2016, 10: 439-481. doi: 10.3934/jmd.2016.10.439

Smooth diffeomorphisms with homogeneous spectrum and disjointness of convolutions

1. 

Department of Mathematics, University of Hamburg, Bundesstraße 55, 20146 Hamburg, Germany

Received  March 2015 Revised  July 2016 Published  October 2016

On any smooth compact connected manifold $M$ of dimension $m\geq 2$ admitting a smooth non-trivial circle action $\mathcal S = \left\{S_t\right\}_{t\in \mathbb{S}^1}$ and for every Liouville number $\alpha \in \mathbb{S}^1$ we prove the existence of a $C^\infty$-diffeomorphism $f \in \mathcal{A}_{\alpha} = \overline{\left\{h \circ S_{\alpha} \circ h^{-1} \;:\;h \in \text{Diff}^{\,\,\infty}\left(M,\nu\right)\right\}}^{C^\infty}$ with a good approximation of type $\left(h,h+1\right)$, a maximal spectral type disjoint with its convolutions and a homogeneous spectrum of multiplicity two for the Cartesian square $f\times f$. This answers a question of Fayad and Katok (10,[Problem 7.11]). The proof is based on a quantitative version of the approximation by conjugation-method with explicitly defined conjugation maps and tower elements.
Citation: Philipp Kunde. Smooth diffeomorphisms with homogeneous spectrum and disjointness of convolutions. Journal of Modern Dynamics, 2016, 10: 439-481. doi: 10.3934/jmd.2016.10.439
References:
[1]

O. N. Ageev, On ergodic transformations with homogeneous spectrum, J. Dynam. Control Systems, 5 (1999), 149-152. doi: 10.1023/A:1021701019156.

[2]

O. N. Ageev, The homogeneous spectrum problem in ergodic theory, Invent. Math., 160 (2005), 417-446. doi: 10.1007/s00222-004-0422-z.

[3]

D. V. Anosov and A. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč., 23 (1970), 3-36.

[4]

M. Benhenda, Non-standard smooth realization of shifts on the torus, J. Modern Dynamics, 7 (2013), 329-367.

[5]

R. Berndt, Einführung in die symplektische Geometrie, Friedr. Vieweg & Sohn, Braunschweig, 1998. doi: 10.1007/978-3-322-80215-6.

[6]

F. Blanchard and M. Lemańczyk, Measure-preserving diffeomorphisms with an arbitrary spectral multiplicity, Topol. Methods Nonlinear Anal., 1 (1993), 275-294.

[7]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.

[8]

G. M. Constantine and T. H. Savits, A multivariate Faà di Bruno formula with applications, Trans. Amer. Math. Soc., 348 (1996), 503-520. doi: 10.1090/S0002-9947-96-01501-2.

[9]

A. Danilenko, A survey on spectral multiplicities of ergodic actions, Ergodic Theory Dynam. Systems, 33 (2013), 81-117. doi: 10.1017/S0143385711000800.

[10]

B. Fayad and A. Katok, Constructions in elliptic dynamics, Ergodic Theory Dynam. Systems, 24 (2004), 1477-1520. doi: 10.1017/S0143385703000798.

[11]

B. Fayad and M. Saprykina, Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary, Ann. Sci. École Norm. Sup. (4), 38 (2005), 339-364. doi: 10.1016/j.ansens.2005.03.004.

[12]

B. Fayad, M. Saprykina and A. Windsor, Non-standard smooth realizations of Liouville rotations, Ergodic Theory Dynam. Systems, 27 (2007), 1803-1818. doi: 10.1017/S0143385707000314.

[13]

R. Gunesch and A. Katok, Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure, Discrete Contin. Dynam. Systems, 6 (2000), 61-88. doi: 10.3934/dcds.2000.6.61.

[14]

G. R. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems, 5 (1999), 173-226. doi: 10.1023/A:1021726902801.

[15]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[16]

A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2), 110 (1979), 529-547. doi: 10.2307/1971237.

[17]

A. Katok, Combinatorical Constructions in Ergodic Theory and Dynamics, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/ulect/030.

[18]

J. Kwiatkowski and M. Lemańczyk, On the multiplicity function of ergodic group extensions. II, Studia Math., 116 (1995), 207-214.

[19]

A. Kriegl and P. Michor, The Convenient Setting of Global Analysis, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/053.

[20]

A. Katok and A. Stepin, Approximations in ergodic theory, Russ. Math. Surveys, 22 (1967), 77-102. doi: 10.1070/RM1967v022n05ABEH001227.

[21]

A. Katok and A. Stepin, Metric properties of measure preserving homeomorphisms, Russ. Math. Surveys, 25 (1970), 191-220. doi: 10.1070/RM1970v025n02ABEH003793.

