T. Ward, Ergodic Theory with A View towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag, London, 2011. |
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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. |
| [5] |
A. S. A. Johnson,
Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers, Israel J. Math., 77 (1992), 211-240.
doi: 10.1007/BF02808018. |
| [6] |
R. Lyons,
On measures simultaneously 2-and 3-invariant, Israel J. Math., 61 (1988), 219-224.
doi: 10.1007/BF02766212. |
| [7] |
D. J. Rudolph,
×2 and×3 invariant measures and entropy, Ergod. Th. and Dynam. Syst., 10 (1990), 395-406.
|
| [8] |
E. A. Sataev,
On measures invariant with respect to polynomial semigoups of circle transformations, Uspehi Mat. Nauk., 30 (1975), 203-204.
|
| [9] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, SpringerVerlag, New York-Berlin, 1982. |
show all references
References:
| [1] |
T. Bowley,
Extension of the Birkhoff and von Neumannn ergodic theorems to semigroup actions, Ann. Inst. Poincaré H., Sect. B (N.S.), 7 (1971), 283-291.
|
| [2] |
M. Einsiedler and A. Fish,
Rigidity of measures invariant under the action of a multiplicative semigroup of polynomial growth on T, Ergodic Theory Dynam. Systems, 30 (2010), 151-157.
doi: 10.1017/S014338570800103X. |
| [3] |
M. Einsiedler and T. Ward, Ergodic Theory with A View towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag, London, 2011. |
| [4] |
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. |
| [5] |
A. S. A. Johnson,
Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers, Israel J. Math., 77 (1992), 211-240.
doi: 10.1007/BF02808018. |
| [6] |
R. Lyons,
On measures simultaneously 2-and 3-invariant, Israel J. Math., 61 (1988), 219-224.
doi: 10.1007/BF02766212. |
| [7] |
D. J. Rudolph,
×2 and×3 invariant measures and entropy, Ergod. Th. and Dynam. Syst., 10 (1990), 395-406.
|
| [8] |
E. A. Sataev,
On measures invariant with respect to polynomial semigoups of circle transformations, Uspehi Mat. Nauk., 30 (1975), 203-204.
|
| [9] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, SpringerVerlag, New York-Berlin, 1982. |
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