arXiv subjects

Fanni M. Sélley

Publications and source records attributed to Fanni M. Sélley.

9 recordsLinked to original sources

arXiv2024

A finiteness condition for complex continued fraction algorithms

It is desirable that a given continued fraction algorithm is simple in the sense that the possible representations can be characterized in an easy way. In this context the so-called finite range condition plays a prominent role. We show that this condition holds for complex $\boldsymbolα$-Hurwitz algorithms with parameters $\boldsymbolα\in\mathbb{Q}^2$. This is equivalent to the existence of certain finite partitions related to these algorithms and lies at the root of explorations into their Diophantine properties. Our result provides a partial answer to a recent question formulated by Lukyanenko and Vandehey.

math.NT↗

arXiv2022

Globally coupled Anosov diffeomorphisms: Statistical properties

We study infinite systems of globally coupled Anosov diffeomorphisms with weak coupling strength. Using transfer operators acting on anisotropic Banach spaces, we prove that the coupled system admits a unique physical invariant state, $h_\varepsilon$. Moreover, we prove exponential convergence to equilibrium for a suitable class of distributions and show that the map $\varepsilon\mapsto h_\varepsilon$ is Lipschitz continuous.

math.DS↗

arXiv2022

Differentiability of the diffusion coefficient for a family of intermittent maps

It is well known that the Liverani-Saussol-Vaienti map satisfies a central limit theorem for Hölder observables in the parameter regime where the correlations are summable. We show that when $C^2$ observables are considered, the variance of the limiting normal distribution is a $C^1$ function of the parameter. We first show this for the first return map to the base of the second branch by studying the Green-Kubo formula, then conclude the result for the original map using Kac's lemma and relying on linear response.

math.DS↗

arXiv2022

Map lattices coupled by collisions: hitting time statistics and collisions per lattice unit

We study map lattices coupled by collision and show how perturbations of transfer operators associated with the spatially periodic approximation of the model can be used to extract information about collisions per lattice unit. More precisely, we study a map on a finite box of $L$ sites with periodic boundary conditions, coupled by collision. We derive, via a non-trivial first order approximation for the leading eigenvalue of the rare event transfer operator, a formula for the first collision rate and a corresponding first hitting time law. For the former we show that the formula scales at the order of $L\cdot\varepsilon^2$, where $\varepsilon$ is the coupling strength, and for the latter, by tracking the $L$ dependency in our arguments, we show that the error in the law is of order $O\left(C(L)\frac{L\varepsilon^2}{ζ(L)}\cdot\left|\ln \frac{L\varepsilon^2}{ζ(L)}\right|\right)$, where $ζ(L)$ is given in terms of the spectral gap of the rare event transfer operator, and $C(L)$ has an explicit expression. Finally, we derive an explicit formula for the first collision rate per lattice unit.

math.DS↗

arXiv2021

Synchronization for Networks of Globally Coupled Maps in the Thermodynamic Limit

We study a network of finitely many interacting clusters where each cluster is a collection of globally coupled circle maps in the thermodynamic (or mean field) limit. The state of each cluster is described by a probability measure, and its evolution is given by a self-consistent transfer operator. A cluster is synchronized if its state is a Dirac measure. We provide sufficient conditions for all clusters to synchronize and we describe setups where the conditions are met thanks to the uncoupled dynamics and/or the (diffusive) nature of the coupling. We also give sufficient conditions for partially synchronized states to arise -- i.e. states where only a subset of the clusters is synchronized -- due to the forcing of a group of cluster on the rest of the network. Lastly, we use this framework to show emergence and stability of chimera states for these systems.

math.DS↗

arXiv2021

Linear Response for a Family of Self-Consistent Transfer Operators

We study a system of all-to-all weakly coupled uniformly expanding circle maps in the thermodynamic limit. The state of the system is described by a probability measure and its evolution is given by the action of a nonlinear operator, also called a self-consistent transfer operator. We prove that when the coupling is sufficiently small, the system has a unique stable state that satisfies a linear response formula when varying the coupling strength.

math.DS↗

arXiv2020

A self-consistent dynamical system with multiple absolutely continuous invariant measures

In this paper we study a class of \emph{self-consistent dynamical systems}, self-consistent in the sense that the discrete time dynamics is different in each step depending on current statistics. The general framework admits popular examples such as coupled map systems. Motivated by an example of M. Blank, we concentrate on a special case where the dynamics in each step is a $β$-map with some $β\geq 2$. Included in the definition of $β$ is a parameter $\varepsilon > 0$ controlling the strength of self-consistency. We show such a self-consistent system which has a unique absolutely continuous invariant measure (acim) for $\varepsilon=0$, but at least two for any $\varepsilon > 0$. With a slight modification, we transform this system into one which produces a phase transition-like behavior: it has a unique acim for $0< \varepsilon < \varepsilon^*$, and multiple for sufficiently large values of $\varepsilon$. We discuss the stability of the invariant measures by the help of computer simulations employing the numerical representation of the self-consistent transfer operator.

math.DS↗

arXiv2018

Synchronization versus stability of the invariant distribution for a class of globally coupled maps

We study a class of globally coupled maps in the continuum limit, where the individual maps are expanding maps of the circle. The circle maps in question are such that the uncoupled system admits a unique absolutely continuous invariant measure (acim), which is furthermore mixing. Interaction arises in the form of diffusive coupling, which involves a function that is discontinuous on the circle. We show that for sufficiently small coupling strength the coupled map system admits a unique absolutely continuous invariant distribution, which depends on the coupling strength $\varepsilon$. Furthermore, the invariant density exponentially attracts all initial distributions considered in our framework. We also show that the dependence of the invariant density on the coupling strength $\varepsilon$ is Lipschitz continuous in the BV norm. When the coupling is sufficiently strong, the limit behavior of the system is more complex. We prove that a wide class of initial measures approach a point mass with support moving chaotically on the circle. This can be interpreted as synchronization in a chaotic state.

math.DS↗

arXiv2018

Symmetry breaking in a globally coupled map of four sites

A system of four globally coupled doubling maps is studied in this paper. It is known that such systems have a unique absolutely continuous invariant measure (acim) for weak interaction, but the case of stronger coupling is still unexplored. As in the case of three coupled sites, we prove the existence of a critical value of the coupling parameter at which multiple acims appear. Our proof has several new ingredients in comparison to the one presented in our previous paper regarding the system of three sites. We strongly exploit the symmetries of the dynamics in the course of the argument. This simplifies the computations considerably, and gives us a precise description of the geometry and symmetry properties of the arising asymmetric invariant sets. Some new phenomena are observed which are not present in the case of three sites. In particular, the asymmetric invariant sets arise in areas of the phase space which are transient for weaker coupling and a nontrivial symmetric invariant set emerges, shaped by an underlying centrally symmetric Lorenz map. We state some conjectures on further invariant sets, indicating that unlike the case of three sites, ergodicity breaks down in many steps, and not all of them are accompanied by symmetry breaking.

math.DS↗