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Ergodic and synthetic Koopman analyses of cat maps onto classical 2-tori

David Viennot

TL;DR

This work develops a Koopman-operator framework for classical cat maps on the torus, revealing how ergodic decomposition and a synthetic spectrum shape the global spectral structure. It constructs full Koopman modes coherently defined on the entire phase space and connects these modes to fixed and cyclic points of the map, with explicit expressions across regimes. The analysis covers cyclic, quasi-cyclic, critical, and chaotic dynamics, showing how regular cases yield wave-like modes and pure-point spectra while chaotic dynamics produce continuous spectra and irregular modes; synthesis can reinforce or alter spectral type. The results provide a countable, globally defined mode basis in many settings and illuminate the role of ergodic Components in shaping observable dynamics for quasilinear maps on the torus.

Abstract

We study classical continuous automorphisms of the torus (cat maps) from the viewpoint of the Koopman theory. We find analytical formulae for Koopman modes defined coherently on the whole of the torus, and their decompositions associated with the partition of the torus into ergodic components. The spectrum of the Koopman operator is studied in four cases of cat maps: cyclic, quasi-cyclic, critical (transition from quasi-cyclic to chaotic behaviour) and chaotic. The synthetic spectrum associated with the ergodic decomposition is also studied.

Ergodic and synthetic Koopman analyses of cat maps onto classical 2-tori

TL;DR

This work develops a Koopman-operator framework for classical cat maps on the torus, revealing how ergodic decomposition and a synthetic spectrum shape the global spectral structure. It constructs full Koopman modes coherently defined on the entire phase space and connects these modes to fixed and cyclic points of the map, with explicit expressions across regimes. The analysis covers cyclic, quasi-cyclic, critical, and chaotic dynamics, showing how regular cases yield wave-like modes and pure-point spectra while chaotic dynamics produce continuous spectra and irregular modes; synthesis can reinforce or alter spectral type. The results provide a countable, globally defined mode basis in many settings and illuminate the role of ergodic Components in shaping observable dynamics for quasilinear maps on the torus.

Abstract

We study classical continuous automorphisms of the torus (cat maps) from the viewpoint of the Koopman theory. We find analytical formulae for Koopman modes defined coherently on the whole of the torus, and their decompositions associated with the partition of the torus into ergodic components. The spectrum of the Koopman operator is studied in four cases of cat maps: cyclic, quasi-cyclic, critical (transition from quasi-cyclic to chaotic behaviour) and chaotic. The synthetic spectrum associated with the ergodic decomposition is also studied.
Paper Structure (20 sections, 2 theorems, 79 equations, 16 figures, 1 table)

This paper contains 20 sections, 2 theorems, 79 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Koopman analysis of classical cat maps
  3. Cat maps and Koopman operators
  4. Ergodic decomposition and synthetic spectrum
  5. Ergodic decomposition:
  6. Koopman modes on a single ergodic component:
  7. Synthetic spectrum:
  8. Full Koopman modes
  9. The Koopman modes with respect to the different classical dynamical regimes
  10. $N$-cyclic cat map case
  11. Quasi-cyclic cat map case
  12. Critical cat map case
  13. Chaotic cat map case
  14. Summary
  15. Conclusion
  16. ...and 5 more sections

Key Result

Theorem 1

If $\theta_a$ is $N$-cyclic ($N \in \mathbb N^*$), i.e. $\varphi^N(\theta_a)=\theta_a \iff \Gamma_a = \mathrm{Orb}(\theta_a) = \{\varphi^n(\theta_a)\}_{n\in\{0,...,N-1\}}$, then $\mathrm{Sp}(\left. \mathcal{K} \right|_{\Gamma_a}) = \mathrm{Sp}_{pp} (\left. \mathcal{K} \right|_{\Gamma_a}) = \{e^{\ima where $f_p \equiv f_{\frac{2\pi}{N} p}$ and $\delta_{\varphi^n(\theta_a)} (\theta) = \delta(\theta-

Figures (16)

  • Figure 1: Graphical representation of $\varphi: \theta \mapsto \Phi \theta + 2\pi m_i$, with $\Phi = \left(-11-10\right)$. $\Gamma$ is represented by $[0,2\pi[^2$.
  • Figure 2: $\Omega$, $\varphi(\Omega)$ and $\varphi^2(\Omega)$ for $\Phi = \left(-11-10\right)$. Right: same as left but with a colouring of the orbits according to the distance from their initial points to the nontrivial fixed points.
  • Figure 3: The real part of the Koopman modes $\Re\mathrm{e}(f_{nm}(\theta))$ eq. (\ref{['modescat1']}) for different values of $n$ and $m$, for the cat system defined by $\Phi = \left(-11-10\right)$ for which $\frac{\omega}{2\pi}=\frac{1}{3}$.
  • Figure 4: Graphical representation of $\varphi: \theta \mapsto \Phi \theta + 2\pi m_i$ (left), with $\Phi = \left(2 \cos \omega1-10\right)$; and for each point $\theta$, the value $N_\theta$ of its first return time in its initial patch $D_{i_\theta}$ (right). $\Gamma$ is represented by $[0,2\pi[^2$.
  • Figure 5: Representation of some orbits $\{\varphi^n(\theta_0)\}_{n \in \mathbb N}$ in $\Gamma$ (with $\Phi = \left(2\cos\omega1-10\right)$), with the position of the fixed points ($*$), 2-cyclic points ($+$), 5-cyclic points ($\times$), and 8-cyclic points ($\circ$) associated with the first return in $D_{i_\theta}$. Right: rough approximation of $\Omega$.
  • ...and 11 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2