We introduce complex cones and associated projective gauges, generalizing a real Birkhoff cone and its Hilbert metric to complex vector spaces. We deduce a variety of spectral gap theorems in complex Banach spaces. We prove a dominated complex cone-contraction Theorem and use it to extend the classical Perron-Frobenius Theorem to complex matrices, Jentzsch's Theorem to complex integral operators, a Krein-Rutman Theorem to compact and quasi-compact complex operators and a Ruelle-Perron-Frobenius Theorem to complex transfer operators in dynamical systems.
In the simplest case of a complex n by n matrix A we have the following statement: Suppose that 0 < c < +infinity is such that |Im A_{ij} \bar{A}_{mn}| < c < Re A_{ij} \bar{A}_{mn} for all indices. Then A has a `spectral gap'.
We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron-Frobenius operator. Our main result states that strong convergence of the powers of the Perron-Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by a uniform mixing-like property of the system.
We characterise amenability of a countable group in terms of the spectral radius of the Perron-Frobenius operator associated to a group extension of a countable Markov shift and a Hölder continuous potential. This extends a result of Day for random walks and recent work of Stadlbauer for dynamical systems. Moreover, we show that, if the potential satisfies a symmetry condition with respect to the group extension, then the logarithm of the spectral radius of the Perron-Frobenius operator is given by the Gurevic pressure of the potential.
The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space $\mathcal{H}$ with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators using a dense locally convex subspace $X$ of $\mathcal{H}$ and its dual space $X'$. The three topological spaces $X \subset \mathcal{H} \subset X'$ is called the rigged Hilbert space or the Gelfand triplet. In this paper, the generalized spectra of the Perron-Frobenius operators of the one-sided and two-sided shifts of finite types (symbolic dynamical systems)…
In this technical report the $C_0$ semigroup of Frobenius-Perron operators on $L^1(X)$ is considered, where the underlying dynamical system is such that trajectories may leave the state space $X$ and terminate. We introduce a discrete infinitesimal generator and show, that the operator semigroup generated by this discrete generator converges in $L^1(X)$ pointwise to the Frobenius-Perron operator of the system.
The goal of this paper is to describe the connections between Perron-Frobenius theory and vertex maps on graphs. In particular, it is shown how Perron-Frobenius theory gives results about the sets of integers that can arise as periods of periodic orbits, about the concepts of transitivity and topological mixing, and about horseshoes and topological entropy. This is a preprint. The final version will appear in the Journal of Difference Equations and Applications.
First approach of invariant densities of a Perron Frobenius operator. Asymptotic behaviours of ODE or PDE, as, are most interesting. The associed infinitesimal iteration is. If is partially linear, a random distribution can be asymptotic solution. Among applications, are asymptotic profiles of of Lorenz, Navier Stokes or Hamilton's équations.
We study the non-uniqueness sets for solutions to the Klein-Gordon equation in 1 space dimension, for solutions whose Fourier transform is a finite complex measure absolutely continuous with respect to arc length. We show that generally, in the non-unique case for lattice-crosses, the space of solutions that vanish there is infinite-dimensional.
It is shown that for certain maps, including concave maps, on the $d$-dimensional lattice of positive integer points, 'approximate' eigenvectors can be found. Applications in epidemiology as well as distributed resource allocation are discussed as examples.
The filtering problems are derived from a sequential minimization of a quadratic function representing a compromise between model and data. In this paper, we use the Perron-Frobenius operator in stochastic process to develop a Perron-Frobenius operator filter. The proposed method belongs to Bayesian filtering and works for non-Gaussian distributions for nonlinear stochastic dynamical systems. The recursion of the filtering can be characterized by the composition of Perron-Frobenius operator and likelihood operator. This gives a significant connection between the Perron-Frobenius operator and B…
We prove a complex Ruelle-Perron-Frobenius theorem for Markov shifts over an infinite alphabet, whence extending results by M. Pollicott from the finite to the infinite alphabet setting. As an application we obtain an extension of renewal theory in symbolic dynamics, as developed by S. P. Lalley and in the sequel generalised by the second author, now covering the infinite alphabet case.
We give a new global presentation of our results on the asymptotic behavior of an iteration. This paper brings many improvements and corrections to our previous preprints on the subject. Among the applications, we use new methods to compute asymptotic results of PDE like Lorenz or Navier-Stokes equations. New questions as the resonance are studied.
We prove that if an $n\times n$ matrix defined over ${\mathbb Q}_p$ (or more generally an arbitrary complete, discretely-valued, non-Archimedean field) satisfies a certain congruence property, then it has a strictly maximal eigenvalue in ${\mathbb Q}_p$, and that iteration of the (normalized) matrix converges to a projection operator onto the corresponding eigenspace. This result may be viewed as a $p$-adic analogue of the Perron-Frobenius theorem for positive real matrices.
