Escape rates and Perron-Frobenius operators: Open and closed dynamical systems
Gary Froyland Ognjen Stancevic
We study the Perron-Frobenius operator $\mathcal{P}$ of closed dynamical systems and certain open dynamical systems. We prove that the presence of a large positive eigenvalue $\rho$ of $\mathcal{P}$ guarantees the existence of a 2-partition of the phase space for which the escape rates of the open systems defined on the two partition sets are both slower than $-\log\rho$. The open systems with slow escape rates are easily identified from the Perron-Frobenius operators of the closed systems. Numerical results are presented for expanding maps of the unit interval. We also apply our technique to shifts of finite type to show that if the adjacency matrix for the shift has a large positive second eigenvalue, then the shift may be decomposed into two disjoint subshifts, both of which have high topological entropies.
keywords: topological entropy. Perron-Frobenius operator open dynamical system escape rate almost-invariant set
On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps
Gary Froyland
Perron-Frobenius operators and their eigendecompositions are increasingly being used as tools of global analysis for higher dimensional systems. The numerical computation of large, isolated eigenvalues and their corresponding eigenfunctions can reveal important persistent structures such as almost-invariant sets, however, often little can be said rigorously about such calculations. We attempt to explain some of the numerically observed behaviour by constructing a hyperbolic map with a Perron-Frobenius operator whose eigendecomposition is representative of numerical calculations for hyperbolic systems. We explicitly construct an eigenfunction associated with an isolated eigenvalue and prove that a special form of Ulam's method well approximates the isolated spectrum and eigenfunctions of this map.
keywords: almost-invariant set Ulam's method isolated spectrum eigenfunction hyperbolic map. Perron-Frobenius operator
Preface: Special issue on the occasion of the 4th International Workshop on Set-Oriented Numerics (SON 13, Dresden, 2013)
Gary Froyland Oliver Junge Kathrin Padberg-Gehle
This issue comprises manuscripts collected on the occasion of the 4th International Workshop on Set-Oriented Numerics which took place at the Technische Universität Dresden in September 2013. The contributions cover a broad spectrum of different subjects in computational dynamics ranging from purely discrete problems on graphs to computer assisted proofs of bifurcations in dissipative PDEs. In many cases, ideas related to set-oriented paradigms turn out to be useful in the computations, for example by quantizing the state space, or by using interval arithmetic to perform rigorous computations.

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Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools
Gary Froyland Cecilia González-Tokman Anthony Quas
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 efficacy 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.
keywords: isolated spectrum Koopman operators mix-norms. Ulam's method Transfer operators metastability
A semi-invertible Oseledets Theorem with applications to transfer operator cocycles
Gary Froyland Simon Lloyd Anthony Quas
Oseledets' celebrated Multiplicative Ergodic Theorem (MET) [V.I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179--210.] is concerned with the exponential growth rates of vectors under the action of a linear cocycle on $\mathbb{R}^d$. When the linear actions are invertible, the MET guarantees an almost-everywhere pointwise splitting of $\mathbb{R}^d$ into subspaces of distinct exponential growth rates (called Lyapunov exponents). When the linear actions are non-invertible, Oseledets' MET only yields the existence of a filtration of subspaces, the elements of which contain all vectors that grow no faster than exponential rates given by the Lyapunov exponents. The authors recently demonstrated [G. Froyland, S. Lloyd, and A. Quas, Coherent structures and exceptional spectrum for Perron--Frobenius cocycles, Ergodic Theory and Dynam. Systems 30 (2010), , 729--756.] that a splitting over $\mathbb{R}^d$ is guaranteed without the invertibility assumption on the linear actions. Motivated by applications of the MET to cocycles of (non-invertible) transfer operators arising from random dynamical systems, we demonstrate the existence of an Oseledets splitting for cocycles of quasi-compact non-invertible linear operators on Banach spaces.
keywords: covariant vector Lyapunov vector Multiplicative ergodic theorem equivariant subspace.
A closing scheme for finding almost-invariant sets in open dynamical systems
Gary Froyland Philip K. Pollett Robyn M. Stuart
We explore the concept of metastability or almost-invariance in open dynamical systems. In such systems, the loss of mass through a ``hole'' occurs in the presence of metastability. We extend existing techniques for finding almost-invariant sets in closed systems to open systems by introducing a closing operation that has a small impact on the system's metastability.
keywords: almostinvariant set metastability conditionally invariant measure Open dynamical system Perron-Frobenius operator Ulam's method.

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