Ergodic Properties of the Fröbenius–Perron Semigroup

C0-continuity of the Fröbenius-perron Semigroup

We consider the Fröbenius-Perron semigroup of linear operators associated to a semidynamical system defined in a topological space X endowed with a finite or a σ -finite regular measure. We prove that if there exists a faithful invariant measure for the semidynamical system, then the Fröbenius-Perron semigroup of linear operators is C0-continuous in the space Lμ(X). We also give a geometrical c...

The infinitestimal operator for the semigroup of the Frobenius-Perron operator from image sequence data: Vector fields and computational measurable dynamics from movies

We present in this paper an approach to approximate the Frobenius-Perron transfer operator from a sequence of time-ordered images, that is, a movie dataset. Unlike time-series data, successive images do not provide a direct access to a trajectory of a point in a phase space; more precisely, a pixel in an image plane. Therefore, we reconstruct the velocity field from image sequences based on the...

Non-linear ergodic theorems in complete non-positive curvature metric spaces

Hadamard (or complete $CAT(0)$) spaces are complete, non-positive curvature, metric spaces. Here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. Our results extend the standard non-linear ergodic theorems for non-expansive maps on real Hilbert spaces, to non-expansive maps on Ha...

Spectrum Properties of the Koopman and Frobenius-Perron Operators

Ergodic properties of contraction semigroups in L p , 1 < p < ∞

Let {T (t) : t > 0} be a strongly continuous semigroup of linear contractions in Lp, 1 < p < ∞, of a σ-finite measure space. In this paper we prove that if there corresponds to each t > 0 a positive linear contraction P (t) in Lp such that |T (t)f | ≤ P (t)|f | for all f ∈ Lp, then there exists a strongly continuous semigroup {S(t) : t > 0} of positive linear contractions in Lp such that |T (t)...