Frobenius Operator Research Articles

Equilibrium states beyond specification and the Bowen property

It is well known that for expansive maps and continuous potential functions, the specification property (for the map) and the Bowen property (for the potential) together imply the existence of a unique equilibrium state. We consider symbolic spaces that may not have specification and potentials that may not have the Bowen property, and give conditions under which uniqueness of the equilibrium state can still be deduced. Our approach is to ask that the collection of cylinders which are obstructions to the specification property or the Bowen property is small in an appropriate quantitative sense. This allows us to construct an ergodic equilibrium state with a weak Gibbs property, which we then use to prove uniqueness. We do not use inducing schemes or the Perron–Frobenius operator, and we strengthen some previous results obtained using these approaches. In particular, we consider β-shifts and show that the class of potential functions with unique equilibrium states strictly contains the set of potentials with the Bowen property. We give applications to piecewise monotonic interval maps, including the family of geometric potentials, for example, which have both indifferent fixed points and a non-Markov structure.

A set oriented definition of finite-time Lyapunov exponents and coherent sets

The problem of phase space transport, which is of interest from both the theoretical and practical point of view, has been investigated extensively using geometric and probabilistic methods. Two important tools to study this problem that have emerged in recent years are finite-time Lyapunov exponents (FTLE) and the Perron–Frobenius operator. The FTLE measures the averaged local stretching around reference trajectories. Regions with high stretching are used to identify phase space transport barriers. One probabilistic method is to consider the spectrum of the Perron–Frobenius operator of the flow to identify almost-invariant densities. These almost-invariant densities are used to identify almost invariant sets. In this paper, a set-oriented definition of the FTLE is proposed which is applicable to phase space sets of finite size and reduces to the usual definition of FTLE in the limit of infinitesimal phase space elements. This definition offers a straightforward connection between the evolution of probability densities and finite-time stretching experienced by phase space curves. This definition also addresses some concerns with the standard computation of the FTLE. For the case of autonomous and periodic vector fields we provide a simplified method to calculate the set-oriented FTLE using the Perron–Frobenius operator. Based on the new definition of the FTLE we propose a simple definition of finite-time coherent sets applicable to vector fields of general time-dependence, which are the analogues of almost-invariant sets in autonomous and time-periodic vector fields. The coherent sets we identify will necessarily be separated from one another by ridges of high FTLE, providing a link between the framework of coherent sets and that of codimension one Lagrangian coherent structures. Our identification of coherent sets is applied to three examples.

Equilibrium states for non-uniformly expanding maps: Decay of correlations and strong stability

We study the rate of decay of correlations for equilibrium states associated to a robust class of non-uniformly expanding maps where no Markov assumption is required. We show that the Ruelle–Perron–Frobenius operator acting on the space of Hölder continuous observables has a spectral gap and deduce the exponential decay of correlations and the central limit theorem. In particular, we obtain an alternative proof for the existence and uniqueness of the equilibrium states and we prove that the topological pressure varies continuously. Finally, we use the spectral properties of the transfer operators in space of differentiable observables to obtain strong stability results under deterministic and random perturbations.

Efficient optimal design of uncertain discrete time dynamical systems

In this paper we consider the problem of optimal design of an uncertain discrete time nonlinear dynamical system. The problem is formulated using an a-posterori design criterion, which can account for uncertainties generated by the dynamics of the system itself as well as parametric uncertainties. In general, for most uncertain complex dynamical systems, this type of method is difficult to solve analytically. A numerical scheme is developed for the optimal design that involves two steps. First, in order to obtain a numerical algorithm for the optimal solution, we apply randomized algorithms for average performance synthesis to approximate the optimal solution. Second, using the properties of the Perron–Frobenius operator we develop an efficient computation approach to calculate the stationary distribution for the uncertain dynamical systems and the average performance criteria.

Graph C⁎-algebras, branching systems and the Perron–Frobenius operator

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 L2(X,μ). These representations are very concrete and, in the case of graphs that satisfy condition (L), we use our techniques to realize the associated graph C⁎-algebra as a subalgebra of the bounded operators in L2(R). We also show how to describe the Perron–Frobenius operator of ergodic theory in terms of the representations we associate to a graph.