[22]

M. G. Nadkarni, Spectral Theory of Dynamical Systems, Birkhäuser Verlag, Basel, 1998. doi: 10.1007/978-3-0348-8841-7.

[23]

H. Omori, Infinite Dimensional Lie Transformation Groups, Springer-Verlag, Berlin-New York, 1974.

[24]

V. I. Oseledets, An automorphism with simple continuous spectrum not having the group property, Mat. Zametki, 5 (1969), 323-326.

[25]

V. V. Ryzhikov, Transformations having homogeneous spectra, J. Dynam. Control Systems, 5 (1999), 145-148. doi: 10.1023/A:1021748902318.

[26]

V. V. Ryzhikov, Homogeneous spectrum, disjointness of convolutions and mixing properties of dynamical systems, Selected Russian Math., 1 (1999), 13-24.

[27]

V. V. Ryzhikov, On the spectral and mixing properties of rank-1 constructions in ergodic theory, Doklady Mathematics, 74 (2006), 545-547.

[28]

A. M. Stepin, Properties of spectra of ergodic dynamical systems with locally compact time, Dokl. Akad. Nauk SSSR, 169 (1966), 773-776.

[29]

A. M. Stepin, Spectral properties of generic dynamical systems, Math. USSR Izv., 29 (1987), 159-192. doi: 10.1070/IM1987v029n01ABEH000965.

show all references

References:
[1]

O. N. Ageev, On ergodic transformations with homogeneous spectrum, J. Dynam. Control Systems, 5 (1999), 149-152. doi: 10.1023/A:1021701019156.

[2]

O. N. Ageev, The homogeneous spectrum problem in ergodic theory, Invent. Math., 160 (2005), 417-446. doi: 10.1007/s00222-004-0422-z.

[3]

D. V. Anosov and A. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč., 23 (1970), 3-36.

[4]

M. Benhenda, Non-standard smooth realization of shifts on the torus, J. Modern Dynamics, 7 (2013), 329-367.

[5]

R. Berndt, Einführung in die symplektische Geometrie, Friedr. Vieweg & Sohn, Braunschweig, 1998. doi: 10.1007/978-3-322-80215-6.

[6]

F. Blanchard and M. Lemańczyk, Measure-preserving diffeomorphisms with an arbitrary spectral multiplicity, Topol. Methods Nonlinear Anal., 1 (1993), 275-294.

[7]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.

[8]

G. M. Constantine and T. H. Savits, A multivariate Faà di Bruno formula with applications, Trans. Amer. Math. Soc., 348 (1996), 503-520. doi: 10.1090/S0002-9947-96-01501-2.

[9]

A. Danilenko, A survey on spectral multiplicities of ergodic actions, Ergodic Theory Dynam. Systems, 33 (2013), 81-117. doi: 10.1017/S0143385711000800.

[10]

B. Fayad and A. Katok, Constructions in elliptic dynamics, Ergodic Theory Dynam. Systems, 24 (2004), 1477-1520. doi: 10.1017/S0143385703000798.

[11]

B. Fayad and M. Saprykina, Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary, Ann. Sci. École Norm. Sup. (4), 38 (2005), 339-364. doi: 10.1016/j.ansens.2005.03.004.

[12]

B. Fayad, M. Saprykina and A. Windsor, Non-standard smooth realizations of Liouville rotations, Ergodic Theory Dynam. Systems, 27 (2007), 1803-1818. doi: 10.1017/S0143385707000314.

[13]

R. Gunesch and A. Katok, Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure, Discrete Contin. Dynam. Systems, 6 (2000), 61-88. doi: 10.3934/dcds.2000.6.61.

[14]

G. R. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems, 5 (1999), 173-226. doi: 10.1023/A:1021726902801.

[15]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[16]

A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2), 110 (1979), 529-547. doi: 10.2307/1971237.

[17]

A. Katok, Combinatorical Constructions in Ergodic Theory and Dynamics, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/ulect/030.

[18]

J. Kwiatkowski and M. Lemańczyk, On the multiplicity function of ergodic group extensions. II, Studia Math., 116 (1995), 207-214.

[19]

A. Kriegl and P. Michor, The Convenient Setting of Global Analysis, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/053.

[20]

A. Katok and A. Stepin, Approximations in ergodic theory, Russ. Math. Surveys, 22 (1967), 77-102. doi: 10.1070/RM1967v022n05ABEH001227.

[21]

A. Katok and A. Stepin, Metric properties of measure preserving homeomorphisms, Russ. Math. Surveys, 25 (1970), 191-220. doi: 10.1070/RM1970v025n02ABEH003793.

