arXiv subjects

Cecilia González-Tokman

Publications and source records attributed to Cecilia González-Tokman.

At least 19 recordsLinked to original sources

arXiv2026

Jumping for diffusion in random metastable systems

Random metastability occurs when an externally forced or noisy system possesses more than one state of apparent equilibrium. This work investigates fluctuations in a class of random dynamical systems, arising from randomly perturbing a piecewise smooth expanding interval map with more than one invariant subinterval. Upon perturbation, this invariance is destroyed, allowing trajectories to switch between subintervals, giving rise to metastable behaviour. We show that the distributions of jumps of a time-homogeneous Markov chain approximate the distributions of jumps for random metastable systems. Additionally, we demonstrate that this approximation extends to the diffusion coefficient for (random) observables of such systems. As an example, our results are applied to Horan's random paired tent maps.

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arXiv2026

The transition between synchronization and chaos for random Blaschke products

In this paper, we establish sufficient conditions for which a class of random circle maps, the so-called random Blaschke products, exhibits either synchronization or chaotic dynamics. We also establish necessary and sufficient conditions for such systems to have a unique random fixed point attractor in the unit disc. As an example, for a class of two map cocycles, we identify the random physical measures as a parameter is varied and investigate the critical value where the dynamics transitions from order to chaos.

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arXiv2026

Universal Gap Growth for Lyapunov Exponents of Perturbed Matrix Products

We study the quantitative simplicity of the Lyapunov spectrum of $d$-dimensional bounded matrix cocycles subjected to additive random perturbations. In dimensions 2 and 3, we establish explicit lower bounds on the gaps between consecutive Lyapunov exponents of the perturbed cocycle, depending only on the scale of the perturbation. In arbitrary dimensions, we show existence of a universal lower bound on these gaps. A novelty of this work is that the bounds provided are uniform over all choices of the original sequence of matrices. Furthermore, we make no stationarity assumptions on this sequence. Hence, our results apply to random and sequential dynamical systems alike.

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arXiv2025

Averaging for random metastable systems

Random metastability occurs when an externally forced or noisy system possesses more than one state of apparent equilibrium. This work investigates a class of random dynamical systems, arising from perturbing a one-dimensional piecewise smooth expanding map of the interval with two invariant subintervals, each supporting a unique ergodic absolutely continuous invariant measure. Upon perturbation, this invariance is destroyed, allowing trajectories to randomly switch between subintervals. We show that the invariant density of the randomly perturbed system may be approximated by an explicit convex combination of the two initially invariant densities, obtained by averaging. Further, we also identify the limit of the second Oseledets space, or coherent structure, as the perturbation shrinks to zero. Our results are applied to random paired tent maps over ergodic, measure-preserving, and invertible driving systems. Finally, we provide generalisations to systems admitting more than two initially invariant sets.

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arXiv2025

An Improved Autoencoder Conjugacy Network to Learn Chaotic Maps

We introduce a method for learning chaotic maps using an improved autoencoder neural network that incorporates a conjugacy layer in the latent space. The added conjugacy layer transforms nonlinear maps into a simple piecewise linear map (the tent map) whilst enforcing dynamical principles of well-known and defective conjugacy functions that increase the accuracy and stability of the learned solution. We demonstrate the method's effectiveness on both continuous and piecewise chaotic one-dimensional maps and numerically illustrate improved performance over related traditional and recently emerged deep learning architectures.

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arXiv2025

Average measure theoretic entropy for a family of expanding on average random Blaschke products

This work gives a computable formula for the average measure theoretic entropy of a family of expanding on average random Blaschke products, generalizing work by Pujals, Roberts and Shub [Expanding maps of the circle revisited: positive Lyapunov exponents in a rich family. $\textit{Ergodic Theory Dynam. Systems.}$ $\textbf{26}(6)$ $(2006),$ $1931$-$1937$] to the random setting. In doing so, we describe the random invariant measure and associated measure theoretic entropy for a class of admissible random Blaschke products, allowing for maps which are not necessarily expanding and may even have an attracting fixed point.

