Eigenfunctions for smooth expanding circle maps

We construct a real-analytic circle map for which the corresponding Perron-Frobenius operator has a real-analytic eigenfunction with an eigenvalue outside the essential spectral radius when acting upon $C^1$-functions.Comment: 10 pages, 2 figure

Mixing property and pseudo random sequences

We will give a summary about the relations between the spectra of the Perron--Frobenius operator and pseudo random sequences for 1-dimensional cases.Comment: Published at http://dx.doi.org/10.1214/074921706000000211 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Data-Driven Approximation of Transfer Operators: Naturally Structured Dynamic Mode Decomposition

In this paper, we provide a new algorithm for the finite dimensional approximation of the linear transfer Koopman and Perron-Frobenius operator from time series data. We argue that existing approach for the finite dimensional approximation of these transfer operators such as Dynamic Mode Decomposition (DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two important properties of these operators, namely positivity and Markov property. The algorithm we propose in this paper preserve these two properties. We call the proposed algorithm as naturally structured DMD since it retains the inherent properties of these operators. Naturally structured DMD algorithm leads to a better approximation of the steady-state dynamics of the system regarding computing Koopman and Perron- Frobenius operator eigenfunctions and eigenvalues. However preserving positivity properties is critical for capturing the real transient dynamics of the system. This positivity of the transfer operators and it's finite dimensional approximation also has an important implication on the application of the transfer operator methods for controller and estimator design for nonlinear systems from time series data

High Temperature Expansions and Dynamical Systems

We develop a resummed high-temperature expansion for lattice spin systems with long range interactions, in models where the free energy is not, in general, analytic. We establish uniqueness of the Gibbs state and exponential decay of the correlation functions. Then, we apply this expansion to the Perron-Frobenius operator of weakly coupled map lattices.Comment: 33 pages, Latex; [email protected]; [email protected]

Escape from noisy intermittent repellers

Intermittent or marginally-stable repellers are commonly associated with a power law decay in the survival fraction. We show here that the presence of weak additive noise alters the spectrum of the Perron - Frobenius operator significantly giving rise to exponential decays even in systems that are otherwise regular. Implications for ballistic transport in marginally stable miscrostructures are briefly discussed.Comment: 3 ps figures include