Perron Frobenius Research Articles

Large deviations, central limit, and dynamical phase transitions in the atom maser

The theory of quantum jump trajectories provides a new framework for understanding dynamical phase transitions in open systems. A candidate for such transitions is the atom maser, which for certain parameters exhibits strong intermittency in the atom detection counts and has a bistable stationary state. Although previous numerical results suggested that the “free energy” may not be a smooth function, we show that the atom detection counts satisfy a large deviations principle and, therefore, we deal with a phase crossover rather than a genuine phase transition. We argue, however, that the latter occurs in the limit of an infinite pumping rate. As a corollary, we obtain the central limit theorem for the counting process. The proof relies on the analysis of a certain deformed generator whose spectral bound is the limiting cumulant generating function. The latter is shown to be smooth so that a large deviations principle holds by the Gärtner–Ellis theorem. One of the main ingredients is the Krein–Rutman theory, which extends the Perron–Frobenius theorem to a general class of positive compact semigroups.

Linear recurrent cryptography: Golden-like cryptography for higher order linear recurrences

We develop matrix cryptography based on linear recurrent sequences of any order that allows securing encryption against brute force and chosen plaintext attacks. In particular, we solve the problem of generalizing error detection and correction algorithms of golden cryptography previously known only for recurrences of a special form. They are based on proving the checking relations (inequalities satisfied by the ciphertext) under the condition that the analog of the golden Q-matrix has the strong Perron–Frobenius property. These algorithms are proved to be especially efficient when the characteristic polynomial of the recurrence is a Pisot polynomial. Finally, we outline algorithms for generating recurrences that satisfy our conditions.

A Fractal Eigenvector

The recursively-constructed family of Mandelbrot matrices for n = 1, 2, have nonnegative entries (indeed just 0 and 1, so each can be called a binary matrix) and have eigenvalues whose negatives give periodic orbits under the Mandelbrot iteration, namely with , and are thus contained in the Mandelbrot set. By the Perron–Frobenius theorem, the matrices have a dominant real positive eigenvalue, which we call ρn . This article examines the eigenvector belonging to that dominant eigenvalue and its fractal-like structure, and similarly examines (with less success) the dominant singular vectors of from the singular value decomposition.

Some Recent Developments IN The Explanation of The Empirical Relationship Between Prices and Distribution

Abstract The paper complements recent contributions towards the explanation of the regularities in the behaviour of prices and capital intensities as an effect of hypothetical changes in the rate of profits in empirical production-price models. It is shown that theoretical price and capital curves, i.e. prices and capital values as a function of the rate of profits, depend on the product of the eigenvalues and what we call the eigenlabours —the representation of the labour vector in the space spanned by the eigenvectors of the input matrix. We report robust evidence that the eigenvalues by themselves cannot produce the curves regularly reported in the literature, but rather it is the joint action of the eigenvalues and the eigenlabours. It is conjectured that the tendency towards zero of the subdominant eigenlabours is driven by the statistical tendency towards proportionality between the labour vector and the Perron–Frobenius eigenvector of the input matrix.

A multiplicative ergodic theoretic characterization of relative equilibrium states

AbstractIn this article, we continue the structural study of factor maps between symbolic dynamical systems and the relative thermodynamic formalism. Here, one is studying a factor map from a shift of finite type X (equipped with a potential function) to a sofic shift Z, equipped with a shift-invariant measure $\nu $ . We study relative equilibrium states, that is, shift-invariant measures on X that push forward under the factor map to $\nu $ which maximize the relative pressure: the relative entropy plus the integral of $\phi $ . In this paper, we establish a new connection to multiplicative ergodic theory by relating these factor triples to a cocycle of Ruelle–Perron–Frobenius operators, and showing that the principal Lyapunov exponent of this cocycle is the relative pressure; and the dimension of the leading Oseledets space is equal to the number of measures of relative maximal entropy, counted with a previously identified concept of multiplicity.

