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. doi: 10.1023/A:1021701019156. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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. doi: 10.1016/j.ansens.2005.03.004. Google Scholar

[12]

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

[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. doi: 10.3934/dcds.2000.6.61. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

A. Katok, Combinatorical Constructions in Ergodic Theory and Dynamics,, American Mathematical Society, (2003). doi: 10.1090/ulect/030. Google Scholar

[18]

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

[19]

A. Kriegl and P. Michor, The Convenient Setting of Global Analysis,, American Mathematical Society, (1997). doi: 10.1090/surv/053. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

H. Omori, Infinite Dimensional Lie Transformation Groups,, Springer-Verlag, (1974). Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[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. doi: 10.1016/j.ansens.2005.03.004. Google Scholar

[12]

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

[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. doi: 10.3934/dcds.2000.6.61. Google Scholar

[14]

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

[15]

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

[16]

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

[17]

A. Katok, Combinatorical Constructions in Ergodic Theory and Dynamics,, American Mathematical Society, (2003). doi: 10.1090/ulect/030. Google Scholar

[18]

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

[19]

A. Kriegl and P. Michor, The Convenient Setting of Global Analysis,, American Mathematical Society, (1997). doi: 10.1090/surv/053. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

H. Omori, Infinite Dimensional Lie Transformation Groups,, Springer-Verlag, (1974). Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

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