We consider Perron-Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations, and establish their Fréchet differentiability with respect to the drift. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular v…
To study the convergence to equilibrium in random maps we developed the spectral theory of the corresponding transfer (Perron-Frobenius) operators acting in a certain Banach space of generalized functions. The random maps under study in a sense fill the gap between expanding and hyperbolic systems since among their (deterministic) components there are both expanding and contracting ones. We prove stochastic stability of the Perron-Frobenius spectrum and developed its finite rank operator approximations by means of a ``stochastically smoothed'' Ulam approximation scheme. A counterexample to the…
We prove a general version of the classical Perron-Frobenius convergence property for reducible matrices. We then apply this result to reducible substitutions and use it to produce limit frequencies for factors and hence invariant measures on the associated subshift. The analogous results are well known for primitive substitutions and have found many applications, but for reducible substitutions the tools provided here were so far missing from the theory.
In this paper we extend work of Kawamura, see kawamura, for Cuntz-Krieger algebras O_A for infinite matrices A. We generalize the definition of branching systems, prove their existence for any given matrix A and show how they induce some very concrete representations of O_A. We use these representations to describe the Perron-Frobenius operator, associated to an nonsingular transformation, as an infinite sum and under some hypothesis we find a matrix representation for the operator. We finish the paper with a few examples.
In this paper we show how to produce a large number of representations of a graph C*-algebra in the space of the bounded linear operators in $L^2(X,μ)$. These representations are very concrete and, in the case of graphs that satisfy condition (K), we use our techniques to realize the associated graph C*-algebra as a subalgebra of the bounded operators in $L^2(R)$. We also show how to describe some Perron-Frobenius operators in $L^1(X,μ)$, in terms of the representations we associate to a graph.
We provide conditions for the existence of measurable solutions to the equation $ξ(Tω)=f(ω,ξ(ω))$, where $T:Ω\rightarrowΩ$ is an automorphism of the probability space $Ω$ and $f(ω,\cdot)$ is a strictly non-expansive mapping. We use results of this kind to establish a stochastic nonlinear analogue of the Perron-Frobenius theorem on eigenvalues and eigenvectors of a positive matrix. We consider a random mapping $D(ω)$ of a random closed cone $K(ω)$ in a finite-dimensional linear space into the cone $K(Tω)$. Under assumptions of monotonicity and homogeneity of $D(ω)$, we prove the existence of sc…
In this paper, we study random Blaschke products, acting on the unit circle, and consider the cocycle of Perron-Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. As a corollary, we obtain a simple random Blaschke product system where the Perron-Frobenius cocycle has infinitely many distinct Lyapunov exponents, but where arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum, except for the exponent 0 associated with the absolutely continuous invariant measure. That…
Given a zero-one matrix A we consider certain one-parameter groups of automorphisms of the Cuntz-Krieger algebra O_A, generalizing the usual gauge group, and depending on a positive continuous function H defined on the Markov space Σ_A. The main result consists of an application of Ruelle's Perron-Frobenius Theorem to show that these automorphism groups admit a single KMS state.
We present an analysis of one-dimensional models of dynamical systems that possess 'coherent structures'; global structures that disperse more slowly than local trajectory separation. We study cocycles generated by expanding interval maps and the rates of decay for functions of bounded variation under the action of the associated Perron-Frobenius cocycles. We prove that when the generators are piecewise affine and share a common Markov partition, the Lyapunov spectrum of the Perron-Frobenius cocycle has at most finitely many isolated points. Moreover, we develop a strengthened version of the M…
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；School of Fundamental Science and Technology, Keio University（庆应义塾大学基础理工学部）；The Institute of Scientific and Industrial Research, Osaka University（大阪大学产业科学研究所）
The development of a metric for structural data is a long-term problem in pattern recognition and machine learning. In this paper, we develop a general metric for comparing nonlinear dynamical systems that is defined with Perron-Frobenius operators in reproducing kernel Hilbert spaces. Our metric includes the existing fundamental metrics for dynamical systems, which are basically defined with principal angles between some appropriately-chosen subspaces, as its special cases. We also describe the estimation of our metric from finite data. We empirically illustrate our metric with an example of …
In this note, we establish an original result for the thermodynamic formalism in the context of expanding circle transformations with an indifferent fixed point. For an observable whose continuity modulus is linked to the dynamics near such a fixed point, by identifying an appropriate linear space to evaluate the action of the transfer operator, we show that there is a strictly positive eigenfunction associated with the maximal eigenvalue given as the exponential of the topological pressure. Taking into account also the corresponding eigenmeasure, the invariant probability thus obtained is pro…
