Generic existence of a nonempty compact set of fixed points

AbstractLet X be a complete metric space, M a set of continuous mappings from X into itself, endowed with a metric topology finer than the compact-open topology. Assuming that there exists a dense subset N contained in M such that for every mapping T in N the set {x ϵ X: Tx = x} is nonempty, it is proved that most mappings (in the Baire category sense) in M do have a nonempty compact set of fixed points. Some applications to α-nonexpansive operators, semiaccretive operators and differential equations in Banach spaces are derived

Continuous Curvelet Transform: I. Resolution of the Wavefront Set

We discuss a Continuous Curvelet Transform (CCT), a transform f → Γf (a, b, θ) of functions f(x1, x2) on R^2, into a transform domain with continuous scale a > 0, location b ∈ R^2, and orientation θ ∈ [0, 2π). The transform is defined by Γf (a, b, θ) = {f, γabθ} where the inner products project f onto analyzing elements called curvelets γ_(abθ) which are smooth and of rapid decay away from an a by √a rectangle with minor axis pointing in direction θ. We call them curvelets because this anisotropic behavior allows them to ‘track’ the behavior of singularities along curves. They are continuum scale/space/orientation analogs of the discrete frame of curvelets discussed in Candès and Donoho (2002). We use the CCT to analyze several objects having singularities at points, along lines, and along smooth curves. These examples show that for fixed (x0, θ0), Γf (a, x0, θ0) decays rapidly as a → 0 if f is smooth near x0, or if the singularity of f at x0 is oriented in a different direction than θ_0. Generalizing these examples, we state general theorems showing that decay properties of Γf (a, x0, θ0) for fixed (x0, θ0), as a → 0 can precisely identify the wavefront set and the H^m- wavefront set of a distribution. In effect, the wavefront set of a distribution is the closure of the set of (x0, θ0) near which Γf (a, x, θ) is not of rapid decay as a → 0; the H^m-wavefront set is the closure of those points (x0, θ0) where the ‘directional parabolic square function’ S^m(x, θ) = ( ʃ|Γf (a, x, θ)|^2 ^(da) _a^3+^(2m))^(1/2) is not locally integrable. The CCT is closely related to a continuous transform used by Hart Smith in his study of Fourier Integral Operators. Smith’s transform is based on strict affine parabolic scaling of a single mother wavelet, while for the transform we discuss, the generating wavelet changes (slightly) scale by scale. The CCT can also be compared to the FBI (Fourier-Bros-Iagolnitzer) and Wave Packets (Cordoba-Fefferman) transforms. We describe their similarities and differences in resolving the wavefront set

Transport in reservoir computing

Reservoir computing systems are constructed using a driven dynamical system in which external inputs can alter the evolving states of a system. These paradigms are used in information processing, machine learning, and computation. A fundamental question that needs to be addressed in this framework is the statistical relationship between the input and the system states. This paper provides conditions that guarantee the existence and uniqueness of asymptotically invariant measures for driven systems and shows that their dependence on the input process is continuous when the set of input and output processes are endowed with the Wasserstein distance. The main tool in these developments is the characterization of those invariant measures as fixed points of naturally defined Foias operators that appear in this context and which have been profusely studied in the paper. Those fixed points are obtained by imposing a newly introduced stochastic state contractivity on the driven system that is readily verifiable in examples. Stochastic state contractivity can be satisfied by systems that are not state-contractive, which is a need typically evoked to guarantee the echo state property in reservoir computing. As a result, it may actually be satisfied even if the echo state property is not present.Comment: 33 pages, 5 figure

Random fixed point theorems under mild continuity assumptions

In this paper, we study the existence of the random fixed points under mild continuity assumptions. The main theorems consider the almost lower semicontinuous operators defined on Frechet spaces and also operators having properties weaker than lower semicontinuity. Our results either extend or improve corresponding ones present in literature.Comment: 15 page