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arXiv2024

Asymptotic limits of transient patterns in a continuous-space interacting particle system

We study a discrete-time interacting particle system with continuous state space which is motivated by a mathematical model for turnover through branching in actin filament networks. It gives rise to transient clusters reminiscent of actin filament assemblies in the cortex of living cells. We reformulate the process in terms of the inter-particle distances and characterise their marginal and joint distributions. We construct a recurrence relation for the associated characteristic functions and pass to the large population limit, reminiscent of the Fleming-Viot super-processes. The precise characterisation of all marginal distributions established in this work opens the way to a detailed analysis of cluster dynamics. We also obtain a recurrence relation which enables us to compute the moments of the asymptotic single particle distribution characterising the transient aggregates. Our results indicate that aggregates have a fat-tailed distribution.

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arXiv2024

Prevalence of stability for smooth Blaschke product cocycles fixing the origin

This work investigates the stability properties of Lyapunov exponents of transfer operator cocycles from a measure-theoretic perspective. Our results focus on so-called Blaschke product cocycles, a class of random dynamical systems amenable to rigorous analysis. We show that prevalence of stability is related to the dimension of the base system's domain, $Ω$. When $Ω=S^1$, we show that stability is prevalent among smooth monic quadratic Blaschke product cocycles fixing the origin by constructing a so-called probe. For higher dimensional $Ω$, we show that a probe does not exist, thus providing strong evidence that stability is not prevalent in this setting. Finally, through a perturbative method we show that almost every smooth Blaschke product cocycle fixing the origin is stable.

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arXiv2024

Compound Poisson statistics for dynamical systems via spectral perturbation

We consider random transformations $T_ω^n:=T_{σ^{n-1}ω}\circ\cdots\circ T_{σω}\circ T_ω,$ where each map $T_ω$ acts on a complete metrizable space $M$. The randomness comes from an invertible ergodic driving map $σ:Ω\toΩ$ acting on a probability space $(Ω,\mathcal{F},m).$ For a family of random target sets $H_{ω, n}\subset M$ that shrink as $n\to\infty$, we consider quenched compound Poisson statistics of returns of random orbits to these random targets. We develop a spectral approach to such statistics: associated with the random map cocycle is a transfer operator cocycle $\mathcal{L}^{n}_{ω,0}:=\mathcal{L}_{σ^{n-1}ω,0}\circ\cdots\circ\mathcal{L}_{σω,0}\circ\mathcal{L}_{ω,0}$, where $\mathcal{L}_{ω,0}$ is the transfer operator for the map $T_ω$. We construct a perturbed cocycle with generator $\mathcal{L}_{ω,n,s}(\cdot):=\mathcal{L}_{ω,0}(\cdot e^{is\mathbb{1}_{H_{ω,n}}})$ and an associated random variable $S_{ω,n,k}(x):=\sum_{j=0}^{k-1}\mathbb{1}_{H_{σ^jω,n}}(T_ω^jx)$, which counts the number of visits to random targets in an orbit of length $k$. Under suitable assumptions, we show that in the $n\to\infty$ limit, the random variables $S_{ω,n,n}$ converge in distribution to a compound Poisson distributed random variable. We provide several explicit examples for piecewise monotone interval maps in both the deterministic and random settings.

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arXiv2023

A patch in time saves nine: Methods for the identification of localised dynamical behaviour and lifespans of coherent structures

We develop a transfer operator-based method for the detection of coherent structures and their associated lifespans. Characterising the lifespan of coherent structures allows us to identify dynamically meaningful time windows, which may be associated with transient coherent structures in the localised phase space, as well as with time intervals within which these structures experience fundamental changes, such as merging or separation events. The localised transfer operator approach we pursue allows one to explore the fundamental properties of a dynamical system without full knowledge of the dynamics. The algorithms we develop prove useful not only in the simple case of a periodically driven double well potential model, but also in more complex cases generated using the rotating Boussinesq equations.

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arXiv2022

Equilibrium states for non-transitive random open and closed dynamical systems

We prove a random Ruelle--Perron--Frobenius theorem and the existence of relative equilibrium states for a class of random open and closed interval maps, without imposing transitivity requirements, such as mixing and covering conditions, which are prevalent in the literature. This theorem provides existence and uniqueness of random conformal and invariant measures with exponential decay of correlations, and allows us to expand the class of examples of (random) dynamical systems amenable to multiplicative ergodic theory and the thermodynamic formalism. Applications include open and closed non-transitive random maps, and a connection between Lyapunov exponents and escape rates through random holes. We are also able to treat random intermittent maps with geometric potentials.

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arXiv2021

Invariant measures for random expanding on average Saussol maps

In this paper, we investigate the existence of random absolutely continuous invariant measures (ACIP) for random expanding on average Saussol maps in higher dimensions. This is done by the establishment of a random Lasota-Yorke inequality for the transfer operators on the space of bounded oscillation. We prove that the number of ergodic skew product ACIPs is finite and provide an upper bound for the number of these ergodic ACIPs. This work can be seen as a generalization of the work in BGT on admissible random Jabłoński maps to a more general class of higher dimensional random maps.

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arXiv2021

Thermodynamic Formalism for Random Weighted Covering Systems

We develop a quenched thermodynamic formalism for random dynamical systems generated by countably branched, piecewise-monotone mappings of the interval that satisfy a random covering condition. Given a random contracting potential $φ$ (in the sense of Liverani-Saussol-Vaienti), we prove there exists a unique random conformal measure $ν_φ$ and unique random equilibrium state $μ_φ$. Further, we prove quasi-compactness of the associated transfer operator cocycle and exponential decay of correlations for $μ_φ$. Our random driving is generated by an invertible, ergodic, measure-preserving transformation $σ$ on a probability space $(Ω,\mathscr{F},m)$; for each $ω\inΩ$ we associate a piecewise-monotone, surjective map $T_ω:I\to I$. We consider general potentials $φ_ω:I\to\mathbb R\cup\{-\infty\}$ such that the weight function $g_ω=e^{φ_ω}$ is of bounded variation. We provide several examples of our general theory. In particular, our results apply to linear and non-linear systems including random $β$-transformations, randomly translated random $β$-transformations, random Gauss-Renyi maps, random non-uniformly expanding maps such as intermittent maps and maps with contracting branches, and a large class of random Lasota-Yorke maps.

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arXiv2021

Thermodynamic Formalism for Random Interval Maps with Holes

We develop a quenched thermodynamic formalism for open random dynamical systems generated by finitely branched, piecewise-monotone mappings of the interval. The openness refers to the presence of holes in the interval, which terminate trajectories once they enter; the holes may also be random. Our random driving is generated by an invertible, ergodic, measure-preserving transformation $σ$ on a probability space $(Ω,\mathscr{F},m)$. For each $ω\inΩ$ we associate a piecewise-monotone, surjective map $T_ω:I\to I$, and a hole $H_ω\subset I$; the map $T_ω$, the random potential $φ_ω$, and the hole $H_ω$ generate the corresponding open transfer operator $\mathcal{L}_ω$. For a contracting potential, under a condition on the open random dynamics in the spirit of Liverani--Maume-Deschamps, we prove there exists a unique random probability measure $ν_ω$ supported on the survivor set ${X}_{ω,\infty}$ satisfying $ν_{σ(ω)}(\mathcal{L}_ωf)=λ_ων_ω(f)$. We also prove the existence of a unique random family of functions $q_ω$ that satisfy $\mathcal{L}_ωq_ω=λ_ωq_{σ(ω)}$. These yield an ergodic random invariant measure $μ=νq$ supported on the global survivor set, while $q$ combined with the random closed conformal measure yields a unique random absolutely continuous conditional invariant measure (RACCIM) $η$ supported on $I$. We prove quasi-compactness of the transfer operator cocycle and exponential decay of correlations for $μ$. Finally, the escape rates of the random closed conformal measure and the RACCIM $η$ coincide, and are given in terms of the expected pressure, as is the Hausdorff dimension of the surviving set $X_{ω,\infty}$. We provide examples of our general theory, including random $β$-transformations and random Lasota-Yorke maps.

