1 Introduction↩︎

Investigating quantitatively how a system responds to perturbations is a key challenge in mathematical, natural, quantitative social sciences, and engineering. In the context of dynamical systems and their statistical properties, the problem can be cast as follows: how do the statistical properties of a system change when we perturb the reference dynamics with a small additional forcing? Is it possible to predict the response from the knowledge of the acting forcing and of the statistical properties of the unperturbed state? In this paper, we will set up a framework where one can answer these questions in the context of non-autonomous, time-dependent systems.

In the context of time-independent deterministic or random systems, a vast body of literature exists on this topic.

In the physical literature, the starting point can be traced to the fluctuation-dissipation theorem (FDT), which establishes a link between unforced fluctuations of a system and its response to perturbations [1]. The linear response of the system can be predicted using Green’s functions, which have particularly intuitive expressions found for systems near thermodynamic equilibrium [2]. Well-established applications of linear response theory include solid state physics and optics [3], plasma physics [4], and stellar dynamics [5], neuroscience [6], multiagent models [7], [8], as well as in many other systems, both near and far from thermodynamic equilibrium [9]–[17], see also the recently dedicated special issue [18].

In the context of deterministic dynamical systems, Ruelle initiated the mathematical investigation of linear response for Axiom A chaotic systems, provided explicit response formulas, and clarified that the classical form of the fluctuation-dissipation theorem does not hold because of the different geometrical properties of the stable vs unstable tangent space [19]–[22]. This leads to major difficulties in directly implementing Ruelle’s formulas [23], even if in recent years much progress has been achieved using adjoint and shadowing methods [24]–[29].

Ruelle’s results have then been first generalized to the some cases of partially hyperbolic systems [30], and, later on, to many other kind of systems (e.g. [31]–[39]) also extending the methods in more abstract terms by taking advantage of the transfer operator formalism [40]. Note that transfer operator methods can also be implemented for the numerical computation of linear response, and can give very precise computations, also with explicit bounds on the approximation errors [41]–[43] however these methods, typically suffer from the curse of dimensionality and are not efficient for performing actual calculations of the response of high-dimensional systems.

Response theory is easier to justify in rigorous mathematical terms when considering random dynamical systems and diffusion processes [44]–[49]. Coming to quenched results, which are nearer to the theme of the present work, rigorous quenched linear response for random dynamical systems has been studied in [50], establishing an abstract quenched response theory for parametrised transfer-operator cocycles and applying it to smooth expanding-on-average cocycles on the circle, while [51] treat random products of nearby Anosov diffeomorphisms in a uniformly hyperbolic setting. Closer to a genuinely nonautonomous sec:sequential perspective, [52] derives linear response for sequential intermittent (LSV-type) systems. The paper treats sequential compositions with a projective-metric method tailored to intermittent maps and obtains quenched response for the corresponding random compositions. We remark that these works on quenched response mostly rely on cocycle-based Oseledets techniques and random dynamical systems methods.

Still related to the random dynamical case, we also mention integral analogs of the perturbed Kolmogorov equation [53] and finite state Markov chains [54]–[56], which have the great advantage of providing an equation-agnostic platform [57] and can be coupled naturally with data-driven reduced-order techniques such as Markov state modeling [58], [59].

Very recently, considerable scientific advances have come, both in terms of theoretical development and of practical use, from the combination of response theory with data driven methods. Specifically, combining response theory with Koopmanism [60]–[63] leads, under suitably hypothesis, to the interpretability of the response operators, as they can be written as the sum of terms, each associated with a specific mode of unforced variability of the system [64]. This approach has the great advantage of allowing the construction of response operators from the properties of the unperturbed system and of identifying which modes of unforced variability modes provide the most important contribution to the response [65], [66] and can be shown to have a clear connection with Markov state modeling [57]. A separate line of fruitful investigation comes from combining response theory with generative modeling [67]–[69] which is providing encouraging results in terms of constructing response operators for high-dimensional systems.

Regardless of the level of mathematical rigor, physical relevance, numerical implementability, or data processing needs, a common major limitation of almost all the references mentioned so far is that they assume that one is perturbing a steady reference state, associated with an invariant measure. But, indeed, many systems of interest evolve according to dynamical laws that are explicitly time-dependent, and, yet, we would like to be able to study the response of such systems to perturbations. The time modulation can be periodic - think of any ecological system influenced by the seasonal cycle - or aperiodic - think of the climate, which is impacted by irregular geochemical forcings, by human activities, on top of more regular astrophysical and astronomical modulations ([70], [71]). Additionally, explicit time-dependence emerges each time we consider a multiscale system and focus our analysis on a restricted range of time scales, see [72] and the physically inspiring discussion in [73].

The extension fluctuation-dissipation results and linear response to aging systems have been presented since a while in the physical literature, see e.g. [74], [75]. Yet aging systems are a very special class of system whose statistical properties properties change with time. In the context of diffusion processes, Branicki and Uda [76] have provided a rigorous generalization of response theory for systems whose reference dynamics is periodic.

Instead, a recent preprint [77] has presented on more heuristic basis general response formulas for a rather general class of time-dependent systems, providing explicit results for time dependent finite Markov chains and diffusion processes. The work [77] also shows numerical evidence of the validity of the proposed nonautonomous linear response formula on a simple yet foundational climate model, namely the celebrated Ghil-Sellers model [78]–[80]. The preprint [77] provides a rather extensive introduction to the practical motivations and challenges behind the development of a response theory for time-dependent systems and provides a sketch of the mathematical framework needed to develop it with a reasonable degree of rigor. A key challenge in this context is to be able to construct a valid way to perturb the equivariant measure of the system supported on its pullback attractor [81], [82].

This work addresses the mathematical gap explicitly highlighted in [77] by providing a rigorous and general foundation for the corresponding response theory, which can be applied both to time dependent deterministic systems or time dependent random ones. This also extends the results of [76] to the case of aperiodic (nonautonomous) reference dynamics. We focus on the case of discrete time dynamics. Our main strategy is to introduce a global transfer operator acting on an extended space of sequences of measures, which allows us to handle the additional difficulties caused by a genuinely time-dependent reference dynamics. The resulting framework applies to nonautonomous systems satisfying assumptions that closely parallel the standard autonomous hypotheses: uniform Lasota-Yorke estimates, differentiability of the perturbation, and sufficiently fast loss of memory.³

We study sequential compositions of transfer operators satisfying the above assumptions. Exponential loss of memory yields a resolvent-type operator for the associated global map, with properties analogous to the classical resolvent in the autonomous case. This leads to our main abstract linear response theorem (Theorem 5), which establishes differentiability of the equivariant family for the operator sequence and applies to both deterministic and random nonautonomous systems. We illustrate the framework on two representative classes of deterministic and random dynamical systems: sequential compositions of expanding maps and random dynamical systems with sufficiently strong additive noise.

The paper is organized as follows. Section 2 introduces the main framework: suitable sequence spaces, the associated global transfer operator, and equivariant measures realized as fixed points (equivalently, eigenvectors for the unit eigenvalue) of this operator. Section 3 presents the key dynamical hypothesis underpinning the theory, namely loss of memory. In Section 4 we establish resolvent estimates for the global map and prove the abstract linear response result for time-dependent systems, obtaining explicit response formulas. Sections 5 and 6 apply the theory to two representative classes of sequential systems of broad interest, one deterministic (expanding maps) and one random (additive-noise dynamics). Section 7 discusses the results and outlines directions for future work. Appendices 8 and 9 collect technical tools used in the proofs.

2 Sequential nonautonomous systems, sequence spaces, and global map↩︎

We address the questions described in the introduction using a transfer-operator approach. Since we also aim to cover deterministic systems, following the standard practice in the literature, we work with transfer operators acting on two Banach spaces, a strong and a weak one. We recast the nonautonomous setting as an autonomous one on an extended phase space, namely a suitable sequence space. In this formulation, equivariant families become fixed points of an associated (global) transfer operator. Passing to the sequence space restores autonomy, at the price of losing the compactness of the strong-to-weak embedding in the corresponding sequence spaces. In this section we introduce this construction and the required preliminary notions.

Let \(X\) be separable metric space. Let \((B_s,\|\cdot\|_s)\) be a Banach space of finite signed Borel measures on \(X\). Let us also consider a "weaker" Banach space of signed measures \((B_w,\|\cdot\|_w)\) such that \(B_s\subseteq B_w\) and \(\|\cdot\|_w\leq \|\cdot\|_s\). ⁴ Write \(\mu(X)\) for the total mass and set \[V_s:=\{\mu\in B_s:\mu(X)=0\}.\] We will also suppose that \(V_s\) is a closed subspace of \(B_s\). Define the sequence space \[\mathcal{B}_s := \ell^\infty(\mathbb{Z};B_s), \qquad \|\boldsymbol{\mu}\|_{\mathcal{B}_s}:=\sup_{n\in\mathbb{Z}}\|\mu_n\|_s,\] and the zero-mass subspace \[\mathcal{V}_s := \ell^\infty(\mathbb{Z};V_s)\subset \mathcal{B}_s.\]

Let \(\mathbb{S}:\mathcal{B}_s\to\mathcal{B}_s\) denote the left shift, \[(\mathbb{S}\boldsymbol{\mu})_n := \mu_{n+1}.\] Then \(\mathbb{S}\) is a linear isometry on \(\mathcal{B}_s\).

Now we introduce the time dependent dynamics by considering a sequence of transfer operators. To formalize the idea of a perturbation in the family, we will introduce a further parameter \(\varepsilon\) to adjust the strength of the perturbation. The operators for \(\varepsilon=0\) will then be considered as unperturbed ones.

Let \(\epsilon_0>0\) and \((L_n^\varepsilon)_{n\in\mathbb{Z}}\) be a family of bounded linear operators \[L_n^\varepsilon:B_s\to B_s, \qquad \varepsilon\in [0, \varepsilon_0),\] which are Markov in the sense that they are positive and they preserve total mass: \[\label{eq:mass95preserving} (L_n^\varepsilon\mu)(X)=\mu(X)\qquad\forall \mu\in B_s,\;\forall n,\;\forall \varepsilon \in [0, \varepsilon_0).\tag{1}\] We define the (parameter-dependent) global map \(\mathbb{F}_\varepsilon:\mathcal{B}_s\to\mathcal{B}_s\) by \[(\mathbb{F}_\varepsilon(\boldsymbol{\mu}))_n := L_n^\varepsilon \mu_n.\] A sequence of probability measures \(\boldsymbol{\mu}^\varepsilon=(\mu_n^\varepsilon)_{n\in\mathbb{Z}}\) in \({\mathcal{B}_s}\) is said to be an equivariant family for \((L_n^\varepsilon)\) if \[\label{eq:equiv} \mu_{n+1}^\varepsilon = L_n^\varepsilon \mu_n^\varepsilon\qquad\forall n\in\mathbb{Z},\tag{2}\] which is equivalent to the fixed point equation on \(\mathcal{B}_s\) \[\label{eq:FP95shift} \mathbb{S}\boldsymbol{\mu}^\varepsilon = \mathbb{F}_\varepsilon(\boldsymbol{\mu}^\varepsilon).\tag{3}\]

In the time-dependent systems we consider, the equivariant measure is supported on the pullback attractor and plays the same role as the invariant measure in the classical autonomous setting.

It is convenient to work on another global map that is the composition \[\mathbb{T}_\varepsilon := \mathbb{S}^{-1}\mathbb{F}_\varepsilon, \qquad (\mathbb{T}_\varepsilon\boldsymbol{v})_n = L_{n-1}^\varepsilon v_{n-1}.\] Then 3 is equivalent to \[\label{eq:FP95T} \boldsymbol{\mu}^\varepsilon = \mathbb{T}_\varepsilon \boldsymbol{\mu}^\varepsilon.\tag{4}\]

Note that in what follows we will consider families of transfer operators parametrized by \(\varepsilon\), where \(\varepsilon=0\) describes our reference state, whilst considering \(\varepsilon\neq0\) is associated with perturbed states.

3 Loss of memory↩︎

In the autonomous setting, linear response results are often proved under a spectral gap assumption or some kind of fast convergence to equilibrium (with notable exceptions such as [35], [83]). In our nonautonomous framework, the analogous role is played by loss of memory. This concept quantifies the speed at which a non autonomous dynamical system forgets its initial condition, and this is also a key ingredient proposed in [77]. In this section, we introduce the concept and derive the first fundamental properties.

The presence of loss of memory ensure the uniqueness of equivariant measures in the strong space.

Proof. Let \(\boldsymbol{\mu},\boldsymbol{\nu}\in\ell^\infty(\mathbb{Z};B_s)\) be two equivariant families and fixed points of \(\mathbb{T}_\varepsilon\). Set \(\Delta_n:=\mu_n-\nu_n\). Then \(\Delta_n\in V_s\) for all \(n\) and \[\Delta_n=\mu_n-\nu_n=L_{n-1}\mu_{n-1}-L_{n-1}\nu_{n-1}=L_{n-1}\Delta_{n-1}.\] Iterating \(k\) steps backwards yields \[\Delta_n = L^{(n-k,n-1)}\Delta_{n-k}.\] Taking \(\|\cdot\|_s\) and applying ?? gives \[\|\Delta_n\|_s \le a(k)\,\|\Delta_{n-k}\|_s \le a(k)\,\sup_{m\in\mathbb{Z}}\|\Delta_m\|_s.\] Letting \(k\to\infty\) and using \(a(k)\to 0\) yields \(\|\Delta_n\|_s=0\) for every \(n\), hence \(\boldsymbol{\mu}=\boldsymbol{\nu}\).  

4 Linear response↩︎

In this section we prove a linear response statement for a sequential family of transfer operators as in the previous setting. We will consider \(\varepsilon_0>0\) and for \(\varepsilon\in [0,\varepsilon_0)\) we consider a sequential family of operators \(L^\varepsilon _n\) with associated global maps \(\mathbb{T}_\varepsilon\) having equivariant measures \(\boldsymbol{\mu}^\varepsilon\). We will also assume the following:

Loosely speaking, Assumption 2 means that, along the reference family \((\mu_n)\), the cocycle admits a first-order expansion in \(\varepsilon\), namely \[L_n^\varepsilon \mu_n \;=\; L_n^0 \mu_n \;+\; \varepsilon\,\dot{L}_n \mu_n \;+\; o(\varepsilon) \quad\text{in }B_s,\;\text{uniformly in }n,\] so that one may heuristically view \(L_n^\varepsilon \approx L_n^0+\varepsilon\,\dot{L}_n\) at first order. We also note that, depending on the perturbation, establishing strong convergence in ?? may require the reference densities \(\mu_n\) to have one additional degree of regularity beyond Assumption 1; see Section 5 for a concrete instance of this issue and how it can be handled.

4.1 Existence of the resolvent on the zero-mass subspace↩︎

In this section we show a kind of Neumann-series based invertibility of \(I-\mathbb{T}_\varepsilon\) on \(\mathcal{V}_s\) and investigate its basic properties, which will be used to establish our Linear Response result.

Proof. Mass preservation 1 implies \(L_n^\varepsilon(V_s)\subset V_s\); hence \(\mathbb{T}_\varepsilon(\mathcal{V}_s)\subset\mathcal{V}_s\). Iterating \((\mathbb{T}_\varepsilon\boldsymbol{v})_n=L_{n-1}^\varepsilon v_{n-1}\) yields \[(\mathbb{T}_\varepsilon^k \boldsymbol{v})_n = L_{n-1}^\varepsilon \cdots L_{n-k}^\varepsilon\, v_{n-k}.\] Since \(v_{n-k}\in V_s\), Assumption 3 gives \[\|(\mathbb{T}_\varepsilon^k \boldsymbol{v})_n\|_s \le C\rho^k \|v_{n-k}\|_s \le C\rho^k \|\boldsymbol{v}\|_{\mathcal{B}_s}.\] Taking \(\sup_n\) yields ?? . Then \(\sum_{k\ge0}\mathbb{T}_\varepsilon^k\) converges in operator norm on \(\mathcal{V}_s\) (because \(\sum_{k\ge0}C\rho^k<\infty\)), and equals \((I-\mathbb{T}_\varepsilon)^{-1}\) by the standard Neumann-series argument, giving ?? and ?? .  

The previous lemma shows the existence of the resolvent operator \(I-\mathbb{T}_\varepsilon\), in the next lemma we prove its stability under perturbation. The approach is general and axiomatic, however almost all the asumptions we will consider for \(\mathbb{T}_\varepsilon\) are direct consequences of the Assumptions  1–3 on the operators \(L_n^\varepsilon\) listed in the previous section.

Proof. We recall that by Assumption (A1) and Lemma 1 we get \(\|\mathbb{T}_\varepsilon^k\|_{\mathcal{V}_s\to\mathcal{V}_s}\le C\rho^k\), hence \(\sum_{k\ge 0}\mathbb{T}_\varepsilon^k\) converges in operator norm on \(\mathcal{V}_s\), and equals \((I-\mathbb{T}_\varepsilon)^{-1}\).

Step 1: Telescoping formula for powers. For every \(k\ge 1\) we have the following alternative expression for \(\mathbb{T}_\varepsilon^k-\mathbb{T}_0^k\) \[\label{eq:power95telescoping} \mathbb{T}_\varepsilon^k-\mathbb{T}_0^k = \sum_{j=0}^{k-1}\mathbb{T}_\varepsilon^{k-1-j}(\mathbb{T}_\varepsilon-\mathbb{T}_0)\mathbb{T}_0^{j}.\tag{5}\] This identity follows by a standard telescoping argument: write \(\mathbb{T}_\varepsilon^k-\mathbb{T}_0^k=(\mathbb{T}_\varepsilon^{k-1}-\mathbb{T}_0^{k-1})\mathbb{T}_\varepsilon+\mathbb{T}_0^{k-1}(\mathbb{T}_\varepsilon-\mathbb{T}_0)\) and iterate (equivalently, insert and subtract \(\mathbb{T}_\varepsilon^{k-1-j}\mathbb{T}_0^{j+1}\) for \(j=0,\dots,k-1\)) to obtain 5 .

Step 2: Truncation of the resolvent series. For \(N\ge 1\) write \[(I-\mathbb{T}_\varepsilon)^{-1}-(I-\mathbb{T}_0)^{-1} = \sum_{k=0}^{N-1}(\mathbb{T}_\varepsilon^k-\mathbb{T}_0^k) \;+\; \sum_{k=N}^{\infty}(\mathbb{T}_\varepsilon^k-\mathbb{T}_0^k).\] We estimate the two sums separately as operators \(\mathcal{V}_s\to\mathcal{V}_w\).

Step 3: Use (A3) to estimate of the finite sum. Consider \(v\in\mathcal{V}_s\). Using 5 , \[\|(\mathbb{T}_\varepsilon^k-\mathbb{T}_0^k)v\|_{\mathcal{B}_w} \le \sum_{j=0}^{k-1} \|\mathbb{T}_\varepsilon^{k-1-j}\|_{\mathcal{V}_w\to\mathcal{V}_w}\, \|\mathbb{T}_\varepsilon-\mathbb{T}_0\|_{\mathcal{B}_s\to\mathcal{B}_w}\, \|\mathbb{T}_0^j v\|_{\mathcal{B}_s}.\] By (A2), (A3), and (A1) for \(\mathbb{T}_0\), \[\|(\mathbb{T}_\varepsilon^k-\mathbb{T}_0^k)v\|_{\mathcal{B}_w} \le \sum_{j=0}^{k-1} M_w\,\delta_\varepsilon\, C\rho^j \|v\|_{\mathcal{B}_s} \le \delta_\varepsilon\,C\,\|v\|_{\mathcal{B}_s}\,\sum_{j=0}^{k-1} M_w\rho^j.\] Hence, for \(k\le N\), \[\|(\mathbb{T}_\varepsilon^k-\mathbb{T}_0^k)v\|_{\mathcal{B}_w} \le \delta_\varepsilon\,C\,\|v\|_{\mathcal{B}_s}\, M_w\cdot \frac{1}{1-\rho}.\] Therefore, \[\left\|\sum_{k=0}^{N-1}(\mathbb{T}_\varepsilon^k-\mathbb{T}_0^k)v\right\|_{\mathcal{B}_w} \le K_1\,\delta_\varepsilon\, N\,\|v\|_{\mathcal{B}_s},\] for a constant \(K_1\) depending only on \(C,\rho,M_w\).

Step 4: Estimate of the tail by the strong decay. For \(k\ge N\) we simply bound \[\|(\mathbb{T}_\varepsilon^k-\mathbb{T}_0^k)v\|_{\mathcal{B}_w} \le \|\mathbb{T}_\varepsilon^k v\|_{\mathcal{B}_w}+\|\mathbb{T}_0^k v\|_{\mathcal{B}_w} \le \bigl(\|\mathbb{T}_\varepsilon^k v\|_{\mathcal{B}_s}+\|\mathbb{T}_0^k v\|_{\mathcal{B}_s}\bigr) \le 2C\rho^k\|v\|_{\mathcal{B}_s},\] using (A1) for both \(\mathbb{T}_0,\mathbb{T}_\varepsilon\) and the embedding \(\|\cdot\|_w\le \|\cdot\|_s\). Summing for \(k\ge N\) yields \[\left\|\sum_{k=N}^{\infty}(\mathbb{T}_\varepsilon^k-\mathbb{T}_0^k)v\right\|_{\mathcal{B}_w} \le K_2\,\rho^{N}\,\|v\|_{\mathcal{B}_s}, \qquad K_2:=\frac{2C}{1-\rho}.\]

Step 5: Optimisation and conclusion. Combining the finite-sum and tail estimates gives \[\|( (I-\mathbb{T}_\varepsilon)^{-1}-(I-\mathbb{T}_0)^{-1})v\|_{\mathcal{B}_w} \le K_1\,\delta_\varepsilon\,N\,\|v\|_{\mathcal{B}_s} + K_2\,\rho^N\,\|v\|_{\mathcal{B}_s}.\] Taking the supremum over \(\|v\|_{\mathcal{B}_s}\le 1\) yields the operator bound \[\|(I-\mathbb{T}_\varepsilon)^{-1}-(I-\mathbb{T}_0)^{-1}\|_{\mathcal{V}_s\to\mathcal{V}_w} \le K_1\,\delta_\varepsilon\,N + K_2\,\rho^N.\] Choose \(N=\left\lceil \frac{\log(1/\delta_\varepsilon)}{-\log\rho}\right\rceil\), so that \(\rho^N \sim \delta_\varepsilon\) and \(N\sim 1+\log(1/\delta_\varepsilon)\). This gives ?? , hence the desired bounds and continuity as \(\delta_\varepsilon\to 0\).  

4.2 Linear response and the series formula↩︎

We are now ready to derive the discrete-time linear response formula as a resolvent applied to a forcing term. The argument is very similar as the ones typically used in the autonomous case, but carried out for the fixed-point equation of the global transfer operator on the sequence space. This leads to obtaining rather compact and interpretable formulas.

Proof. From \(\mathbb{S}\boldsymbol{\mu}^\varepsilon=\mathbb{F}_\varepsilon(\boldsymbol{\mu}^\varepsilon)\) and \(\mathbb{S}\boldsymbol{\mu}=\mathbb{F}(\boldsymbol{\mu})\), subtract and divide by \(\varepsilon\): \[\label{eq:quotient95identity} \mathbb{S}\boldsymbol{h}^\varepsilon = \mathbb{F}_\varepsilon(\boldsymbol{h}^\varepsilon) + \boldsymbol{g}^\varepsilon, \qquad \boldsymbol{g}^\varepsilon:=(g_n^\varepsilon)_{n\in \mathbb{Z}}, \; (g_n^\varepsilon):=\frac{L_n^\varepsilon-L_n}{\varepsilon}\,\mu_n.\tag{6}\] Applying \(\mathbb{S}^{-1}\) gives \[\label{eq:Res95eps95again} (I-\mathbb{T}_\varepsilon)\boldsymbol{h}^\varepsilon = \mathbb{S}^{-1}\boldsymbol{g}^\varepsilon, \qquad \mathbb{T}_\varepsilon=\mathbb{S}^{-1}\mathbb{F}_\varepsilon.\tag{7}\]

Note that by Assumption 2, \[\|\boldsymbol{g}^\varepsilon-\boldsymbol{g}\|_{\mathcal{B}_s} = \sup_n\left\| \frac{L_n^\varepsilon\mu_n-L_n \mu_n}{\varepsilon}-\dot{L}_n\mu_n \right\|_s \to 0\] where \(g_n=\dot{L}_n\mu_n\) and \(\boldsymbol{g}:=(g_n)_{n\in \mathbb{Z}}\).

Moreover, 1 implies \((\dot{L}_n\mu_n)(X)=0\), hence \(\boldsymbol{g}^\varepsilon\in\mathcal{V}_s\) and similarly \(\boldsymbol{g}\in\mathcal{V}_s\) for \(\varepsilon\) small.

Step 1: uniform invertibility on \(\mathcal{V}_s\) and boundedness of \(\boldsymbol{h}^\varepsilon\). By Lemma 1, for each \(|\varepsilon|\le\varepsilon_0\) the inverse \((I-\mathbb{T}_\varepsilon)^{-1}\) exists on \(\mathcal{V}_s\) and \[\|(I-\mathbb{T}_\varepsilon)^{-1}\|_{\mathcal{V}_s\to\mathcal{V}_s}\le \frac{C}{1-\rho}.\] Applying this to 7 yields \[\|\boldsymbol{h}^\varepsilon\|_{\mathcal{B}_s} \le \|(I-\mathbb{T}_\varepsilon)^{-1}\|\,\|\mathbb{S}^{-1}\boldsymbol{g}^\varepsilon\|_{\mathcal{B}_s} \le \frac{C}{1-\rho}\,\|\boldsymbol{g}^\varepsilon\|_{\mathcal{B}_s}.\] Since \(\boldsymbol{g}^\varepsilon\to \boldsymbol{g}\) in \(\mathcal{B}_s\), the family \((\boldsymbol{h}^\varepsilon)\) is bounded in \(\mathcal{B}_s\).

Step 2: identification of the limit and convergence of \(\boldsymbol{h}^\varepsilon\).

From 7 we get \[(I-\mathbb{T}_\varepsilon)\boldsymbol{h}^\varepsilon = \mathbb{S}^{-1}(\boldsymbol{g}^\varepsilon- \boldsymbol{g})+ \mathbb{S}^{-1} \boldsymbol{g}\]

and

\[\boldsymbol{h}^\varepsilon = (I-\mathbb{T}_\varepsilon)^{-1} \mathbb{S}^{-1}(\boldsymbol{g}^\varepsilon- \boldsymbol{g})+ (I-\mathbb{T}_\varepsilon)^{-1} \mathbb{S}^{-1} \boldsymbol{g} .\] Since \((I-\mathbb{T}_\varepsilon)^{-1}\) is uniformly bounded and \((\boldsymbol{g}^\varepsilon- \boldsymbol{g})\to 0\) then \((I-\mathbb{T}_\varepsilon)^{-1} \mathbb{S}^{-1}(\boldsymbol{g}^\varepsilon- \boldsymbol{g})\to 0\) in the strong norm. By Lemma 2 we also have that \((I-\mathbb{T}_\varepsilon)^{-1} \mathbb{S}^{-1} \boldsymbol{g} \to (I-\mathbb{T}_0)^{-1} \mathbb{S}^{-1} \boldsymbol{g}\) in the weak norm. All together this implies \(\boldsymbol{h}^{\varepsilon} \to \boldsymbol{\eta}\) in the weak norm as claimed.

Step 3: Neumann series and the coordinate-wise formula. Equation ?? implies ?? by Lemma 1 at \(\varepsilon=0\). Taking the \(n\)-th coordinate, note that \((\mathbb{S}^{-1}\boldsymbol{g})_n=g_{n-1}=\dot{L}_{n-1}\mu_{n-1}\) and \[(\mathbb{T}^k \boldsymbol{v})_n = L_{n-1}\cdots L_{n-k}\, v_{n-k}.\] Therefore \[\eta_n = \sum_{k=0}^{\infty} L_{n-1}\cdots L_{n-k}\, (\mathbb{S}^{-1}\boldsymbol{g})_{n-k} = \sum_{k=1}^{\infty} L_{n-1}\cdots L_{n-k}\, \dot{L}_{n-k-1}\mu_{n-k-1},\] which is ?? .

Finally, absolute convergence in \(B_s\) follows from Assumptions 3, 1, and ?? : each summand has zero mass and \[\|L_{n-1}\cdots L_{n-k}\dot{L}_{n-k-1}\mu_{n-k-1}\|_s \le C\rho^k\,\|\dot{L}_{n-k-1}\|_{s\to s}\,\|\mu_{n-k-1}\|_s \le C\rho^k \Big(\sup_j\|\dot{L}_j\|_{s\to s}\Big) M_s,\] so the series is dominated by a geometric series uniformly in \(n\).  

5 Linear response for sequential composition of expanding maps↩︎

In this section we apply Theorem 5 to establish linear response for sequential composition of deterministic expanding maps. The kind systems we will consider are sequential composition of maps that are close to a certain expanding map \(T_0\) (the admitted distance will be estimated by the properties of \(T_0\)), and we will consider additive deterministic perturbations of the maps. In the following subsections we will see how to verify the assumptions needed to apply Theorem 5, starting from the exponential loss of memory, which will be provided by the results of Appendix 8. The results of Appendix 8 would also allow one to treat similar examples in which the expanding maps vary slowly along the sequence (see Theorem 8). To avoid a repetition of nearly identical arguments, we do not spell out this extension in detail.

5.1 Uniform exponential loss of memory for sequential composition of expanding maps near \(T_0\)↩︎

In this section we set and work with \[B_s = W^{1,1}(\mathbb{S}^1),\qquad B_w = L^1(\mathbb{S}^1).\]

Let \(T_0:\mathbb{S}^1\to\mathbb{S}^1\) be a \(C^3\) expanding map of degree \(n\ge2\) such that \[\lambda_0:=\inf_{\mathbb{S}^1}|T_0'|>1,\qquad M_0:=\|T_0'\|_\infty<\infty,\qquad M_2:=\|T_0''\|_\infty<\infty,\] and assume that \(T_0\) is mixing. We recall that under these assumptions the transfer operator \(L_0\) associated to \(T_0\) has a spectral gap on \(W^{1,1}\) (equivalently, exponential contraction on \(V_s:=\{f\in W^{1,1}:\int f\,dm=0\}\)).

Fix constants \(\lambda_1\in(0,1)\) and \(B\ge0\) such that the one-step Lasota–Yorke inequality holds for \(L_0\) on \(B_s=W^{1,1}(\mathbb{S}^1)\) and \(B_w=L^1(\mathbb{S}^1)\): \[\label{eq:LY95T095W11} \|L_0 f\|_{L^1}\le \|f\|_{L^1},\qquad \|L_0 f\|_{W^{1,1}}\le \lambda_1\|f\|_{W^{1,1}}+B\|f\|_{L^1},\qquad f\in W^{1,1}.\tag{8}\] In particular in this case, the assumption(A2) is satisfied with \(M_w=1\). For expanding maps as above such constants are explicitly available (see for example [84]).

Let \(M\in\mathbb{N}\) be such that \[\label{eq:M95choice95ML2} \lambda_1^{M}\le \frac{1}{10\left(\frac{B}{1-\lambda_1}+1\right)} \quad\text{and}\quad \|L_0^{M}v\|_{L^1}\le \frac{1}{10B}\|v\|_{W^{1,1}}\qquad \forall v\in V_s.\tag{9}\] (Existence of such an \(M\) follows for example from [84])

Let \[\label{eq:CT095explicit95final} C(T_0)=2 n\bigl(\|T_0'\|_\infty+\lambda_0-1\bigr)\left(\frac{1}{\lambda_0^2}+\frac{M_2}{\lambda_0^3}+\frac{1}{\lambda_0}\right)\tag{10}\] be the mixed-norm constant from Proposition 11 in Appendix B, namely for any \(T\) such that \(\|T_0-T\|_{C^2} \leq \lambda_0 -1\)

\[\label{eq:mixed95bound95CT0} \|L_T-L_0\|_{W^{1,1}\to L^1}\le C(T_0)\,\|T-T_0\|_{C^2}.\tag{11}\]

Proof. The proof will follow by applying Lemma 10 combined with explicit estimates of the constants involved in the mixed norm distance between transfer operators associated with nearby maps provided by Proposition 11. We now verify that the assumptions (ML1),...,(ML3) hold under the above assumptions.

For each \(T\in\mathcal{S}_{T_0}(\delta_\ast)\), denote by \(L_T\) its Perron–Frobenius operator, considered as usual with \[B_s=W^{1,1}(\mathbb{S}^1),\qquad B_w=L^1(\mathbb{S}^1),\qquad V_s=\Bigl\{f\in W^{1,1}:\int f\,dm=0\Bigr\}.\]

The following holds:

(ML1): Uniform one-step Lasota–Yorke constants on \(\mathcal{S}_{T_0}(\delta_\ast)\). Every \(T\in\mathcal{S}_{T_0}(\delta_\ast)\) satisfies \(\inf|T'|\ge \lambda_0-\delta_\ast>1\) and \(\|T''\|_\infty\le M_2+\delta_\ast\). Consequently, the one-step Lasota–Yorke inequality holds with the same constants \[\label{eq:LY95uniform95constants95explicit} \lambda_1:=\frac{1}{\lambda_0-\delta_\ast}\in(0,1), \qquad B:=\frac{M_2+\delta_\ast}{(\lambda_0-\delta_\ast)^2},\tag{12}\] namely for all \(f\in W^{1,1}\) and all \(T\in\mathcal{S}_{T_0}(\delta_\ast)\), \[\label{eq:ML195uniform95explicit} \|L_T f\|_{L^1}\le \|f\|_{L^1},\qquad \|L_T f\|_{W^{1,1}}\le \lambda_1\|f\|_{W^{1,1}}+B\|f\|_{L^1}.\tag{13}\] In particular, \((ML1)\) holds uniformly for any sequence of operators associated to maps in \(\mathcal{S}_{T_0}(\delta_\ast)\).

Choice of \(M\) and (ML2). By the choice of \(M\) made at beginning of the subsection it immediately follows that (ML2) is verified for \(L_0\).

Nearby operators and (ML3). For each \(i\), since \(T_i\in\mathcal{S}_{T_0}(\delta_\ast)\), Proposition 11 gives \(\|L_i-L_0\|_{W^{1,1}\to L^1}\le C(T_0)\delta_\ast\). Condition ?? is exactly the numerical bound needed to satisfy \((ML3)\). Since the assumptions are verified, the application of Lemma 10 then directly lead to the statement.  

5.2 Perturbations of the sequential composition and verification of the Assumptions 1, 2.↩︎

In this section we define the kinds of perturbations we mean to apply to a sequential composition of expanding maps and verify that Assumptions 1, 2 hold. The kind of perturbations we consider are given by the application of a further diffeomorphism after the application of the dynamics, adding in some sense a further "small kick" to the result of the unperturbed dynamics. This will lead to the definition of a certain kind of family of sequential maps \(T_n^\varepsilon\) parametrized by \(\varepsilon \in [0,\varepsilon_0 )\).

Throughout this section, we continue working with the general setting and notations stated at beginning of Section 5.1. We suppose \(T_0\), \(\delta_\ast\) and the set \(\mathcal{S}_{T_0}(\delta_\ast)\) being fixed.

5.2.1 Post-composition perturbations.↩︎

Let \((T_n)_{n\in\mathbb{Z}}\) be a sequence of \(C^3\) expanding covering maps in \(\mathcal{S}_{T_0}(\delta_\ast)\). Let \(X\in C^3(\mathbb{S}^1)\) be a smooth vector field on the circle, and let \((h_\varepsilon)_{\varepsilon \in [0, \varepsilon_0)}\) be a \(C^3\) family of \(C^3\) diffeomorphisms of \(\mathbb{S}^1\) with \(h_0=\mathrm{id}\) and \[\label{eq:kick95family} h_\varepsilon(x)=x+\varepsilon X(x)+r^\varepsilon(x), \qquad \frac{\|r^\varepsilon\|_{C^3}}{|\varepsilon|}\xrightarrow[\varepsilon\to0]{}0.\tag{14}\] Define the perturbed sequential maps by post-composition \[\label{eq:T95kick} T_n^\varepsilon := h_\varepsilon\circ T_n.\tag{15}\] Let \(L_n\) and \(L_n^\varepsilon\) be the Perron–Frobenius operators associated to \(T_n\) and \(T_n^\varepsilon\), respectively. \(L_n^\varepsilon = L_n\,L_{h_\varepsilon}\) where \(L_{h_\varepsilon}\) is the operator associated to \(h_\varepsilon\).

5.2.2 Uniform bounds for the equivariant family (Assumption 1)↩︎

Under the following assumption on the maps \(T_n^\varepsilon\) we can establish Assumption 1 for the associated system of transfer operators.

The following global regularity property for the maps \(T_n^\varepsilon\) directly follows by the way the perturbations are constructed.