Dynamical system generated by algebraic method and low discrepancy sequences

For higher dimensional cases, the essential spectrum radius of the Perron–Frobenius operator is usually greater than (the definition is given below) even if we restrict its domain to a suitable space. In this article, using algebraic method, we construct a transformation whose essential spectrum radius equals . By this transformation, we can construct low discrepancy sequences.

Spectral degeneracy and escape dynamics for intermittent maps with a hole

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron–Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds and demonstrate L1 convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.

Invariant densities and escape rates: Rigorous and computable approximations in the [formula omitted]-norm

In this article, we study piecewise linear discretization schemes for transfer operators (Perron–Frobenius operators) associated with interval maps. We show how these can be used to provide rigorous pointwise approximations for invariant densities of Markov interval maps. We also derive the order of convergence of the approximate invariant density to the real one in the L ∞ -norm. The outcome of this paper complements recent results on the formulae of escape rates of open dynamical systems, (Keller and Liverani, 2009) [7]. In particular, the novelty of our work over previous results on B V and L ∞ approximations is that it provides a method for explicit computation of the approximation error.

INVARIANT MEASURES OF STOCHASTIC PERTURBATIONS OF DYNAMICAL SYSTEMS USING FOURIER APPROXIMATIONS

We consider dynamical system τ : [0, 1] → [0, 1] and its stochastic perturbations [Formula: see text], N ≥ 1. Using Fourier approximation, we construct a finite dimensional approximation PN to a perturbed Perron–Frobenius operator. Let [Formula: see text] be an invariant density of τ and [Formula: see text] be a fixed point of PN. We show that [Formula: see text] converges in L1 to [Formula: see text].

Higher topological cyclic homology and the Segal conjecture for tori

We investigate higher topological cyclic homology as an approach to studying chromatic phenomena in homotopy theory. Higher topological cyclic homology is constructed from the fixed points of a version of topological Hochschild homology based on the n-dimensional torus, and we propose it as a computationally tractable cousin of n-fold iterated algebraic K-theory. The fixed points of toral topological Hochschild homology are related to one another by restriction and Frobenius operators. We introduce two additional families of operators on fixed points, the Verschiebung, indexed on self-isogenies of the n-torus, and the differentials, indexed on n-vectors. We give a detailed analysis of the relations among the restriction, Frobenius, Verschiebung, and differentials, producing a higher analog of the structure Hesselholt and Madsen described for 1-dimensional topological cyclic homology. We calculate two important pieces of higher topological cyclic homology, namely topological restriction homology and topological Frobenius homology, for the sphere spectrum. The latter computation allows us to establish the Segal conjecture for the torus, which is to say to completely compute the cohomotopy type of the classifying space of the torus.

Ulam method and fractal Weyl law for Perron--Frobenius operators

Ulam method and fractal Weyl law for Perron--Frobenius operators

Cuntz–Krieger Algebras and Wavelets on Fractals

We consider representations of Cuntz–Krieger algebras on the Hilbert space of square integrable functions on the limit set, identified with a Cantor set in the unit interval. We use these representations and the associated Perron–Frobenius and Ruelle operators to construct families of wavelets on these Cantor sets.

Frobenius modules and Galois representations

Frobenius modules are difference modules with respect to a Frobenius operator. Here we show that over non-archimedean complete differential fields Frobenius modules define differential modules with the same Picard-Vessiot ring and the same Galois group schemes up to extension by constants. Moreover, these Frobenius modules are classified by unramified Galois representations over the base field. This leads among others to the solution of the inverse differential Galois problem for p-adic differential equations with (strong) Frobenius structure over p-adic differential fields with algebraically closed residue field.

