Smooth diffeomorphisms with homogeneous spectrum and disjointness of convolutions

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

Smooth diffeomorphisms with homogeneous spectrum and disjointness of convolutions

Philipp Kunde ^1,

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

Received March 2015 Revised July 2016 Published October 2016

Abstract
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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.

Keywords: Smooth ergodic theory, homogeneous spectrum, disjointness of convolutions, periodic approximation..

Mathematics Subject Classification: Primary: 37A05, 37A30, 37C40; Secondary: 37C0.

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.

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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.

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