Perron–Frobenius theory for some classes of nonnegative tensors in the max algebra

An analog of Perron–Frobenius theory is proposed for some classes of nonnegative tensors in the max algebra. In the first part some important characterizations of nonnegative matrices can be extended to nonnegative tensors over max algebra, especially the Perron–Frobenius theorem for weakly irreducible nonnegative tensors and the Collatz–Wielandt minimax theorem for nonnegative tensors. Then, in the second part, an iterative method is proposed for finding the largest max eigenvalue of a nonnegative tensor based on diagonal similar tensors. The iterative method is convergent for weakly irreducible nonnegative tensors. Some numerical results are provided to illustrate the efficiency of the iterative method.

Two asymptotic distributions related to Rényi-type continued fraction expansions

We attempt to investigate a two-dimensional Gauss–Kuzmin theorem for Rényi-type continued fraction expansions. More precisely, our focus is to obtain specific lower and upper bounds for the error term considered which imply the convergence rate of the distribution function involved to its limit. To achieve our goal, we exploit the significant properties of the Perron–Frobenius operator of the Rényi-type map under its invariant measure on the Banach space of functions of bounded variation. Finally, we give some numerical calculations to conclude the paper.

A probabilistic proof of the Perron–Frobenius theorem

AbstractThe Perron–Frobenius theorem plays an important role in many areas of management science and operations research. This article provides a probabilistic perspective on the theorem, by discussing a proof that exploits a probabilistic representation of the Perron–Frobenius eigenvalue and eigenvectors in terms of the dynamics of a Markov chain. The proof recovers conditions in both the finite‐dimensional and infinite‐dimensional settings under which the Perron–Frobenius eigenvalue and eigenvectors have been shown to exist by other methods. In addition to providing new insights, the probabilistic representations that arise can be used to produce a Monte–Carlo algorithm for computing the Perron–Frobenius eigenvalue and eigenvectors that will be explored elsewhere.

Hidden positivity and a new approach to numerical computation of Hausdorff dimension: higher order methods

In 2018, the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend this approach to incorporate high order approximation methods. We again rely on the fact that we can associate to the IFS a parametrized family of positive, linear, Perron–Frobenius operators L_s , an idea known in varying degrees of generality for many years. Although L_s is not compact in the setting we consider, it possesses a strictly positive C^m eigenfunction v_s with eigenvalue R(L_s) for arbitrary m and all other points z in the spectrum of L_s satisfy |z| \le b for some constant b < R(L_s) . Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value s=s_* for which R(L_s) =1 . This eigenvalue problem is then approximated by a collocation method at the extended Chebyshev points of each subinterval using continuous piecewise polynomials of arbitrary degree r . Using an extension of the Perron theory of positive matrices to matrices that map a cone K to its interior and explicit a priori bounds on the derivatives of the strictly positive eigenfunction v_s , we give rigorous upper and lower bounds for the Hausdorff dimension s_* , and these bounds converge rapidly to s_* as the mesh size decreases and/or the polynomial degree increases.

Spectrum of weighted isometries: C*-algebras, transfer operators and topological pressure

