Convergence of dynamics and the Perron–Frobenius operator

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 uniform mixing-like properties of the system.     

Vertex maps on graphs – Perron–Frobenius theory

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.     

Cell Mapping for Random Transformations

The method of Cell Mapping is a numerical tool to analyze the long-term behavior of dynamical systems. For deterministic systems, which are described by nonsingular transformations, Cell Mapping is characterized by a certain discretization of the Frobenius–Perron operator first proposed by Ulam [3]. Our purpose is to extend the concept to dynamical systems which are generated by random transformations. At this, time evolution of absolutely continuous measures and the corresponding densities will be described by Markov operators whose fixed points refer to invariant measures and densities respectively. A discretization of the Markov operator on densities leads directly to the reformulation of Cell Mapping in the stochastic context. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Frobenius–Perron theory of modified ADE bound quiver algebras

The Frobenius–Perron dimension for an abelian category was recently introduced in [5]. We apply this theory to the category of representations of the finite-dimensional radical square zero algebras associated to certain modified ADE graphs. In particular, we take an ADE quiver with arrows in a certain orientation and an arbitrary number of loops at each vertex. We show that the Frobenius–Perron dimension of this category is equal to the maximum number of loops at a vertex. Along the way, we introduce a result which can be applied in general to calculate the Frobenius–Perron dimension of a radical square zero bound quiver algebra. We use this result to introduce a family of abelian categories which produce arbitrarily large irrational Frobenius–Perron dimensions.     

On Perron–Frobenius property of matrices having some negative entries

We extend the theory of nonnegative matrices to the matrices that have some negative entries. We present and prove some properties which give us information, when a matrix possesses a Perron–Frobenius eigenpair. We apply also this theory by proposing the Perron–Frobenius splitting for the solution of the linear system Ax = b by classical iterative methods. Perron–Frobenius splittings constitute an extension of the well known regular splittings, weak regular splittings and nonnegative splittings. Convergence and comparison properties are given and proved.     

The analyticity of a generalized Ruelle’s operator

In this work we propose a generalization of the concept of Ruelle’s operator for one dimensional lattices used in thermodynamic formalism and ergodic optimization, which we call generalized Ruelle’s operator. Our operator generalizes both the Ruelle operator proposed in [2] and the Perron Frobenius operator defined in [7]. We suppose the alphabet is given by a compact metric space, and consider a general a-priori measure to define the operator. We also consider the case where the set of symbols that can follow a given symbol of the alphabet depends on such symbol, which is an extension of the original concept of transition matrices from the theory of subshifts of finite type. We prove the analyticity of the Ruelle’s operator and present some examples.     

Risk-Sensitive Control and an Abstract Collatz–Wielandt Formula

The ‘value’ of infinite horizon risk-sensitive control is the principal eigenvalue of a certain positive operator. For the case of compact domain, Chang has built upon a nonlinear version of the Krein–Rutman theorem to give a ‘min–max’ characterization of this eigenvalue which may be viewed as a generalization of the classical Collatz–Wielandt formula for the Perron–Frobenius eigenvalue of a nonnegative irreducible matrix. We apply this formula to the Nisio semigroup associated with risk-sensitive control and derive a variational characterization of the optimal risk-sensitive cost. For the linear, i.e., uncontrolled case, this is seen to reduce to the celebrated Donsker–Varadhan formula for principal eigenvalue of a second-order elliptic operator.     

Absolute Logarithmic Norm

In this paper we introduce and study a new concept of the absolute logarithmic norm, which has much in common with the classical definition of the logarithmic norm by S. M. Lozinskii. The the theory that we develop allows to obtain new facts from the Lyapunov stability theory for the systems of linear differential equations with constant coefficients. The presentation of the material relies heavily on the theory of off-diagonally nonnegative matrices arising from the Perron–Frobenius theory for nonnegative matrices.     

Finite approximations of Markov operators

In this paper we survey some recent developments in the numerical analysis of Markov operators, and in particular Frobenius–Perron operators associated with chaotic discrete dynamical systems.     

New eigenvalue inclusion sets for tensors

Two new eigenvalue inclusion sets for tensors are established. It is proved that the new eigenvalue inclusion sets are tighter than that in Qi's paper “Eigenvalues of a real supersymmetric tensor”. As applications, upper bounds for the spectral radius of a nonnegative tensor are obtained, and it is proved that these upper bounds are sharper than that in Yang's paper “Further results for Perron–Frobenius theorem for nonnegative tensors”. And some sufficient conditions of the positive definiteness for an even‐order real supersymmetric tensor are given. Copyright © 2012 John Wiley & Sons, Ltd.     

