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.

[2]

O. N. Ageev, The homogeneous spectrum problem in ergodic theory,, Invent. Math., 160 (2005), 417. 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.

[4]

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

[5]

R. Berndt, Einführung in die symplektische Geometrie,, Friedr. Vieweg & Sohn, (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.

[7]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory,, Springer-Verlag, (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. 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. doi: 10.1017/S0143385711000800.

[10]

B. Fayad and A. Katok, Constructions in elliptic dynamics,, Ergodic Theory Dynam. Systems, 24 (2004), 1477. 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. 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. 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. 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. doi: 10.1023/A:1021726902801.

[15]

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

[16]

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

[17]

A. Katok, Combinatorical Constructions in Ergodic Theory and Dynamics,, American Mathematical Society, (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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

V. V. Ryzhikov, Transformations having homogeneous spectra,, J. Dynam. Control Systems, 5 (1999), 145. 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.

[27]

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

[28]

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

[29]

A. M. Stepin, Spectral properties of generic dynamical systems,, Math. USSR Izv., 29 (1987), 159. 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. doi: 10.1023/A:1021701019156.

[2]

O. N. Ageev, The homogeneous spectrum problem in ergodic theory,, Invent. Math., 160 (2005), 417. 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.

[4]

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

[5]

R. Berndt, Einführung in die symplektische Geometrie,, Friedr. Vieweg & Sohn, (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.

[7]

I. P. Cornfeld, S. V. Fomin and Ya. G. Sinaĭ, Ergodic Theory,, Springer-Verlag, (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. 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. doi: 10.1017/S0143385711000800.

[10]

B. Fayad and A. Katok, Constructions in elliptic dynamics,, Ergodic Theory Dynam. Systems, 24 (2004), 1477. 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. 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. 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. 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. doi: 10.1023/A:1021726902801.

[15]

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

[16]

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

[17]

A. Katok, Combinatorical Constructions in Ergodic Theory and Dynamics,, American Mathematical Society, (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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

V. V. Ryzhikov, Transformations having homogeneous spectra,, J. Dynam. Control Systems, 5 (1999), 145. 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.

[27]

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

[28]

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

[29]

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

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