This allows to prove the following uniform bound on the regularity of the equivariant measures \(\mu^\varepsilon_n\).

Proof. By Lemma 4 the maps \(T_n^\varepsilon\) have uniform \(C^3\) regularity, furthermore, if \(\varepsilon_0\) is small enough they are also uniformly expanding. It is well known (see e.g. [84], Lemma 18) that such a set of maps satisfy uniformly Lasota Yorke inequalities on \(W^{i,1}\), in particular for \(i=1,2\): \[\|L_n^\varepsilon f\|_{W^{1,1}}\le a\|f\|_{W^{1,1}}+b\|f\|_{L^1},\qquad \|L_n^\varepsilon f\|_{W^{2,1}}\le a_2\|f\|_{W^{2,1}}+b_2\|f\|_{W^{1,1}},\] with constants independent of \(n\) and \(\varepsilon\). Fix a reference density \(f_\ast\in W^{2,1}\) with \(\int f_\ast\,dm=1\), and define pullback iterates \(\mu_{n}^{\varepsilon,(k)}:=L_{n-k,n-1}^\varepsilon f_\ast\). The Lasota-Yorke bounds give uniform \(W^{2,1}\) bounds for \(\mu_{n}^{\varepsilon,(k)}\). Loss of memory on the zero-mean subspace proved in Lemma 3 implies \(\mu_{n}^{\varepsilon,(k)}\) is Cauchy in \(W^{1,1}\) as \(k\to\infty\), hence converges to some \(\mu_n^\varepsilon \in W^{1,1}\) satisfying equivariance. Proposition 2 ensures the system has a unique equivariant family \((\mu_n^\varepsilon)_{n\in\mathbb{Z}}\in W^{1,1}\). The uniform bounds on the \(W^{2,1}\) norm of \(\mu_n^\varepsilon\) again follows from the uniform Lasota Yorke inequalities.  

We remark that the regularity established above for the family \(\mu_{n}^{\varepsilon}\) is more than necessary to verify Assumption 1, taking \(W^{1,1}\) as a strong space. However this will be useful to verfy Assumption 2, as we will see in the next section.

5.2.3 Strong differentiability (Assumption 2)↩︎

In this subsection we verify Assumption 2 for the above family of perturbed sequential transfer operators \(L_n^\varepsilon\).

5.2.3.1 Derivative of the diffeomorphism transfer operator.

For a \(C^2\) diffeomorphism \(h\) of \(\mathbb{S}^1\), the Perron–Frobenius operator is \[(L_h u)(x)=\frac{u(h^{-1}(x))}{h'(h^{-1}(x))}.\] The following is a "uniform on \(W^{2,1}\)" version of a well known result, establishing a formula for the "derivative" at \(0\) of the family of operators \(L_{h_\varepsilon}\). We include the simple proof for completeness.

Proof. (i) Since \(Du=-(Xu)'=-X'u-Xu'\), one has \(\|Du\|_{L^1}\le \|X'\|_\infty\|u\|_{L^1}+\|X\|_\infty\|u'\|_{L^1}\). Moreover \((Du)'=-(X''u+2X'u'+Xu'')\), hence \(\|(Du)'\|_{L^1}\le \|X''\|_\infty\|u\|_{L^1}+2\|X'\|_\infty\|u'\|_{L^1}+\|X\|_\infty\|u''\|_{L^1}\). This yields the stated bound.

(ii) This is a standard linearisation of the pushforward by a near-identity diffeomorphism. Using \(h_\varepsilon^{-1}(x)=x-\varepsilon X(x)+o_{C^2}(\varepsilon)\) and \(h_\varepsilon'(x)=1+\varepsilon X'(x)+o_{C^2}(\varepsilon)\) (both consequences of 14 ), expand \[\frac{u(h_\varepsilon^{-1}(x))}{h_\varepsilon'(h_\varepsilon^{-1}(x))} = u(x)-\varepsilon\big(X(x)u'(x)+X'(x)u(x)\big)+o_{W^{1,1}}(\varepsilon),\] where the remainder estimate uses \(u\in W^{2,1}\) and the uniform \(o_{C^3}(\varepsilon)\) in 14 . This gives ?? . Uniformity on bounded subsets of \(W^{2,1}\) follows because the remainder is controlled by \(\|u\|_{W^{2,1}}\) times \(\|r^\varepsilon\|_{C^3}/|\varepsilon|\to0\).  

5.2.3.2 Verification of Assumption 2.

Let \(\boldsymbol{\mu}=(\mu_n)_{n\in\mathbb{Z}}\) be the unperturbed equivariant family for \((L_n)\). Define the forcing sequence \[\label{eq:forcing95kick} g_n := D(L_n(\mu_n))\in W^{1,1}.\tag{16}\]

Proof. Since the perturbation is a post-composition kick, \(T_n^\varepsilon=h_\varepsilon\circ T_n\), the Perron–Frobenius operators satisfy \[L_n^\varepsilon = L_{h_\varepsilon\circ T_n}=L_{h_\varepsilon}\,L_{T_n}=L_{h_\varepsilon}L_n.\] Hence \[\frac{L_n^\varepsilon\mu_n-L_n\mu_n}{\varepsilon} = \frac{L_{h_\varepsilon}(L_n\mu_n)-L_n\mu_n}{\varepsilon} = \frac{L_{h_\varepsilon}\mu_{n+1}-\mu_{n+1}}{\varepsilon}, \qquad\text{since }\mu_{n+1}=L_n\mu_n.\] By Lemma 6(ii), for \(u\in W^{2,1}\), \[\frac{L_{h_\varepsilon}u-u}{\varepsilon}\to Du \quad\text{in }W^{1,1}.\] Applying this with \(u=\mu_{n+1}\) and using \(\sup_n\|\mu_n\|_{W^{2,1}}<\infty\) (hence also for \(\mu_{n+1}\)) yields \[\sup_{n\in\mathbb{Z}}\left\| \frac{L_n^\varepsilon\mu_n-L_n\mu_n}{\varepsilon}-D\mu_{n+1} \right\|_{W^{1,1}} = \sup_{n\in\mathbb{Z}}\left\| \frac{L_{h_\varepsilon}\mu_{n+1}-\mu_{n+1}}{\varepsilon}-D\mu_{n+1} \right\|_{W^{1,1}} \to 0,\] which proves (a) with \(g_n=D\mu_{n+1}\) (equivalently \(g_n=D(L_n\mu_n)\)). Moreover, the uniform bound \(\sup_n\|g_n\|_{W^{1,1}}<\infty\) follows from Lemma 6(i) and the uniform \(W^{2,1}\) bound on \(\mu_n\). Item (b) follows from Proposition 11.  