Attracting Sets for Increasing Co‐Radiant and Topical Operators

A generalization of the Perron‐Frobenius theorem to increasing positively homogeneous of degree one operators is extended to increasing co‐radiant and topical operators, which are of interest in mathematical economics. In particular, small attracting sets containing the limit points of all sequences generated by iteration of such operators are determined.     

A Perron–Frobenius theorem for a class of positive quasi-polynomial matrices

In this work, we give an extension of the classical Perron–Frobenius theorem to positive quasi-polynomial matrices. Then the result obtained is applied to derive necessary and sufficient conditions for the exponential stability of positive linear time-delay differential systems.     

Vecteurs de Perron–Frobenius et produits Gamma

This Note presents formulas to express the coordinates of Perron–Frobenius vectors of Cartan matrices (finite or affine) as products of Gamma values, for each finite irreducible root system of rank r.     

Optimal multilevel matrix algebra operators

We study the optimal Frobenius operator in a general matrix vector space and in particular in the multilevel trigonometric matrix vector spaces, by emphasizing both the algebraic and geometric properties. These general results are used to extend the Korovkin matrix theory for the approximation of block Toeplitz matrices via trigonometric vector spaces. The abstract theory is then applied to the analysis of the approximation properties of several sine and cosine based vector spaces. Few numerical experiments are performed to give evidence of the theoretical results.     

An age‐dependent population equation with delayed birth process

Within a semigroup framework, we discuss well posedness and qualitative behaviour of an age‐dependent population equation with delay in the birth process. Using positivity and Perron–Frobenius theory we obtain an explicit stability criterion. Copyright © 2004 John Wiley & Sons, Ltd.     

Harmonic analysis on perturbed Cayley Trees

We study some spectral properties of the adjacency operator of non-homogeneous networks. The graphs under investigation are obtained by adding density zero perturbations to the homogeneous Cayley Trees. Apart from the natural mathematical meaning, such spectral properties are relevant for the Bose Einstein Condensation for the pure hopping model describing arrays of Josephson junctions on non-homogeneous networks. The resulting topological model is described by a one particle Hamiltonian which is, up to an additive constant, the opposite of the adjacency operator on the graph. It is known that the Bose Einstein condensation already occurs for unperturbed homogeneous Cayley Trees. However, the particles condensate on the perturbed graph, even in the configuration space due to non-homogeneity. Even if the graphs under consideration are exponentially growing, we show that it is enough to perturb in a negligible way the original graph in order to obtain a new network whose mathematical and physical properties dramatically change. Among the results proved in the present paper, we mention the following ones. The appearance of the Hidden Spectrum near the zero of the Hamiltonian, or equivalently below the norm of the adjacency. The latter is related to the value of the critical density and then with the appearance of the condensation phenomena. The investigation of the recurrence/transience character of the adjacency, which is connected to the possibility to construct locally normal states exhibiting the Bose Einstein condensation. Finally, the study of the volume growth of the wave function of the ground state of the Hamiltonian, which is nothing but the generalized Perron Frobenius eigenvector of the adjacency. This Perron Frobenius weight describes the spatial distribution of the condensate and its shape is connected with the possibility to construct locally normal states exhibiting the Bose Einstein condensation at a fixed density greater than the critical one.     

A perron theorem for the existence of invariant subspaces

Summary An extended notion of cone K in a linear vector space is introduced and certain properties of characteristic values and eigenvectors of a linear operator mapping K {0} into its interior are derived by proving a theorem which extends the classical results of Perron on positive matrices.     

算子概率范数与共鸣定理

提出概率赋范线性空间上集合有界性的简化定义,利用算子概率范数概念。进一步研究概率赋范线性空间上的线性算子理论,并在算子概率赋范空间上,建立了概率有界、概率半有界、非概率无界意义下的共鸣定理。     

On Frobenius incidence varieties of linear subspaces over finite fields

We define Frobenius incidence varieties by means of the incidence relation of Frobenius images of linear subspaces in a fixed vector space over a finite field, and investigate their properties such as supersingularity, Betti numbers and unirationality. These varieties are variants of the Deligne–Lusztig varieties. We then study the lattices associated with algebraic cycles on them. We obtain a positive-definite lattice of rank 84 that yields a dense sphere packing from a 4-dimensional Frobenius incidence variety in characteristic 2.