American Institute of Mathematical Sciences

Advanced
  2017, 11: 551-562. doi: 10.3934/jmd.2017021

Fourier coefficients of $\times p$-invariant measures

Huichi Huang

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

Received  August 28, 2016 Revised  September 03, 2017 Published  November 2017

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 $S_\tau$.

Keywords: Fourier coefficients, density, ×p-invariant measure, ergodic, weakly mixing, strongly mixing.
Mathematics Subject Classification: Primary: 37A25, 28D05, 37B99.
Citation: Huichi Huang. Fourier coefficients of $\times p$-invariant measures. Journal of Modern Dynamics, 2017, 11: 551-562. doi: 10.3934/jmd.2017021
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.   Google Scholar

[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.  Google Scholar

[3]

M. Einsiedler and T. Ward, Ergodic Theory with A View towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag, London, 2011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. Lyons, On measures simultaneously 2-and 3-invariant, Israel J. Math., 61 (1988), 219-224.  doi: 10.1007/BF02766212.  Google Scholar

[7]

D. J. Rudolph, ×2 and×3 invariant measures and entropy, Ergod. Th. and Dynam. Syst., 10 (1990), 395-406.   Google Scholar

[8]

E. A. Sataev, On measures invariant with respect to polynomial semigoups of circle transformations, Uspehi Mat. Nauk., 30 (1975), 203-204.   Google Scholar

[9]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, SpringerVerlag, New York-Berlin, 1982.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[3]

M. Einsiedler and T. Ward, Ergodic Theory with A View towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag, London, 2011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. Lyons, On measures simultaneously 2-and 3-invariant, Israel J. Math., 61 (1988), 219-224.  doi: 10.1007/BF02766212.  Google Scholar

[7]

D. J. Rudolph, ×2 and×3 invariant measures and entropy, Ergod. Th. and Dynam. Syst., 10 (1990), 395-406.   Google Scholar

[8]

E. A. Sataev, On measures invariant with respect to polynomial semigoups of circle transformations, Uspehi Mat. Nauk., 30 (1975), 203-204.   Google Scholar

[9]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, SpringerVerlag, New York-Berlin, 1982.  Google Scholar

[1]

Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61
[2]

Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082
[3]

Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079
[4]

François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275
[5]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817
[6]

Ian Melbourne, Dalia Terhesiu. Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure. Journal of Modern Dynamics, 2018, 12: 285-313. doi: 10.3934/jmd.2018011
[7]

Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315
[8]

Roland Gunesch, Philipp Kunde. Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1615-1655. doi: 10.3934/dcds.2018067
[9]

A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195
[10]

Zhi Lin, Katarína Boďová, Charles R. Doering. Models & measures of mixing & effective diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 259-274. doi: 10.3934/dcds.2010.28.259
[11]

Nir Avni. Spectral and mixing properties of actions of amenable groups. Electronic Research Announcements, 2005, 11: 57-63.

[12]

Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181
[13]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547
[14]

Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175
[15]

Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657
[16]

Dmitri Scheglov. Absence of mixing for smooth flows on genus two surfaces. Journal of Modern Dynamics, 2009, 3 (1) : 13-34. doi: 10.3934/jmd.2009.3.13
[17]

Jian Li. Localization of mixing property via Furstenberg families. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 725-740. doi: 10.3934/dcds.2015.35.725
[18]

Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33
[19]

Makoto Mori. Higher order mixing property of piecewise linear transformations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 915-934. doi: 10.3934/dcds.2000.6.915
[20]

Benoît Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 259-267. doi: 10.3934/dcds.2006.15.259

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (34)
  • HTML views (126)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences