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Building on tools that have been successfully used in the study of rational billiards, such as induced maps and interval exchange transformations, we provide a construction of a one-parameter family of isosceles triangles exhibiting non-periodic trajectories that are not everywhere dense. This provides, by elementary means, a definitive negative answer to a long-standing open question on the density of non-periodic trajectories in triangular billiards., Comment: 15 pages, 2 figures

Extended dynamic mode decomposition (EDMD) is a data-driven algorithm for approximating spectral data of the Koopman operator associated to a dynamical system, combining a Galerkin method of order N and collocation method of order M. Spectral convergence of this method subtly depends on appropriate choice of the space of observables. For chaotic analytic full branch maps of the interval, we derive a constraint between M and N guaranteeing spectral convergence of EDMD., Comment: 24 pages

We show that spectral data of the Koopman operator arising from an analytic expanding circle map $\tau$ can be effectively calculated using an EDMD-type algorithm combining a collocation method of order m with a Galerkin method of order n. The main result is that if $m \geq \delta n$, where $\delta$ is an explicitly given positive number quantifying by how much $\tau$ expands concentric annuli containing the unit circle, then the method converges and approximates the spectrum of the Koopman operator, taken to be acting on a space of analytic hyperfunctions, exponentially fast in n. Additionally, these results extend to more general expansive maps on suitable annuli containing the unit circle., Comment: 18 pages, 4 figures

A complete description of resonances for rational toral Anosov diffeomorphisms preserving certain Reinhardt domains is presented. As a consequence it is shown that every homotopy class of two-dimensional Anosov diffeomorphisms contains maps with the sequence of resonances decaying stretched-exponentially. This is achieved by introducing a certain group of rational toral diffeomorphisms and computing the resonances of the respective composition operator considered on suitable anisotropic spaces of hyperfunctions. The class of examples is sufficiently rich to also include non-linear Anosov maps with trivial resonances, or resonances decaying exponentially, as well as with or without area-preservation or reversing symmetries., Comment: 46 pages, 4 figures

We introduce compactness classes of Hilbert space operators by grouping together all operators for which the associated singular values decay at a certain speed and establish upper bounds for the norm of the resolvent of operators belonging to a particular compactness class. As a consequence we obtain explicitly computable upper bounds for the Hausdorff distance of the spectra of two operators belonging to the same compactness class in terms of the distance of the two operators in operator norm., Comment: 26 pages

We show that spectral data of transfer operators given by holomorphic data can be approximated using an effective numerical scheme based on Lagrange interpolation. In particular, we show that for one-dimensional systems satisfying certain complex contraction properties, spectral data of the approximants converge exponentially to the spectral data of the transfer operator with the exponential rate determined by the respective complex contraction ratios of the underlying systems. We demonstrate the effectiveness of this scheme by numerically computing eigenvalues of transfer operators arising from interval and circle maps, as well as Lyapunov exponents of (positive) random matrix products and iterated function systems, based on examples taken from the literature., Comment: 22 pages, 1 figure

Given a domain $D$ in $\mathbb{C}^n$ and $K$ a compact subset of $D$, the set $\mathcal{A}_K^D$ of all restrictions of functions holomorphic on $D$ the modulus of which is bounded by $1$ is a compact subset of the Banach space $C(K)$ of continuous functions on $K$. The sequence $(d_m(\mathcal{A}_K^D))_{m\in \mathbb{N}}$ of Kolmogorov $m$-widths of $\mathcal{A}_K^D$ provides a measure of the degree of compactness of the set $\mathcal{A}_K^D$ in $C(K)$ and the study of its asymptotics has a long history, essentially going back to Kolmogorov's work on $\epsilon$-entropy of compact sets in the 1950s. In the 1980s Zakharyuta showed that for suitable $D$ and $K$ the asymptotics \begin{equation*} \lim_{m\to \infty}\frac{- \log d_m(\mathcal{A}_K^D)}{m^{1/n}} = 2\pi \left ( \frac{n!}{C(K,D)}\right ) ^{1/n}\,, \end{equation*} where $C(K,D)$ is the Bedford-Taylor relative capacity of $K$ in $D$ is implied by a conjecture, now known as Zakharyuta's Conjecture, concerning the approximability of the regularised relative extremal function of $K$ and $D$ by certain pluricomplex Green functions. Zakharyuta's Conjecture was proved by Nivoche in 2004 thus settling the asymptotics above at the same time. We shall give a new proof of the asymptotics above for $D$ strictly hyperconvex and $K$ non-pluripolar which does not rely on Zakharyuta's Conjecture. Instead we proceed more directly by a two-pronged approach establishing sharp upper and lower bounds for the Kolmogorov widths. The lower bounds follow from concentration results of independent interest for the eigenvalues of a certain family of Toeplitz operators, while the upper bounds follow from an application of the Bergman-Weil formula together with an exhaustion procedure by special holomorphic polyhedra., Comment: 34 pages; strengthened result: compact $K$ now only assumed to be non-pluripolar