[22]

M. G. Nadkarni, Spectral Theory of Dynamical Systems, Birkhäuser Verlag, Basel, 1998. doi: 10.1007/978-3-0348-8841-7.

[23]

H. Omori, Infinite Dimensional Lie Transformation Groups, Springer-Verlag, Berlin-New York, 1974.

[24]

V. I. Oseledets, An automorphism with simple continuous spectrum not having the group property, Mat. Zametki, 5 (1969), 323-326.

[25]

V. V. Ryzhikov, Transformations having homogeneous spectra, J. Dynam. Control Systems, 5 (1999), 145-148. doi: 10.1023/A:1021748902318.

[26]

V. V. Ryzhikov, Homogeneous spectrum, disjointness of convolutions and mixing properties of dynamical systems, Selected Russian Math., 1 (1999), 13-24.

[27]

V. V. Ryzhikov, On the spectral and mixing properties of rank-1 constructions in ergodic theory, Doklady Mathematics, 74 (2006), 545-547.

[28]

A. M. Stepin, Properties of spectra of ergodic dynamical systems with locally compact time, Dokl. Akad. Nauk SSSR, 169 (1966), 773-776.

[29]

A. M. Stepin, Spectral properties of generic dynamical systems, Math. USSR Izv., 29 (1987), 159-192. doi: 10.1070/IM1987v029n01ABEH000965.

[1]

Wen Huang, Zhiren Wang, Guohua Zhang. Möbius disjointness for topological models of ergodic systems with discrete spectrum. Journal of Modern Dynamics, 2019, 14: 277-290. doi: 10.3934/jmd.2019010

[2]

Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757

[3]

Thierry de la Rue. An introduction to joinings in ergodic theory. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 121-142. doi: 10.3934/dcds.2006.15.121

[4]

Gary Froyland. On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 671-689. doi: 10.3934/dcds.2007.17.671

[5]

Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22

[6]

Cristina Lizana, Vilton Pinheiro, Paulo Varandas. Contribution to the ergodic theory of robustly transitive maps. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 353-365. doi: 10.3934/dcds.2015.35.353

[7]

Dmitry Kleinbock, Barak Weiss. Dirichlet's theorem on diophantine approximation and homogeneous flows. Journal of Modern Dynamics, 2008, 2 (1) : 43-62. doi: 10.3934/jmd.2008.2.43

[8]

Maxim Sølund Kirsebom. Extreme value theory for random walks on homogeneous spaces. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4689-4717. doi: 10.3934/dcds.2014.34.4689

[9]

Manfred Denker, Samuel Senti, Xuan Zhang. Fluctuations of ergodic sums on periodic orbits under specification. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4665-4687. doi: 10.3934/dcds.2020197

[10]

Hebai Chen, Jaume Llibre, Yilei Tang. Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6495-6509. doi: 10.3934/dcdsb.2019150

[11]

Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198

[12]

El Houcein El Abdalaoui, Joanna Kułaga-Przymus, Mariusz Lemańczyk, Thierry de la Rue. The Chowla and the Sarnak conjectures from ergodic theory point of view. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2899-2944. doi: 10.3934/dcds.2017125

[13]

Nimish Shah, Lei Yang. Equidistribution of curves in homogeneous spaces and Dirichlet's approximation theorem for matrices. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5247-5287. doi: 10.3934/dcds.2020227

[14]

Xu Xu, Xin Zhao. Exponential upper bounds on the spectral gaps and homogeneous spectrum for the non-critical extended Harper's model. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4777-4800. doi: 10.3934/dcds.2020201

[15]

Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006

[16]

Hajnal R. Tóth. Infinite Bernoulli convolutions with different probabilities. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 595-600. doi: 10.3934/dcds.2008.21.595

[17]

Sergei A. Nazarov, Rafael Orive-Illera, María-Eugenia Pérez-Martínez. Asymptotic structure of the spectrum in a Dirichlet-strip with double periodic perforations. Networks and Heterogeneous Media, 2019, 14 (4) : 733-757. doi: 10.3934/nhm.2019029

[18]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807

[19]

Fabien Durand, Alejandro Maass. A note on limit laws for minimal Cantor systems with infinite periodic spectrum. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 745-750. doi: 10.3934/dcds.2003.9.745

[20]

Günter Leugering, Sergei A. Nazarov, Jari Taskinen. The band-gap structure of the spectrum in a periodic medium of masonry type. Networks and Heterogeneous Media, 2020, 15 (4) : 555-580. doi: 10.3934/nhm.2020014

2020 Impact Factor: 0.848

Metrics

  • PDF downloads (162)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]