arXiv subjects

Vincent P. H. Goverse

Publications and source records attributed to Vincent P. H. Goverse.

5 recordsLinked to original sources

arXiv2026

Atomic physical measures for non-invertible random dynamical systems

We construct an example of a random dynamical system on the circle, formed by maps that are only locally invertible, which possesses an atomic stationary measure $ν$. Moreover, this measure is physical: for Lebesgue-almost every initial point $x_0$, the Cesàro averages of its random trajectory almost surely converge to $ν$. This shows that the Hölder regularity of stationary measures, known for (non-measure-preserving) random dynamical systems formed by diffeomorphisms, cannot be generalized to this class of systems. We also provide some related examples, including ones where a stationary measure charges a proper submanifold, despite the absence of a closed common invariant submanifold.

math.DS↗

arXiv2026

Intermittent two-point dynamics at the transition to chaos for random circle endomorphisms

We establish the existence of intermittent two-point dynamics and infinite stationary measures for a class of random circle endomorphisms with zero Lyapunov exponent, as a dynamical characterisation of the transition from synchronisation (negative Lyapunov exponent) to chaos (positive Lyapunov exponent).

math.DS↗

arXiv2025

Analytic Dependence of the Lyapunov Moment Function and the Projective Stationary Measure for Random Matrix Products

We consider the product of i.i.d. random matrices sampled according to a probability measure $μ$ supported on a strongly irreducible and proximal subset of a compact set $S\subset GL(d,\mathbb{R})$. We establish the local analyticity of the Lyapunov moment function and the unique stationary measure on the projective space with respect to $μ$ in the total variation topology. As a consequence, we obtain the analyticity of the asymptotic variance and all higher-order Lyapunov moments.

math.DS↗

arXiv2024

On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape

In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $μ$ on $M$ where the transition kernel $\mathcal P$ admits an eigenfunction $0\leq η\in L^1(M,μ)$. We find conditions on the transition densities of $\mathcal P$ with respect to $μ$ which ensure that $η(x) μ(\mathrm d x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $μ$-almost surely. We apply this result to the random logistic map $X_{n+1} = ω_n X_n (1-X_n)$ absorbed at $\mathbb R \setminus [0,1],$ where $ω_n$ is an i.i.d sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a <4$ and $b>4.$

math.PR↗

arXiv2023

Optimal Approximation Complexity of High-Dimensional Functions with Neural Networks

We investigate properties of neural networks that use both ReLU and $x^2$ as activation functions and build upon previous results to show that both analytic functions and functions in Sobolev spaces can be approximated by such networks of constant depth to arbitrary accuracy, demonstrating optimal order approximation rates across all nonlinear approximators, including standard ReLU networks. We then show how to leverage low local dimensionality in some contexts to overcome the curse of dimensionality, obtaining approximation rates that are optimal for unknown lower-dimensional subspaces.

cs.LG↗