American Institute of Mathematical Sciences

Advanced
February  2007, 1(1): 189-215. doi: 10.3934/ipi.2007.1.189

Zeros of OPUC and long time asymptotics of Schur and related flows

Barry Simon 1,

1. 

Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, United States

Received  September 2006 Revised  October 2006 Published  January 2007

We provide a complete analysis of the asymptotics for the semi-infinite Schur flow: $\alpha_j(t)=(1-|\alpha_j(t)|^2) (\alpha_{j+1}(t)-\alpha_{j-1}(t))$ for $\alpha_{-1}(t)= 1$ boundary conditions and $n=0,1,2,...$, with initial condition $\alpha_j(0)\in (-1,1)$. We also provide examples with $\alpha_j(0)\in\bbD$ for which $\alpha_0(t)$ does not have a limit. The proofs depend on the solution via a direct/inverse spectral transform.
Keywords: Toda flows., orthogonal polynomials, Schur flows.
Mathematics Subject Classification: 05E35, 33D45, 37K1.
Citation: Barry Simon. Zeros of OPUC and long time asymptotics of Schur and related flows. Inverse Problems & Imaging, 2007, 1 (1) : 189-215. doi: 10.3934/ipi.2007.1.189
[1]

Kurt Vinhage. On the rigidity of Weyl chamber flows and Schur multipliers as topological groups. Journal of Modern Dynamics, 2015, 9: 25-49. doi: 10.3934/jmd.2015.9.25
[2]

Benjamin Letson, Jonathan E. Rubin. Local orthogonal rectification: Deriving natural coordinates to study flows relative to manifolds. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3725-3747. doi: 10.3934/dcdsb.2020088
[3]

Livio Flaminio, Giovanni Forni. Orthogonal powers and Möbius conjecture for smooth time changes of horocycle flows. Electronic Research Announcements, 2019, 26: 16-23. doi: 10.3934/era.2019.26.002
[4]

He Zhang, John Harlim, Xiantao Li. Estimating linear response statistics using orthogonal polynomials: An RKHS formulation. Foundations of Data Science, 2020, 2 (4) : 443-485. doi: 10.3934/fods.2020021
[5]

Alfonso Artigue. Rescaled expansivity and separating flows. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4433-4447. doi: 10.3934/dcds.2018193
[6]

Alfonso Artigue. Expansive flows of surfaces. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 505-525. doi: 10.3934/dcds.2013.33.505
[7]

Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201
[8]

Michael Cranston, Benjamin Gess, Michael Scheutzow. Weak synchronization for isotropic flows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3003-3014. doi: 10.3934/dcdsb.2016084
[9]

Jinqiao Duan, Vincent J. Ervin, Daniel Schertzer. Dispersion in flows with obstacles and uncertainty. Conference Publications, 2001, 2001 (Special) : 131-136. doi: 10.3934/proc.2001.2001.131
[10]

Michel Benaim, Morris W. Hirsch. Chain recurrence in surface flows. Discrete & Continuous Dynamical Systems, 1995, 1 (1) : 1-16. doi: 10.3934/dcds.1995.1.1
[11]

Alexandre I. Danilenko, Mariusz Lemańczyk. Spectral multiplicities for ergodic flows. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4271-4289. doi: 10.3934/dcds.2013.33.4271
[12]

Kristian Bjerklöv, Russell Johnson. Minimal subsets of projective flows. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 493-516. doi: 10.3934/dcdsb.2008.9.493
[13]

R. H.W. Hoppe, William G. Litvinov. Problems on electrorheological fluid flows. Communications on Pure & Applied Analysis, 2004, 3 (4) : 809-848. doi: 10.3934/cpaa.2004.3.809
[14]

W. Wei, Yin Li, Zheng-An Yao. Decay of the compressible viscoelastic flows. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1603-1624. doi: 10.3934/cpaa.2016004
[15]

Klaus-Jochen Engel, Marjeta Kramar Fijavž, Rainer Nagel, Eszter Sikolya. Vertex control of flows in networks. Networks & Heterogeneous Media, 2008, 3 (4) : 709-722. doi: 10.3934/nhm.2008.3.709
[16]

Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345
[17]

Jayadev S. Athreya, Gregory A. Margulis. Logarithm laws for unipotent flows, Ⅱ. Journal of Modern Dynamics, 2017, 11: 1-16. doi: 10.3934/jmd.2017001
[18]

Se-Hyun Ku. Expansive flows on uniform spaces. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021165
[19]

Darren C. Ong. Orthogonal polynomials on the unit circle with quasiperiodic Verblunsky coefficients have generic purely singular continuous spectrum. Conference Publications, 2013, 2013 (special) : 605-609. doi: 10.3934/proc.2013.2013.605
[20]

Junyu Lin. Uniqueness of harmonic map heat flows and liquid crystal flows. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 739-755. doi: 10.3934/dcds.2013.33.739

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy