# American Institute of Mathematical Sciences

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.

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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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