Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2310.19191

Optimal linear response for expanding circle maps

Gary Froyland·Stefano Galatolo
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the problem of optimal linear response for deterministic expanding maps of the circle. To each infinitesimal perturbation $\dot{T}$ of a circle map $T$ we consider (i) the response of the expectation of an observation function and (ii) the response of isolated spectral points of the transfer operator of $T$. In each case, under mild conditions on the set of feasible perturbations $\dot{T}$ we show there is a unique optimal feasible infinitesimal perturbation $\dot{T}_{\rm optimal}$, maximising the increase of the expectation of the given observation function or maximising the increase of the spectral gap of the transfer operator associated to the system. We derive expressions for the unique maximiser $\dot{T}_{\rm optimal}$ in terms of its Fourier coefficients. We also devise a Fourier-based computational scheme and apply it to illustrate our theory.

Explore related subjects

math.DSnlin.CD

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Gary Froyland, Stefano Galatolo. 2023-10-29. Optimal linear response for expanding circle maps. https://arxiv.org/abs/2310.19191

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Equidistribution of saddle periodic points for Hénon-like maps

We prove that under a natural assumption on the dynamical degrees, the saddle periodic points of a Hénon-like map in any dimension equidistribute with respect to the equilibrium measure. Our work is a generalization of the results of Bedford-Lyubich-Smillie, Dujardin, and Dinh-Sibony along with improvements of their techniques. We also investigate some fine properties of Green currents associated with the map.

math.DS↗
arXiv2026

A flux-based approach for analyzing the disguised toric locus of reaction networks

Dynamical systems with polynomial right-hand sides are very important in various applications, e.g., in biochemistry and population dynamics. The mathematical study of these dynamical systems is challenging due to the possibility of multistability, oscillations, and chaotic dynamics. One important tool for this study is the concept of reaction systems, which are dynamical systems generated by reaction networks for some choices of parameter values. Among these, disguised toric systems are remarkably stable: they have a unique attracting fixed point, and cannot give rise to oscillations or chaotic dynamics. The computation of the set of parameter values for which a network gives rise to disguised toric systems (i.e., the disguised toric locus of the network) is an important but difficult task. We introduce new ideas based on network fluxes for studying the disguised toric locus. We prove, under mild assumptions, that the disguised toric locus of any network $G$ is a contractible manifold with boundary, and introduce an associated graph $G^{\max}$ that characterizes its interior. These theoretical tools allow us, for the first time, to compute the full disguised toric locus for many networks of interest.

math.DS↗
arXiv2026

On dissonance and orthogonal projections of self-conformal measures

Let $μ$ be a self-conformal measure on $\mathbb{R}^d$. We establish conditions for $μ$ under which $\dim(μ*ν) = \min\lbrace d,\dimμ+\dimν\rbrace$ holds when $ν$ is any Ahlfors-regular or self-conformal measure on $\mathbb{R}^d$. Our main result states the following sufficient condition: $μ$ is totally non-linear and not supported on a smooth hypersurface. We also establish sufficient (likely non-sharp) algebraic conditions for self-conformal measures which are not totally non-linear. In addition, we show that $\dim μ\circπ^{-1} = \min\{ k, \dim μ\}$ for every ortohogonal projection $π:\mathbb{R}^d\to\mathbb{R}^k$, $0<k<d$, when either $d=2$ and $μ$ is not self-similar and not supported on a line, or $d\geq 3$ and $μ$ is totally non-linear and not supported on a smooth hypersurface.

math.DS↗