Extended dynamic mode decomposition (EDMD) provides a class of algorithms to identify patterns and effective degrees of freedom in complex dynamical systems. We show that the modes identified by EDMD correspond to those of compact Perron-Frobenius and Koopman operators defined on suitable Hardy-Hilbert spaces when the method is applied to classes of analytic maps. Our findings elucidate the interpretation of the spectra obtained by EDMD for complex dynamical systems. We illustrate our results by numerical simulations for analytic maps., Comment: 14 pages. 3 figures

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., Comment: 19 pages, 4 figures

We study the dependence of the topological entropy of piecewise monotonic maps with holes under perturbations, for example sliding a hole of fixed size at uniform speed or expanding a hole with uniform expansion. We show that under suitable conditions the topological entropy varies locally Hoelder continuously with the local Hoelder exponent depending itself on the value of the topological entropy., Comment: 23 pages; section 6 has been considerably simplified following suggestions of a referee; to appear in Ergodic Theory and Dynamical Systems

Given two trace class operators A and B on a separable Hilbert space we provide an upper bound for the Hausdorff distance of their spectra involving only the distance of A and B in operator norm and the singular values of A and B. By specifying particular asymptotics of the singular values our bound reproduces or improves existing bounds for the spectral distance. The proof is based on lower and upper bounds for determinants of trace class operators of independent interest., Comment: 11 pages

We explicitly determine the spectrum of transfer operators (acting on spaces of holomorphic functions) associated to analytic expanding circle maps arising from finite Blaschke products. This is achieved by deriving a convenient natural representation of the respective adjoint operators., Comment: 13 pages

We show that for any $\lambda \in \mathbb{C}$ with $|\lambda|<1$ there exists an analytic expanding circle map such that the eigenvalues of the associated transfer operator (acting on holomorphic functions) are precisely the nonnegative powers of $\lambda$ and $\bar{\lambda}$. As a consequence we obtain a counterexample to a variant of a conjecture of Mayer on the reality of spectra of transfer operators., Comment: 15 pages, 3 figures

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. For piecewise linear expanding Markov maps observed via piecewise analytic functions we provide explicit bounds of the decay rate in terms of the Lyapunov exponent. In addition, we comment on similar relations for general piecewise smooth expanding maps., Comment: 19 pages, 6 figures

Let $\Omega$ be an open set in $\R^d$ $(d > 1)$ and $h(\Omega)$ the Fr\'echet space of harmonic functions on $\Omega$. Given a bounded linear operator $L :h(\Omega)\to h(\Omega)$, we show that its eigenvalues $\lambda_n$, arranged in decreasing order and counting multiplicities, satisfy $|\lambda_n|\leq K\exp(-cn^{1/(d-1)})$, where $K$ and $c$ are two explicitly computable positive constants., Comment: AMS-LaTeX, 14 pages

For $a,\alpha>0$ let $E(a,\alpha)$ be the set of all compact operators $A$ on a separable Hilbert space such that $s_n(A)=O(\exp(-an^\alpha))$, where $s_n(A)$ 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 $E(a,\alpha)$., Comment: AMS-LaTeX, 20 pages

We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if D is any open set in C^d, and L is a suitable transfer operator acting on Bergman space A^2(D), its eigenvalue sequence lambda_n(L) is bounded by |lambda_n(L)| \leq A\exp(-a n^{1/d}), where a, A are explicitly given., Comment: 19 pages, to appear in Advances in Mathematics

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., Comment: 9 pages, to appear in Ergod. Th. & Dyn. Sys.

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