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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.

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 the 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.

We study cocycles of compact operators acting on a separable Hilbert space and investigate the stability of the Lyapunov exponents and Oseledets spaces when the operators are subjected to additive Gaussian noise. We show that as the noise is shrunk to 0, the Lyapunov exponents of the perturbed cocycle converge to those of the unperturbed cocycle, and the Oseledets spaces converge in probability to those of the unperturbed cocycle. This is, to our knowledge, the first result of this type with cocycles taking values in operators on infinite-dimensional spaces. The infinite dimensionality gives rise to a number of substantial difficulties that are not present in the finite-dimensional case.

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.

The paper by Froyland, González-Tokman and Quas [Stability and approximation of random invariant densities for Lasota–Yorke map cocycles.Nonlinearity27(4) (2014), 647] established fibrewise stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota–Yorke maps under a variety of perturbations, including ‘Ulam’s method’, a popular numerical method for approximating acims. The expansivity requirements of Froylandet alwere that the cocycle (or powers of the cocycle) should be ‘expanding on average’ before applying a perturbation, such as Ulam’s method. In the present work, we make a significant theoretical and computational weakening of the expansivity hypotheses of Froylandet al, requiring only that the cocycle be eventually expanding on average, and importantly,allowing the perturbation to be applied after each single step of the cocycle. The family of random maps that generate our cocycle need not be close to a fixed map and our results can handle very general driving mechanisms. We provide a detailed numerical example of a random Lasota–Yorke map cocycle with expanding and contracting behaviour and illustrate the extra information carried by our fibred random acims, when compared to annealed acims or ‘physical’ random acims.

We review state-of-the-art results on multiplicative ergodic theory for operators, with a view towards applications to the analysis of transport phenomena in non-autonomous dynamical systems, such as geophysical flows. The focus of this work is on ideas and motivation, rather than on proofs and technical aspects.

Environmental conditions shape entire communities by driving microbial interactions. These interactions then find their reflection in the evolutionary outcome of microbial competition. In static, homogeneous environments a robust, or evolutionary stable, outcome in microbial communities is reachable, if it exists. However, introducing heterogeneity and time-dependence in microbial ecology leads to stochastic evolutionary outcomes determined by specific environmental changes. We utilise evolutionary game theory to provide insight into phenotypic competition in dynamic environments. We capture these effects in a perturbed evolutionary game describing microbial interactions at a phenotypic level. We show that under regular periodic environmental fluctuations a stable state that preserves dominant phenotypes is reached. However, rapid environmental shifts, especially in a cyclic interactions, can lead to critical shifts in the evolutionary balance among phenotypes. Our analysis suggests that an understanding of the robustness of the systems current state is necessary to understand when system will shift to the new equilibrium. This can be done by understanding the systems overall margin of safety, that is, what level of perturbations it can take before its equilibrium changes. In particular, the extent to which an environmental shift affects the system’s behaviour.

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., To appear in the Journal of Computational Dynamics

In [22], Jabłoński proved that a piecewise expanding \begin{document}$ C^{2} $\end{document} multidimensional Jabłoński map admits an absolutely continuous invariant probability measure (ACIP). In [6], 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 ACIPs, motivated by the works of Buzzi [9] in one dimension and Góra, Boyarsky and Proppe [19] in higher dimensions.

International audience; We extend the recent spectral approach for quenched limit theorems developed for piecewise expanding dynamics under general random driving [9] to quenched random piecewise hyperbolic dynamics including some classes of billiards. For general ergodic sequences of maps in a neighbourhood of a hyperbolic map we prove a quenched large deviations principle (LDP), central limit theorem (CLT), and local central limit theorem (LCLT).

We give a streamlined proof of the multiplicative ergodic theorem for quasi-compact operators on Banach spaces with a separable dual., 18 pages

We establish stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota-Yorke maps under a variety of perturbations. Our family of random maps need not be close to a fixed map; thus, our results can handle very general driving mechanisms. We consider (i) perturbations via convolutions, (ii) perturbations arising from finite-rank transfer operator approximation schemes and (iii) static perturbations, perturbing to a nearby cocycle of Lasota-Yorke maps. The former two results provide a rigorous framework for the numerical approximation of random acims using a Fourier-based approach and Ulam's method, respectively; we also demonstrate the efficacy of these schemes.