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arXiv2021

On the number of invariant measures for random expanding maps in higher dimensions

In \cite{J}, Jabłoński proved that a piecewise expanding $C^{2}$ multidimensional Jabłoński map admits an absolutely continuous invariant probability measure (ACIP). In \cite{BL}, Boyarsky and Lou extended this result to the case of i.i.d. compositions of the above maps, with an on average expanding condition. We generalize these results to the (quenched) setting of random Jabłoński maps, where the randomness is governed by an ergodic, invertible and measure preserving transformation. We prove that the skew product associated to this random dynamical system admits a finite number of ergodic ACIPs. Furthermore, we provide two different upper bounds on the number of mutually singular ergodic ACIP's, motivated by the works of Buzzi \cite{B} in one dimension and Gora, Boyarsky and Proppe \cite{GBP} in higher dimensions.

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arXiv2021

Lyapunov exponents for transfer operator cocycles of metastable maps: a quarantine approach

This works investigates the Lyapunov-Oseledets spectrum of transfer operator cocycles associated to one-dimensional random paired tent maps depending on a parameter $ε$, quantifying the strength of the \emph{leakage} between two nearly invariant regions. We show that the system exhibits metastability, and identify the second Lyapunov exponent $λ_2^ε$ within an error of order $ε^2|\log ε|$. This approximation agrees with the naive prediction provided by a time-dependent two-state Markov chain. Furthermore, it is shown that $λ_1^ε=0$ and $λ_2^ε$ are simple, and are the only exceptional Lyapunov exponents of magnitude greater than $-\log2+ O(\log\log\tfrac 1ε/\log\tfrac 1ε)$.

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arXiv2020

A tale of two vortices: how numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems

Coherent structures are spatially varying regions which disperse minimally over time and organise motion in non-autonomous systems. This work develops and implements algorithms providing multilayered descriptions of time-dependent systems which are not only useful for locating coherent structures, but also for detecting time windows within which these structures undergo fundamental structural changes, such as merging and splitting events. These algorithms rely on singular value decompositions associated to Ulam type discretisations of transfer operators induced by dynamical systems, and build on recent developments in multiplicative ergodic theory. Furthermore, they allow us to investigate various connections between the evolution of relevant singular value decompositions and dynamical features of the system. The approach is tested on models of periodically and quasi-periodically driven systems, as well as on a geophysical dataset corresponding to the splitting of the Southern Polar Vortex.

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arXiv2018

Stability and Collapse of the Lyapunov spectrum for Perron-Frobenius Operator cocycles

In this paper, we study random Blaschke products, acting on the unit circle, and consider the cocycle of Perron-Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. As a corollary, we obtain a simple random Blaschke product system where the Perron-Frobenius cocycle has infinitely many distinct Lyapunov exponents, but where arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum, except for the exponent 0 associated with the absolutely continuous invariant measure. That is, under perturbations, the Lyapunov exponents become 0 with multiplicity 1, and $-\infty$ with infinite multiplicity. This is superficially similar to the finite-dimensional phenomenon, discovered by Bochi \cite{Bochi-thesis}, that away from the uniformly hyperbolic setting, small perturbations can lead to a collapse of the Lyapunov spectrum to zero. In this paper, however, the cocycle and its perturbation are explicitly described; and further, the mechanism for collapse is quite different. We study stability of the Perron-Frobenius cocycles arising from general random Blaschke products. We give a necessary and sufficient criterion for stability of the Lyapunov spectrum in terms of the derivative of the random Blaschke product at its random fixed point, and use this to show that an open dense set of Blaschke product cocycles have hyperbolic Perron-Frobenius cocycles. In the final part, we prove a relationship between the Lyapunov spectrum of a single cocycle acting on two different Banach spaces, allowing us to draw conclusions for the same cocycles acting on $C^r$ functions spaces.

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