We study the spectrum of operators $$aT \in {\cal B}(H)$$ on a Hilbert space H where T is an isometry and a belongs to a commutative C*-subalgebra $$C(X) \cong A \subseteq {\cal B}(H)$$ such that the formula L(a) = T*aT defines a faithful transfer operator on A. Based on the analysis of the C*-algebra C* (A, T) generated by the operators aT, a ∈ A, we give dynamical conditions implying that the spectrum σ(aT) is invariant under rotation around zero, σ(aT) coincides with the essential spectrum σess (aT) or that σ(aT) is the disc {z ∈ ℂ: ∣z∣ ≤ r(aT)}. We get the best results when the underlying mapping φ: X → X is expanding and open. We prove for any such map and a continuous map c: X → [0, ∞) that the spectral logarithm of a Ruelle—Perron—Frobenius operator $${{\cal L}_c}f(y) = \sum\nolimits_{x \in {\varphi ^{ - 1}}(y)} {c(x)f(x)} $$ is equal to the topological pressure P(ln c, φ). This extends Ruelle’s classical result and implies the variational principle for the spectral radius: $$r(aT) = \mathop {\max }\limits_{\mu \in {\rm{Erg}}(X,\varphi )} {\rm{exp}}\left( {\int_X {\ln (\left| a \right|\sqrt \varrho )} \,d\mu + {{{h_\varphi }(\mu )} \over 2}} \right),$$ where Erg(X, φ) is the set of ergodic Borel probability measures, hφ(μ) is the Kolmogorov—Sinai entropy, and ϱ: X → [0, 1] is the cocycle associated to L. In particular, we clarify the relationship between the Kolmogorov—Sinai entropy and t-entropy introduced by Antonevich, Bakhtin and Lebedev.

Eigenfunctions of the Perron–Frobenius operators for generalized beta-maps

For every generalized β-map τ introduced by Góra [P. Góra, Invariant densities for generalized β-maps, Ergod. Theory Dyn. Syst. 27 (2007), pp. 1583–1598], we find an explicit formula for a basis of the (generalized) eigenspace corresponding to an isolated eigenvalue of its Perron–Frobenius operator on the space of functions of bounded variation. From this formula, we see that any (generalized) eigenfunction is a singular function related to the orbit at 1 by the map τ. In addition, as a consecutive work of the paper [S. Suzuki, Artin-Mazur zeta functions of generalized β-transformations, Kyushu J. Math. 71 (2017), pp. 85–103], the analytic continuation of its lap-counting function is given by the generating function for the coefficient sequence of the τ-expansion of 1.

Invariant probabilities for discrete time linear dynamics via thermodynamic formalism * *AOL partially supported by CNPq. AM partially supported by CNPq, project 311018/2018-1, and FAPESP, project 2013/24541-0. MS partially supported by CNPq, project 312632/2018-5. VV supported by INCTMat.

We show the existence of invariant ergodic σ-additive probability measures with full support on X for a class of linear operators L : X → X, where L is a weighted shift operator and X either is the Banach space or for 1 ⩽ p < ∞. In order to do so, we adapt ideas from thermodynamic formalism as follows. For a given bounded Hölder continuous potential , we define a transfer operator which acts on continuous functions on X and prove that this operator satisfies a Ruelle–Perron–Frobenius theorem. That is, we show the existence of an eigenfunction for which provides us with a normalised potential and an action of the dual operator on the one-Wasserstein space of probabilities on X with a unique fixed point, to which we refer to as Gibbs probability. It is worth noting that the definition of requires an a priori probability on the kernel of L. These results are extended to a wide class of operators with a non-trivial kernel defined on separable Banach spaces.

Existence of Gibbs States and Maximizing Measures on a General One-Dimensional Lattice System with Markovian Structure

Consider a compact metric space $(M, d_M)$ and $X = M^{\mathbb{N}}$. We prove a Ruelle's Perron Frobenius Theorem for a class of compact subshifts with Markovian structure introduced in [Bull. Braz. Math. Soc. 45 (2014), pp. 53-72] which are defined from a continuous function $A : M \times M \to \mathbb{R}$ that determines the set of admissible sequences. In particular, this class of subshifts includes the finite Markov shifts and models where the alphabet is given by the unit circle $S^1$. Using the involution Kernel, we characterize the normalized eigenfunction of the Ruelle operator associated to its maximal eigenvalue and present an extension of its corresponding Gibbs state to the bilateral approach. From these results, we prove existence of equilibrium states and accumulation points at zero temperature in a particular class of countable Markov shifts.