Let $C$ be a closed cone with nonempty interior $C^\circ$ in a Banach space. Let $f:C^\circ \rightarrow C^\circ$ be an order-preserving subhomogeneous function with a fixed point in $C^\circ$. We introduce a condition which guarantees that the iterates $f^k(x)$ converge to a fixed point for all $x \in C^\circ$. This condition generalizes the notion of type K order-preserving for maps on $\mathbb{R}^n_{>0}$. We also prove that when iterates converge to a fixed point, the rate of convergence is always R-linear in two special cases: for piecewise affine maps and also for order-preserving, homogen…
The paper is about the data-driven computation of optimal control for a class of control affine deterministic nonlinear systems. We assume that the control dynamical system model is not available, and the only information about the system dynamics is available in the form of time-series data. We provide a convex formulation for the optimal control problem of the nonlinear system. The convex formulation relies on the duality result in the dynamical system's stability theory involving density function and Perron-Frobenius operator. We formulate the optimal control problem as an infinite-dimensio…
Given a discrete-time random dynamical system represented by a cocycle of non-singular measurable maps, we may obtain information on dynamical quantities by studying the cocycle of Perron-Frobenius operators associated to the maps. Of particular interest is the second-largest Lyapunov exponent, $λ_2$, which can tell us about mixing rates and decay of correlations in the system. We prove a generalized Perron-Frobenius theorem for cocycles of bounded linear operators on Banach spaces that preserve and occasionally contract a cone; this theorem shows that the top Oseledets space for the cocycle i…
We consider discrete-time dynamical systems with a linear relaxation dynamics that are driven by deterministic chaotic forces. By perturbative expansion in a small time scale parameter, we derive from the Perron-Frobenius equation the corrections to ordinary Fokker-Planck equations in leading order of the time scale separation parameter. We present analytic solutions to the equations for the example of driving forces generated by N-th order Chebychev maps. The leading order corrections are universal for N larger or equal to 4 but different for N=2 and N=3. We also study diffusively coupled Che…
The study of escape rates for a ball in a dynamical systems has been much studied. Understanding the asymptotic behavior of the escape rate as the radius of the ball tends to zero is an especially subtle problem. In the case of hyperbolic conformal systems this has been addressed by various authors. In this paper we consider a far more general realm of conformal maps where the analysis is correspondingly more complicated. We prove the existence of escape rates and calculate them in the context of countable alphabets, either finite or infinite, uniformly contracting conformal graph directed Mar…
We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (Information on the existence of an SRB measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension $d=2$ we show that the transfer operator associated to …
We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent $ν=d-1$, where $d$ is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we find the Weyl exponent $ν=d/2$ where $d$ is the fractal dimension of strange attractor. The Weyl exponent can be…
We establish a generalized Perron-Frobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any nonlinear order-preserving and positively homogeneous map $f$ acting on the open orthant $\mathbb{R}_{\scriptscriptstyle >0}^n$. This criterion involves dominions, i.e., sets of states that can be made invariant by one player in a two-person game that only depends on the behavior of $f$ "at infinity". In this way, we characterize the situation in which for all $α, β> 0$, the "slice space" $\mathcal{S}_α^β:= \{ x \in \mathbb{R}_{\scriptscriptstyle >0}^n \…
Let $Γ$ be a smooth curve or finite disjoint union of smooth curves in the plane and $Λ$ be any subset of the plane. Let $\mathcal X(Γ)$ be the space of all finite complex-valued Borel measures in the plane which are supported on $Γ$ and are absolutely continuous with respect to the arc length measure on $Γ.$ Let $\mathcal{AC}(Γ,Λ)=\{μ\in \mathcal{X}(Γ) : \hatμ|_Λ=0\},$ then we prove the following results: \begin{enumerate}[(a)] \item For a rational perturbation of $Λ_β$ namely, $Λ_β^θ=\left((\mathbb Z+\{θ\})\times\{0\}\right)\cup\left(\{0\}\timesβ\mathbb Z\right),$ where $θ=1/{p},~\text{for s…
A Perron-Frobenius strong threshold theorem for (A, B, P, ϕ) balanced bilinear models, and the role of left and right Perron eigenvectors in mathematical epidemiology
We explore the concept of metastability in random dynamical systems, focussing on connections between random Perron-Frobenius operator cocycles and escape rates of random maps, and on topological entropy of random shifts of finite type. The Lyapunov spectrum of the random Perron-Frobenius cocycle and the random adjacency matrix cocycle is used to decompose the random system into two disjoint random systems with rigorous upper and lower bounds on (i) the escape rate in the setting of random maps, and (ii) topological entropy in the setting of random shifts of finite type, respectively.
Finite time coherent sets [8] have recently been defined by a measure based objective function describing the degree that sets hold together, along with a Frobenius-Perron transfer operator method to produce optimally coherent sets. Here we present an extension to generalize the concept to hierarchially defined relatively coherent sets based on adjusting the finite time coherent sets to use relative mesure restricted to sets which are developed iteratively and hierarchically in a tree of partitions. Several examples help clarify the meaning and expectation of the techniques, as they are the no…