A local criterion for the Saito-Kurokawa lifting of cuspforms with characters

Let $f$ be a holomorphic degree-2 Siegel cuspform of weight $\kappa$, level $N$, and nebentype a primitive Dirichlet character $\chi$. Let $f$ be an eigenfunction of the regular Hecke operators $T_p,T_{p^2}$ at primes $p\nmid N$ and an eigenfunction of the Frobenius operators $\Pi_p$ and their duals $\Pi^*_p$ at primes $p|N$. For certain $\chi$, we give conditions on the Satake parameters of $f$ which imply that $f$ is lifted from an elliptic cuspform $\phi$ of weight $2\kappa-2$, level $N$, and nebentype $\chi^2$. We also show that for such $f$ and $\phi$, the eigenvalues of the Frobenius operators on $f$ are eigenvalues of Hecke operators on $\phi$.

Perron–Frobenius operators and representations of the Cuntz–Krieger algebras for infinite matrices

In this paper we extend the work of Kawamura, see [K. Kawamura, The Perron–Frobenius operators, invariant measures and representations of the Cuntz–Krieger algebras, J. Math. Phys. 46 (2005)], 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 a 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.

Rigorous numerical approximation of Ruelle–Perron–Frobenius operators and topological pressure of expanding maps

It is well known that for different classes of transformations, including the class of piecewise C2 expanding maps T : [0, 1] ↺, Ulam's method is an efficient way to numerically approximate the absolutely continuous invariant measure of T. We develop a new extension of Ulam's method and prove that this extension can be used for the numerical approximation of the Ruelle–Perron–Frobenius operator associated with T and the potential ϕβ = −β log |T∣|, where . In particular, we prove that our extended Ulam's method is a powerful tool for computing the topological pressure P(T, ϕβ) and the density of the equilibrium state.

Continuity properties of transport coefficients in simple maps

We consider families of dynamics that can be described in terms of Perron–Frobenius operators with exponential mixing properties. For piecewise C2 expanding interval maps we rigorously prove continuity properties of the drift J(λ) and of the diffusion coefficient D(λ) under parameter variation. Our main result is that D(λ) has a modulus of continuity of order , i.e. D(λ) is Lipschitz continuous up to quadratic logarithmic corrections. For a special class of piecewise linear maps we provide more precise estimates at specific parameter values. Our analytical findings are quantified numerically for the latter class of maps by using exact series expansions for the transport coefficients that can be evaluated numerically. We numerically observe strong local variations of all continuity properties.

Decay of correlations in suspension semi-flows of angle-multiplying maps

AbstractWe consider suspension semi-flows of angle-multiplying maps on the circle for Cr ceiling functions with r≥3. Under a Crgeneric condition on the ceiling function, we show that there exists a Hilbert space (anisotropic Sobolev space) contained in the L2 space such that the Perron–Frobenius operator for the time-t-map acts naturally on it and that the essential spectral radius of that action is bounded by the square root of the inverse of the minimum expansion rate. This leads to a precise description of decay of correlations. Furthermore, the Perron–Frobenius operator for the time-t-map is quasi-compact for a Cr open and dense set of ceiling functions.

Spectra of Perron–Frobenius operator and new construction of two dimensional low discrepancy sequences

For high dimensional transformations, the essential spectral radius of the Perron–Frobenius operator restricted to some natural space is generally smaller than the reciprocals of the radius of convergence of the dynamical zeta function. We will construct transformations whose essential spectral radius coincide with the reciprocal of the radius of convergence of the dynamical zeta functions, and construct new type of 2 dimensional low discrepancy sequences.

Unwrapping eigenfunctions to discover the geometry of almost-invariant sets in hyperbolic maps

The numerical approximation of Perron–Frobenius operators allows efficient determination of the physical invariant measure of chaotic dynamical systems as a fixed point of the operator. Eigenfunctions of the Perron–Frobenius operator corresponding to large subunit eigenvalues have been shown to describe “almost-invariant” dynamics in one-dimensional expanding maps. We extend these ideas to hyperbolic maps in higher dimensions. While the eigendistributions of the operator are relatively uninformative, applying a new procedure called “unwrapping” to regularised versions of the eigendistributions clearly reveals the geometric structures associated with almost-invariant dynamics. This unwrapping procedure is applied to a uniformly hyperbolic map of the unit square to discover this map’s dominant underlying dynamical structure, and to the standard map to pinpoint clusters of period 6 orbits.