Pandemic spread in communities via random graphs

Working in the multi-type Galton–Watson branching-process framework we analyse the spread of a pandemic via a general multi-type random contact graph. Our model consists of several communities, and takes, as input, parameters that outline the contacts between individuals in distinct communities. Given these parameters, we determine whether there will be an outbreak and if yes, we calculate the size of the giant-connected-component of the graph, thereby, determining the fraction of the population of each type that would be infected before it ends. We show that the pandemic spread has a natural evolution direction given by the Perron–Frobenius eigenvector of a matrix whose entries encode the average number of individuals of one type expected to be infected by an individual of another type. The corresponding eigenvalue is the basic reproduction number of the pandemic. We perform numerical simulations that compare homogeneous and heterogeneous spread graphs and quantify the difference between them. We elaborate on the difference between herd immunity and the end of the pandemic and the effect of countermeasures on the fraction of infected population.

Cocycles on groupoids arising from -actions

AbstractWe consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space and study generalized Ruelle operators and $ C^{\ast } $ -algebras associated to these groupoids. We provide a new characterization of $ 1 $ -cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $ -tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle–Perron–Frobenius theory for dynamical systems of several commuting operators ( $ k $ -Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast } $ -algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence and uniqueness results for KMS states associated to the graphs.

Contaminant Source Identification from Finite Sensor Data: Perron–Frobenius Operator and Bayesian Inference

Sensors in the built environment ensure safety and comfort by tracking contaminants in the occupied space. In the event of contaminant release, it is important to use the limited sensor data to rapidly and accurately identify the release location of the contaminant. Identification of the release location will enable subsequent remediation as well as evacuation decision-making. In previous work, we used an operator theoretic approach—based on the Perron–Frobenius (PF) operator—to estimate the contaminant concentration distribution in the domain given a finite amount of streaming sensor data. In the current work, the approach is extended to identify the most probable contaminant release location. The release location identification is framed as a Bayesian inference problem. The Bayesian inference approach requires considering multiple release location scenarios, which is done efficiently using the discrete PF operator. The discrete PF operator provides a fast, effective and accurate model for contaminant transport modeling. The utility of our PF-based Bayesian inference methodology is illustrated using single-point release scenarios in both two and three-dimensional cases. The method provides a fast, accurate, and efficient framework for real-time identification of contaminant source location.

Minimum Length Scheduling for Multi-Cell Full Duplex Wireless Powered Communication Networks.

Wireless powered communication networks (WPCNs) will be a major enabler of massive machine type communications (MTCs), which is a major service domain for 5G and beyond systems. These MTC networks will be deployed by using low-power transceivers and a very limited set of transmission configurations. We investigate a novel minimum length scheduling problem for multi-cell full-duplex wireless powered communication networks to determine the optimal power control and scheduling for constant rate transmission model. The formulated optimization problem is combinatorial in nature and, thus, difficult to solve for the global optimum. As a solution strategy, first, we decompose the problem into the power control problem (PCP) and scheduling problem. For the PCP, we propose the optimal polynomial time algorithm based on the evaluation of Perron–Frobenius conditions. For the scheduling problem, we propose a heuristic algorithm that aims to maximize the number of concurrently transmitting users by maximizing the allowable interference on each user without violating the signal-to-noise-ratio (SNR) requirements. Through extensive simulations, we demonstrate a 50% reduction in the schedule length by using the proposed algorithm in comparison to unscheduled concurrent transmissions.

A Convex Data-Driven Approach for Nonlinear Control Synthesis

We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples.

Perron’s Theorem in an Hour

We give a short and elementary proof of the Perron–Frobenius theorem. It uses Gelfand’s formula for the spectral radius of a matrix, which is also given an elementary proof.

Perron and Frobenius Meet Carathéodory

We present a new approach of proving certain Carathéodory-type theorems using the Perron–Frobenius Theorem, a classical result in matrix theory describing the largest eigenvalue of a matrix with positive entries.&#x0D; At the end, we list some results and conjectures that we hope can be approached with this method.