Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools

The isolated spectrum of transfer operators is known to play a critical role in determining mixing properties of piecewise smooth dynamical systems. The so-called Dellnitz-Froyland ansatz places isolated eigenvalues in correspondence with structures in phase space that decay at rates slower than local expansion can account for. Numerical approximations of transfer operator spectrum are often insufficient to distinguish isolated spectral points, so it is an open problem to decide to which eigenvectors the ansatz applies. We propose a new numerical technique to identify the isolated spectrum and large-scale structures alluded to in the ansatz. This harmonic analytic approach relies on new stability properties of the Ulam scheme for both transfer and Koopman operators, which are also established here. We demonstrate the ecacy of this scheme in metastable one- and two-dimensional dynamical systems, including those with both expanding and contracting dynamics, and explain how the leading eigenfunctions govern the dynamics for both real and complex isolated eigenvalues.

[1]  The Diffusion Coefficient For Piecewise Expanding Maps Of The Interval With Metastable States , 2010, 1011.5330.

[2]  B. Saussol,et al.  Absolutely continuous invariant measures for multidimensional expanding maps , 2000 .

[3]  S. Ulam.  A collection of mathematical problems , 1960 .

[4]  Marek R Rychlik,et al.  Bounded variation and invariant measures , 1983 .

[5]  Piecewise expanding maps on the plane with singular ergodic properties , 1998, Ergodic Theory and Dynamical Systems.

[6]  On the number of invariant measures for higher-dimensional chaotic transformations , 1991 .

[7]  Franz Hofbauer,et al.  Ergodic properties of invariant measures for piecewise monotonic transformations , 1982 .

[8]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[9]  Gerhard Keller,et al.  Stability of the spectrum for transfer operators , 1999 .

[10]  Andrzej Banaszuk,et al.  Comparison of systems with complex behavior , 2004 .

[11]  V. Baladi.  Positive transfer operators and decay of correlations , 2000 .

[12]  Igor Mezić,et al.  Ergodic theory and visualization. I. Mesochronic plots for visualization of ergodic partition and invariant sets. , 2008, Chaos.

[13]  Tosio Kato,et al.  Perturbation theory for nullity, deficiency and other quantities of linear operators , 1958 .

[14]  Michael Dellnitz,et al.  On the isolated spectrum of the Perron-Frobenius operator , 2000 .

[15]  G. Froyland.  Unwrapping eigenfunctions to discover the geometry of almost-invariant sets in hyperbolic maps , 2008 .

[16]  I. Mezić,et al.  Applied Koopmanism. , 2012, Chaos.

[17]  Jerrold E. Marsden,et al.  Uncertainty in the dynamics of conservative maps , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[18]  Brian R. Hunt,et al.  Approximating invariant densities of metastable systems , 2009, Ergodic Theory and Dynamical Systems.

[19]  O. Junge,et al.  On the Approximation of Complicated Dynamical Behavior , 1999 .

[20]  Gerhard Keller,et al.  On the rate of convergence to equilibrium in one-dimensional systems , 1984 .

[21]  C. Gonz'alez-Tokman,et al.  A semi-invertible operator Oseledets theorem , 2011, Ergodic Theory and Dynamical Systems.

[22]  P. Deuflhard,et al.  A Direct Approach to Conformational Dynamics Based on Hybrid Monte Carlo , 1999 .

[23]  J. Thiffeault,et al.  Using multiscale norms to quantify mixing and transport , 2011, 1105.1101.

[24]  G. Keller,et al.  Rare Events, Escape Rates and Quasistationarity: Some Exact Formulae , 2008, 0810.2229.

[25]  G. Froyland,et al.  Almost-invariant sets and invariant manifolds — Connecting probabilistic and geometric descriptions of coherent structures in flows , 2009 .

[26]  Gary Froyland,et al.  Spectral degeneracy and escape dynamics for intermittent maps with a hole , 2010, 1012.2149.

[27]  Absolutely continuous invariant measures for most piecewise smooth expanding maps , 2002, Ergodic Theory and Dynamical Systems.

[28]  No or Infinitely Many A.C.I.P.¶for Piecewise Expanding Cr Maps¶in Higher Dimensions , 2001 .

[29]  Mark F. Demers,et al.  Stability of statistical properties in two-dimensional piecewise hyperbolic maps , 2006, math/0602067.

[30]  G. Froyland,et al.  ESCAPE RATES AND PERRON-FROBENIUS OPERATORS: OPEN AND CLOSED DYNAMICAL SYSTEMS , 2010 .

[31]  V. Baladi,et al.  Good Banach spaces for piecewise hyperbolic maps via interpolation , 2007, 0711.1960.

[32]  Anisotropic Sobolev spaces and dynamical transfer operators: C , 2004, math/0408430.

[33]  Gerhard Keller,et al.  Ruelle?Perron?Frobenius spectrum for Anosov maps , 2002 .

[34]  G. Froyland,et al.  A semi-invertible Oseledets Theorem with applications to transfer operator cocycles , 2010, 1001.5313.

[35]  Hubert Hennion,et al.  Sur un théorème spectral et son application aux noyaux lipchitziens , 1993 .

[36]  Eigenfunctions for smooth expanding circle maps , 2004, math/0405055.

[37]  I. Mezić,et al.  A multiscale measure for mixing , 2005 .

[38]  G. Froyland,et al.  Detection of coherent oceanic structures via transfer operators. , 2007, Physical review letters.

[39]  Metastability of certain intermittent maps , 2011, 1105.0538.