# 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 & Continuous Dynamical Systems - A, 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. Google Scholar [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. Google Scholar [3] H. Anzai, Ergodic skew product transformations on the torus, Osaka Journal of Mathematics, 3 (1951), 83-99. Google Scholar [4] J. Brown, Ergodic Theory and Topological Dynamics Academic Press, 1976. Google Scholar [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. Google Scholar [6] H. Furstenberg, Strict Ergodicity and transformation of the torus, American Journal of Mathematics, 83 (1961), 573-601. doi: 10.2307/2372899. Google Scholar [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. Google Scholar [8] H. Helson, Cocyles on the circle, Journal of Operator Theory, 16 (1986), 189-199. Google Scholar [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. Google Scholar [10] A. Iwanik, Spectral properties of skew-product diffeomorphisms of tori, Colloquium Mathematicum, 72 (1997), 223-235. Google Scholar [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. Google Scholar [12] A. G. Kushnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Vestnik Moskov. Univ. Ser. I Mat. Meh., 29 (1974), 101-108. Google Scholar [13] M. Lemańczyk, Spectral theory of dynamical systems, Encyclopedia of Complexity and Systems Science, (2009), 8554-8575. Google Scholar [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. Google Scholar [15] C. C. Moore, Ergodicity of flows on homogeneous spaces, American Journal of Mathematics, 88 (1966), 154-178. doi: 10.2307/2373052. Google Scholar [16] E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Communications in Mathematical Phsyics, 78 (1980/81), 391-408. Google Scholar [17] S. Richard and R. Tiedra de Aldecoa, Commutator criteria for strong mixing Ⅱ, preprint, arXiv: 1510.00201Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

show all references

##### References:
 [1] W. O Amrein, Hilbert Space Methods in Quantum Mechanics, Fundamental Sciences, EPFL Press Lausanne, 2009. Google Scholar [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. Google Scholar [3] H. Anzai, Ergodic skew product transformations on the torus, Osaka Journal of Mathematics, 3 (1951), 83-99. Google Scholar [4] J. Brown, Ergodic Theory and Topological Dynamics Academic Press, 1976. Google Scholar [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. Google Scholar [6] H. Furstenberg, Strict Ergodicity and transformation of the torus, American Journal of Mathematics, 83 (1961), 573-601. doi: 10.2307/2372899. Google Scholar [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. Google Scholar [8] H. Helson, Cocyles on the circle, Journal of Operator Theory, 16 (1986), 189-199. Google Scholar [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. Google Scholar [10] A. Iwanik, Spectral properties of skew-product diffeomorphisms of tori, Colloquium Mathematicum, 72 (1997), 223-235. Google Scholar [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. Google Scholar [12] A. G. Kushnirenko, Spectral properties of certain dynamical systems with polynomial dispersal, Vestnik Moskov. Univ. Ser. I Mat. Meh., 29 (1974), 101-108. Google Scholar [13] M. Lemańczyk, Spectral theory of dynamical systems, Encyclopedia of Complexity and Systems Science, (2009), 8554-8575. Google Scholar [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. Google Scholar [15] C. C. Moore, Ergodicity of flows on homogeneous spaces, American Journal of Mathematics, 88 (1966), 154-178. doi: 10.2307/2373052. Google Scholar [16] E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Communications in Mathematical Phsyics, 78 (1980/81), 391-408. Google Scholar [17] S. Richard and R. Tiedra de Aldecoa, Commutator criteria for strong mixing Ⅱ, preprint, arXiv: 1510.00201Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
 [1] Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325 [2] Kim Dang Phung. Carleman commutator approach in logarithmic convexity for parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 899-933. doi: 10.3934/mcrf.2018040 [3] Yoonsang Lee, Bjorn Engquist. Variable step size multiscale methods for stiff and highly oscillatory dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1079-1097. doi: 10.3934/dcds.2014.34.1079 [4] Zoltán Horváth, Yunfei Song, Tamás Terlaky. Steplength thresholds for invariance preserving of discretization methods of dynamical systems on a polyhedron. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2997-3013. doi: 10.3934/dcds.2015.35.2997 [5] Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641 [6] Rémi Leclercq. Spectral invariants in Lagrangian Floer theory. Journal of Modern Dynamics, 2008, 2 (2) : 249-286. doi: 10.3934/jmd.2008.2.249 [7] Barry Simon. Equilibrium measures and capacities in spectral theory. Inverse Problems & Imaging, 2007, 1 (4) : 713-772. doi: 10.3934/ipi.2007.1.713 [8] Dung Le. Global existence and regularity results for strongly coupled nonregular parabolic systems via iterative methods. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 877-893. doi: 10.3934/dcdsb.2017044 [9] Michael Ghil. The wind-driven ocean circulation: Applying dynamical systems theory to a climate problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 189-228. doi: 10.3934/dcds.2017008 [10] Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761 [11] Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703 [12] Robert Carlson. Spectral theory for nonconservative transmission line networks. Networks & Heterogeneous Media, 2011, 6 (2) : 257-277. doi: 10.3934/nhm.2011.6.257 [13] Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22 [14] Leonid Golinskii, Mikhail Kudryavtsev. An inverse spectral theory for finite CMV matrices. Inverse Problems & Imaging, 2010, 4 (1) : 93-110. doi: 10.3934/ipi.2010.4.93 [15] J. G. Ollason, N. Ren. A general dynamical theory of foraging in animals. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 713-720. doi: 10.3934/dcdsb.2004.4.713 [16] Eugene Kashdan, Dominique Duncan, Andrew Parnell, Heinz Schättler. Mathematical methods in systems biology. Mathematical Biosciences & Engineering, 2016, 13 (6) : i-ii. doi: 10.3934/mbe.201606i [17] Krešimir Burazin, Marko Vrdoljak. Homogenisation theory for Friedrichs systems. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1017-1044. doi: 10.3934/cpaa.2014.13.1017 [18] Roman Shvydkoy, Eitan Tadmor. Eulerian dynamics with a commutator forcing Ⅱ: Flocking. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5503-5520. doi: 10.3934/dcds.2017239 [19] Lijian Jiang, Yalchin Efendiev, Victor Ginting. Multiscale methods for parabolic equations with continuum spatial scales. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 833-859. doi: 10.3934/dcdsb.2007.8.833 [20] Rúben Sousa, Semyon Yakubovich. The spectral expansion approach to index transforms and connections with the theory of diffusion processes. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2351-2378. doi: 10.3934/cpaa.2018112

2018 Impact Factor: 1.143