The isolated spectrum of transfer operators is known to play a critical role in determining mixing properties of piecewise smooth dynamical systems. The so-called Dellnitz-Froyland ansatz places isolated eigenvalues in correspondence with structures in phase space that decay at rates slower than local expansion can account for. Numerical approximations of transfer operator spectrum are often insufficient to distinguish isolated spectral points, so it is an open problem to decide to which eigenvectors the ansatz applies. We propose a new numerical technique to identify the isolated spectrum and large-scale structures alluded to in the ansatz. This harmonic analytic approach relies on new stability properties of the Ulam scheme for both transfer and Koopman operators, which are also established here. We demonstrate the ecacy of this scheme in metastable one- and two-dimensional dynamical systems, including those with both expanding and contracting dynamics, and explain how the leading eigenfunctions govern the dynamics for both real and complex isolated eigenvalues.

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 $T_{\sigma^{n-1}\omega}\circ\cdots\circ T_{\sigma\omega}\circ T_\omega$. An important aspect of our results is that we only assume ergodicity and invertibility of the random driving $\sigma:\Omega\to\Omega$; in particular no expansivity or mixing properties are, Comment: revised version taking into account referee's comments. Accepted for publication in Communications in Mathematical Physics

Ensemble based methods are now widely used in applications such as weather prediction, but there are few rigorous results regarding their application. The broad goal of this paper is to provide some theoretical evidence of their applicability in the computational study of dynamical systems in some idealized, yet interesting setting. The specific goal of this paper is to investigate a data assimilation procedure (DAP), an ensemble Kalman filter (EKF), in the context of hyperbolic systems. We show that with appropriate assumptions on observations, for every trajectory on an attractor, the predictions produced by the DAP remain close to the truth for all time provided the ensemble is properly initialized, making the DAP reliable. We deal with the case of a one-dimensional unstable direction first, and later extend to higher dimensional unstable spaces. A feature of this approach is that no model linearizations are involved, making it efficient and potentially of interest for applications in high dimensional systems. Lyapunov exponents are also investigated.

International audience; We prove a fiberwise almost sure invariance principle for random piecewise expanding transformations in one and higher dimensions using recent developments on martingale techniques.

We establish rigorous scaling laws for the average bursting time for bubbling bifurcations of an invariant manifold, assuming the dynamics within the manifold to be uniformly hyperbolic. This type of global bifurcation appears in nearly synchronized systems, and is conjectured to be typical among those breaking the invariance of an asymptotically stable hyperbolic invariant manifold. We consider bubbling precipitated by generic bifurcations of a fixed point in both symmetric and non-symmetric systems with a codimension one invariant manifold, and discuss their extension to bifurcations of periodic points. We also discuss generalizations to invariant manifolds with higher codimension, and to systems with random noise.

Ulam's method is a rigorous numerical scheme for approximating invariant densities of dynamical systems. The phase space is partitioned into connected sets and an inter-set transition matrix is computed from the dynamics; an approximate invariant density is read off as the leading left eigenvector of this matrix. When a hole in phase space is introduced, one instead searches for \emph{conditional} invariant densities and their associated escape rates. For Lasota-Yorke maps with holes we prove that a simple adaptation of the standard Ulam scheme provides convergent sequences of escape rates (from the leading eigenvalue), conditional invariant densities (from the corresponding left eigenvector), and quasi-conformal measures (from the corresponding right eigenvector). We also immediately obtain a convergent sequence for the invariant measure supported on the survivor set. Our approach allows us to consider relatively large holes. We illustrate the approach with several families of examples, including a class of Lorenz maps., Comment: 20 pages, 6 figures, added section on Lorenz-like maps

Semi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we establish a semi-invertible multiplicative ergodic theorem that for the first time can be applied to the study of transfer operators associated to the composition of piecewise expanding interval maps randomly chosen from a set of cardinality of the continuum., 45 pages

We consider a piecewise smooth expanding map of the interval possessing two invariant subsets of positive Lebesgue measure and exactly two ergodic absolutely continuous invariant probability measures (ACIMs). When this system is perturbed slightly to make the invariant sets merge, we describe how the unique ACIM of the perturbed map can be approximated by a convex combination of the two initial ergodic ACIMs., 19 pages, 6 figures

We consider finite-state Markov chains driven by stationary ergodic invertible processes representing random environments. Our main result is that the invariant measures of Markov chains in random environments (MCREs) are stable under a wide variety of perturbations. We prove stability in the sense of convergence in probability of the invariant measure of the perturbed MCRE to the original invariant measure. We also develop a new numerical scheme to construct rigorous approximations of the invariant measures, which converge in probability as the resolution of the scheme increases. This numerical approach is illustrated with an example of a random walk in a random environment.