Oscar F. Bandtlow

On the Ruelle eigenvalue sequence

For certain real analytic data, we show that the eigenvalue sequence of the associated transfer operator L is insensitive to the holomorphic function space on which L acts. Explicit bounds on this eigenvalue sequence are established.

Rigged Hilbert spaces for chaotic dynamical systems

We consider the problem of rigging for the Koopman operators of the Renyi and the baker maps. We show that the rigged Hilbert space for the Renyi maps has some of the properties of a strict inductive limit and give a detailed description of the rigged Hilbert space for the baker maps.

Resolvent Estimates for Operators Belonging to Exponential Classes

Abstract.For a, α > 0 let E(a, α) be the set of all compact operators A on a separable Hilbert space such that sn(A) = O(exp(-anα)), where sn(A) denotes the n-th singular number of A. We provide upper bounds for the norm of the resolvent (zI − A)−1 of A in terms of a quantity describing the departure from normality of A and the distance of z to the spectrum of A. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in E(a, α).

Explicit A Priori Bounds on Transfer Operator Eigenvalues

We provide explicit bounds on the eigenvalues of transfer operators defined in terms of holomorphic data.

On the discrete time version of the Brussels formalism

We develop a discrete time version of the so-called Brussels formalism in nonequilibrium statistical mechanics for continuous endomorphisms of a Banach space. We show that, if the evolution operator U and the projector P are such that PU is a compact operator and the spectral radius of (I-P)U(I-P) is strictly less than the spectral radius of U, then the formalism holds and the evolution operator is quasicompact.

On the relation between Lyapunov exponents and exponential decay of correlations

Chaotic dynamics with sensitive dependence on initial conditions may result in exponential decay of correlation functions. We show that for one-dimensional interval maps the corresponding quantities, that is, Lyapunov exponents and exponential decay rates, are related. More specifically, for piecewise linear expanding Markov maps observed via piecewise analytic functions, we show that the decay rate is bounded above by twice the Lyapunov exponent, that is, we establish lower bounds for the subleading eigenvalue of the corresponding Perron‐Frobenius operator. In addition, we comment on similar relations for general piecewise smooth expanding maps.

Resonances of dynamical systems and Fredholm-Riesz operators on rigged Hilbert spaces☆

Abstract Resonances of dynamical systems are defined as the singularities of the analytically continued resolvent of the restriction of the Frobenius-Perron operator to suitable test-function spaces. A sufficient condition for resonances to arise from a meromorphic continuation to the entire plane is that the Frobenius-Perron operator is a Fredholm-Riesz operator on a rigged Hilbert space. After a discussion of spectral theory in locally convex topological vector spaces, we illustrate the approach for a simple chaotic system, namely the Renyi map.

Complete spectral data for analytic Anosov maps of the torus

Using analytic properties of Blaschke factors we construct a family of analytic hyperbolic diffeomorphisms of the torus for which the spectral properties of the associated transfer operator acting on a suitable Hilbert space can be computed explicitly. As a result, we obtain explicit expressions for the decay of correlations of analytic observables without resorting to any kind of perturbation argument.

Mapping between dynamic markings and performed loudness: A machine learning approach

Loudness variation is one of the foremost tools for expressivity in music performance. Loudness is frequently notated as dynamic markings such as (piano, meaning soft) or (forte, meaning loud). While dynamic markings in music scores are important indicators of how music pieces should be interpreted, their meaning is less straightforward than it may seem, and depends highly on the context in which they appear. In this article, we investigate the relationship between dynamic markings in the score and performed loudness by applying machine learning techniques – decision trees, support vector machines, artificial neural networks, and a k-nearest neighbor method – to the prediction of loudness levels corresponding to dynamic markings, and to the classification of dynamic markings given loudness values. The methods are applied to 44 recordings of performances of Chopins Mazurkas, each by 8 pianists. The results show that loudness values and markings can be predicted relatively well when trained across recordings of the same piece, but fail dismally when trained across the pianists recordings of other pieces, demonstrating that score features may trump individual style when modeling loudness choices. Evidence suggests that all the features chosen for the task are relevant, and analysis of the results reveals the forms (such as the return of the theme) and structures (such as dynamic-marking repetitions) that influence the predictability of loudness and markings. Modeling of loudness trends in expressive performance appears to be a delicate matter, and sometimes loudness expression can be a matter of the performers idiosyncracy.

Asymptotics of the Farey Fraction Spin Chain Free Energy at the Critical Point

AbstractWe consider the Farey fraction spin chain in an external field h. Using ideas from dynamical systems and functional analysis, we show that the free energy f in the vicinity of the second-order phase transition is given, exactly, by