arXiv subjects

Cecilia Gonzalez-Tokman

Publications and source records attributed to Cecilia Gonzalez-Tokman.

4 recordsLinked to original sources

arXiv2025

Linear response for random and sequential intermittent maps

This work establishes a quenched (trajectory-wise) linear response formula for random intermittent dynamical systems, consisting of Liverani-Saussol-Vaienti maps with varying parameters. This result complements recent annealed (averaged) results in the i.i.d setting. As an intermediate step, we show existence, uniqueness and statistical stability of the random absolutely continuous invariant probability measure (a.c.i.m.) for such non-uniformly expanding systems. Furthermore, we investigate sequential intermittent dynamical systems of this type and establish a linear response formula. Our arguments rely on the cone technique introduced by Baladi and Todd and further developed by Lepp{ä}nen. We also demonstrate that sequential systems exhibit a subtle distinction from both random and autonomous settings: they may possess infinitely many sequential absolutely continuous equivariant densities. However, only one of these corresponds to an SRB state in the sense of Ruelle.

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arXiv2024

Thermodynamic Formalism and Perturbation Formulae for Quenched Random Open Dynamical Systems

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. Our random driving is generated by an invertible, ergodic, measure-preserving transformation on a probability space $(Ω,\mathscr{F},m)$. For each $ω$ we associate a piecewise-monotone, surjective map $T_ω$ and a hole $H_ω\subset [0,1]$; the map and the hole generate the corresponding open transfer operator. In the first chapter we prove, for a contracting potential, that there exists a unique random conformal measure $ν_ω$ supported on the survivor set. We also prove the existence of a unique random invariant density $ϕ_ω$. These provide an ergodic random invariant measure $μ=νϕ$ supported on the global survivor set. Further, we prove quasi-compactness of the transfer operator cocycle and exponential decay of correlations for $μ$. The escape rate of $ν$ is given by the difference of the expected pressures for the closed and open random systems. Finally, we prove that the Hausdorff dimension of the surviving set is equal to the unique zero of the expected pressure function for almost every fiber. In the second chapter we consider quasi-compact linear operator cocycles and their small perturbations. We prove an abstract fiberwise first-order formula for the leading Lyapunov multipliers. Our new machinery is deployed to create a spectral approach for a quenched extreme value theory that considers random dynamics and random observations. Finally, we prove quenched statistical limit theorems for random equilibrium states arising from contracting potentials. We illustrate the above theory with a variety of examples.

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arXiv2022

Perturbation formulae for quenched random dynamics with applications to open systems and extreme value theory

We consider quasi-compact linear operator cocycles $\mathcal{L}^{n}_ω:=\mathcal{L}_{σ^{n-1}ω}\circ\cdots\circ\mathcal{L}_{σω}\circ \mathcal{L}_ω$ driven by an invertible ergodic process $σ:Ω\toΩ$, and their small perturbations $\mathcal{L}_{ω,ε}^{n}$. We prove an abstract $ω$-wise first-order formula for the leading Lyapunov multipliers. We then consider the situation where $\mathcal{L}_ω^{n}$ is a transfer operator cocycle for a random map cocycle $T_ω^{n}:=T_{σ^{n-1}ω}\circ\cdots\circ T_{σω}\circ T_ω$ and the perturbed transfer operators $\mathcal{L}_{ω,ε}$ are defined by the introduction of small random holes $H_{ω,ε}$ in $[0,1]$, creating a random open dynamical system. We obtain a first-order perturbation formula in this setting, which reads $λ_{ω,ε}=λ_ω-θ_ωμ_ω(H_{ω,ε})+o(μ_ω(H_{ω,ε})),$ where $μ_ω$ is the unique equivariant random measure (and equilibrium state) for the original closed random dynamics. Our new machinery is then deployed to create a spectral approach for a quenched extreme value theory that considers random dynamics with general ergodic invertible driving, and random observations. An extreme value law is derived using the first-order terms $θ_ω$. Further, in the setting of random piecewise expanding interval maps, we establish the existence of random equilibrium states and conditionally invariant measures for random open systems via a random perturbative approach. Finally we prove quenched statistical limit theorems for random equilibrium states arising from contracting potentials. We illustrate the theory with a variety of explicit examples.

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arXiv2017

A spectral approach for quenched limit theorems for random expanding dynamical systems

We prove quenched versions of (i) a large deviations principle (LDP), (ii) a central limit theorem (CLT), and (iii) a local central limit theorem (LCLT) for non-autonomous dynamical systems. A key advance is the extension of the spectral method, commonly used in limit laws for deterministic maps, to the general random setting. We achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of Lyapunov exponents of \emph{twisted transfer operator cocycles} with respect to a twist parameter. While some versions of the LDP and CLT have previously been proved with other techniques, the local central limit theorem is, to our knowledge, a completely new result, and one that demonstrates the strength of our method. Applications include non-autonomous (piecewise) expanding maps, defined by random compositions of the form $T_{σ^{n-1}ω}\circ\cdots\circ T_{σω}\circ T_ω$. An important aspect of our results is that we only assume ergodicity and invertibility of the random driving $σ:Ω\toΩ$; in particular no expansivity or mixing properties are

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