American Institute of Mathematical Sciences

January  2018, 38(1): 263-292. doi: 10.3934/dcds.2018013

Absolutely continuous spectrum for parabolic flows/maps

 Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy

Received  November 2016 Revised  August 2017 Published  September 2017

We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures. We use these general conditions to derive results for spectral properties of time-changes of unipotent flows on homogeneous spaces of semisimple groups regarding absolutely continuous spectrum as well as maximal spectral type; the time-changes of the horocycle flow are special cases of this general category of flows. In addition we use the general conditions to derive spectral results for twisted horocycle flows and to rederive certain spectral results for skew products over translations and Furstenberg transformations.

Citation: Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013
References:
 [1] W. O Amrein, Hilbert Space Methods in Quantum Mechanics, Fundamental Sciences, EPFL Press Lausanne, 2009. [2] W. O. Amrein, A. Boutet de Monvel and V. Georgescu, $C_0$-groups, Commutator Methods and Spectral Theory of N-body Hamiltonians, Progress in Math Birkhäuser, Basel, 1996. doi: 10.1007/978-3-0348-7762-6. [3] H. Anzai, Ergodic skew product transformations on the torus, Osaka Journal of Mathematics, 3 (1951), 83-99. [4] J. Brown, Ergodic Theory and Topological Dynamics Academic Press, 1976. [5] G. Forni and C. Ulcigrai, Time-changes of horocycle flows, Journal of Modern Dynamics, 6 (2012), 251-273.  doi: 10.3934/jmd.2012.6.251. [6] H. Furstenberg, Strict Ergodicity and transformation of the torus, American Journal of Mathematics, 83 (1961), 573-601.  doi: 10.2307/2372899. [7] H. Furstenberg, The unique ergodicity of the horocycle flow, Recent advances in topological dynamics (Proc. Conf. , Yale Univ. , New Haven, Conn. , 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Mathematics, Springer, Berlin, 318 (1972), 95–115. [8] H. Helson, Cocyles on the circle, Journal of Operator Theory, 16 (1986), 189-199. [9] A. Iwanik, Anzai skew products with Lebesgue with Lebesgue component of infinite multiplicity, Bulletin of the London Mathematical Society, 29 (1997), 195-199.  doi: 10.1112/S0024609396002147. [10] A. Iwanik, Spectral properties of skew-product diffeomorphisms of tori, Colloquium Mathematicum, 72 (1997), 223-235. [11] A. Iwanik, M. Lemańczyk and D. Rudolph, Absolutely continuous cocycles over irrational rotations, Israel Journal of Mathematics, 83 (1993), 73-95.  doi: 10.1007/BF02764637. [12] A. G. Kushnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Vestnik Moskov. Univ. Ser. I Mat. Meh., 29 (1974), 101-108. [13] M. Lemańczyk, Spectral theory of dynamical systems, Encyclopedia of Complexity and Systems Science, (2009), 8554-8575. [14] B. Marcus, Ergodic properties of horocycle flows for surfaces of negative curvature, Annals of Mathematics, Second Series 105 (1977), 81–105. doi: 10.2307/1971026. [15] C. C. Moore, Ergodicity of flows on homogeneous spaces, American Journal of Mathematics, 88 (1966), 154-178.  doi: 10.2307/2373052. [16] E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Communications in Mathematical Phsyics, 78 (1980/81), 391-408. [17] S. Richard and R. Tiedra de Aldecoa, Commutator criteria for strong mixing Ⅱ, preprint, arXiv: 1510.00201 [18] R. Tiedra de Aldecoa, Spectral analysis of time-changes of the horocycle flow, Journal of Modern Dynamics, 6 (2012), 275-285.  doi: 10.3934/jmd.2012.6.275. [19] R. Tiedra de Aldecoa, Commutator methods for the spectral analysis of uniquely ergodic dynamical systems, Ergodic Theory and Dynamical Systems, 35 (2015), 944-967.  doi: 10.1017/etds.2013.76. [20] R. Tiedra de Aldecoa, Commutator criteria for strong mixing, Ergodic Theory and Dynamical Systems, 37 (2017), 308-323.  doi: 10.1017/etds.2015.47.

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References:
 [1] W. O Amrein, Hilbert Space Methods in Quantum Mechanics, Fundamental Sciences, EPFL Press Lausanne, 2009. [2] W. O. Amrein, A. Boutet de Monvel and V. Georgescu, $C_0$-groups, Commutator Methods and Spectral Theory of N-body Hamiltonians, Progress in Math Birkhäuser, Basel, 1996. doi: 10.1007/978-3-0348-7762-6. [3] H. Anzai, Ergodic skew product transformations on the torus, Osaka Journal of Mathematics, 3 (1951), 83-99. [4] J. Brown, Ergodic Theory and Topological Dynamics Academic Press, 1976. [5] G. Forni and C. Ulcigrai, Time-changes of horocycle flows, Journal of Modern Dynamics, 6 (2012), 251-273.  doi: 10.3934/jmd.2012.6.251. [6] H. Furstenberg, Strict Ergodicity and transformation of the torus, American Journal of Mathematics, 83 (1961), 573-601.  doi: 10.2307/2372899. [7] H. Furstenberg, The unique ergodicity of the horocycle flow, Recent advances in topological dynamics (Proc. Conf. , Yale Univ. , New Haven, Conn. , 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Mathematics, Springer, Berlin, 318 (1972), 95–115. [8] H. Helson, Cocyles on the circle, Journal of Operator Theory, 16 (1986), 189-199. [9] A. Iwanik, Anzai skew products with Lebesgue with Lebesgue component of infinite multiplicity, Bulletin of the London Mathematical Society, 29 (1997), 195-199.  doi: 10.1112/S0024609396002147. [10] A. Iwanik, Spectral properties of skew-product diffeomorphisms of tori, Colloquium Mathematicum, 72 (1997), 223-235. [11] A. Iwanik, M. Lemańczyk and D. Rudolph, Absolutely continuous cocycles over irrational rotations, Israel Journal of Mathematics, 83 (1993), 73-95.  doi: 10.1007/BF02764637. [12] A. G. Kushnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Vestnik Moskov. Univ. Ser. I Mat. Meh., 29 (1974), 101-108. [13] M. Lemańczyk, Spectral theory of dynamical systems, Encyclopedia of Complexity and Systems Science, (2009), 8554-8575. [14] B. Marcus, Ergodic properties of horocycle flows for surfaces of negative curvature, Annals of Mathematics, Second Series 105 (1977), 81–105. doi: 10.2307/1971026. [15] C. C. Moore, Ergodicity of flows on homogeneous spaces, American Journal of Mathematics, 88 (1966), 154-178.  doi: 10.2307/2373052. [16] E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Communications in Mathematical Phsyics, 78 (1980/81), 391-408. [17] S. Richard and R. Tiedra de Aldecoa, Commutator criteria for strong mixing Ⅱ, preprint, arXiv: 1510.00201 [18] R. Tiedra de Aldecoa, Spectral analysis of time-changes of the horocycle flow, Journal of Modern Dynamics, 6 (2012), 275-285.  doi: 10.3934/jmd.2012.6.275. [19] R. Tiedra de Aldecoa, Commutator methods for the spectral analysis of uniquely ergodic dynamical systems, Ergodic Theory and Dynamical Systems, 35 (2015), 944-967.  doi: 10.1017/etds.2013.76. [20] R. Tiedra de Aldecoa, Commutator criteria for strong mixing, Ergodic Theory and Dynamical Systems, 37 (2017), 308-323.  doi: 10.1017/etds.2015.47.
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