5.3 Conclusion: linear response for sequential expanding maps with post-composition kicks↩︎

Proof. The exponential loss of memory, Assumption 3, is verified under the hipotesis of the statement in Section 5.1. The uniform strong bounds required in Assumption 3 was verified in Section 5.2.2. The differentiability, Assumption 2 was verified in Section 5.2.3. The application of Theorem 5 directly lead to the statement.  

6 Sequential composition of systems with additive noise↩︎

In this section we apply the abstract linear response theorem (Theorem 5) to sequential random maps on \(\mathbb{S}^1\) with i.i.d.additive noise, in the annealed setting.

We introduce the kind of systems we are considering. Let \((\xi_n)_{n\in\mathbb{Z}}\) be i.i.d.random variables on \(\mathbb{S}^1\) with common Lipschitz density \(q\in Lip(\mathbb{S}^1)\). We will assume that the noise density is uniformly positive: there exists some \(\alpha >0\) for which \[\label{eq:noise95positive} q(y)\ge \alpha>0\quad\text{for m-a.e.\;}y\in\mathbb{S}^1.\tag{17}\]

Let \((f_n^\varepsilon)_{n\in\mathbb{Z}}\) be a family of measurable maps \(f_n^\varepsilon:\mathbb{S}^1\to\mathbb{S}^1\).

We now impose a differentiability assumption ensuring Assumption 2 with \(B_s=B_w=L^1\). Assume that for each \(n\in\mathbb{Z}\) the perturbation \(\varepsilon\mapsto f_n^\varepsilon\) is uniformly differentiable in the sense that there exists \(\dot{f}_n\in L^\infty(\mathbb{S}^1)\) and a remainder \(r_n^\varepsilon\) such that \[\label{eq:fn95eps} f_n^\varepsilon = f_n^0 + \varepsilon\,\dot{f}_n + r_n^\varepsilon, \qquad \sup_n\|\dot{f}_n\|_{L^\infty}<\infty, \qquad \sup_n\frac{\|r_n^\varepsilon\|_{L^\infty}}{|\varepsilon|}\xrightarrow[\varepsilon\to0]{}0.\tag{18}\]

Consider the random dynamical system \[X_{n+1} = f_n^\varepsilon(X_n) + \xi_n \quad (\mathrm{mod}\;1).\]

The associated annealed (Markov) kernel is \[K_n^\varepsilon(x,dy) = q\bigl(y-f_n^\varepsilon(x)\bigr)\,m(dy),\] and the corresponding annealed transfer operator \(L_n^\varepsilon:L^1(m)\to L^1(m)\) is \[\label{eq:annealed95L} (L_n^\varepsilon \varphi)(y) := \int_{\mathbb{S}^1} \varphi(x)\,q\bigl(y-f_n^\varepsilon(x)\bigr)\,dm(x).\tag{19}\] Each \(L_n^\varepsilon\) is positive and preserves mass: \[\label{eq:mass95preserving95noise} \int L_n^\varepsilon \varphi\,dm = \int \varphi\,dm,\qquad \varphi\in L^1(m).\tag{20}\] We set \(B_s=B_w=L^1(m)\) and \(V_s:=\{\varphi\in L^1:\int\varphi\,dm=0\}\).

Since \((\xi_n)\) is i.i.d.and independent of the past, \((X_n)\) is a (time-inhomogeneous) Markov chain with one-step kernel \(K^\varepsilon_n(x,A)=\mathbb{P}(f^\varepsilon_n(x)+\xi_n\in A)\); hence, by Chapman–Kolmogorov, the time-marginals satisfy \(\mu^\varepsilon_{n+1}=L^\varepsilon_n\mu^\varepsilon_n\), i.e.the natural nonautonomous stationary (reference) state is the equivariant sequence \(\mu^\varepsilon_{n+1}=L^\varepsilon_n\mu^\varepsilon_n\). This state is also unique, since, as we will see in the following, under our assumptions, the system has rapid loss of memory (Proposition 2).

Let us now we show that the system has exponential loss of memory, and verify the other assumptions in order to apply Theorem 5. Since \(q(y)\) is uniformly positive by assumption, the Doeblin minorization ?? holds with the same \(\alpha\) for every \(n\) and every \(\varepsilon\) (because \(k_n^\varepsilon(x,y)=q(y-f_n^\varepsilon(x))\ge \alpha\)), hence by Theorem 9 (Appendix A) we obtain for all \(m<n\) and all \(h\in L^1_0(m)\), \[\label{eq:ELoM95noise} \|L_{n-1}^\varepsilon\cdots L_m^\varepsilon\,h\|_{L^1} \le (1-\alpha)^{n-m}\,\|h\|_{L^1}.\tag{21}\] Thus Assumption 3 holds with \(C=1\) and \(\rho=1-\alpha\) (uniformly in \(\varepsilon\in[0,\varepsilon_0\))).

An equivariant family \(\boldsymbol{\mu}^\varepsilon=(\mu_n^\varepsilon)_{n\in\mathbb{Z}}\) is a in this case is a sequence of densities \(\mu_n^\varepsilon\in L^1(m)\) such that \(\mu_n^\varepsilon\ge 0\), \(\int\mu_n^\varepsilon dm=1\), and \[\mu_{n+1}^\varepsilon = L_n^\varepsilon \mu_n^\varepsilon.\] When 21 holds, the pullback construction yields existence and uniqueness of such an equivariant family (with boundedness immediate since \(\|\mu_n^\varepsilon\|_{L^1}=1\)). In particular, Assumption 1 holds with \(M_s=1\). Now we address Assumption 2 and the differentiability of the family of operators. Since the density \(q\) is Lipschitz then translations are \(L^1\)-Lipschitz: \[\label{eq:translation95L1} \|q(\cdot-a)-q(\cdot-b)\|_{L^1}\le \|q'\|_{L^1}\,|a-b|.\tag{22}\]

We have by 18 and 22 that \[\label{eq:dotL95along95mu95noise} g_n := \lim_{\varepsilon\to0}\frac{L_n^\varepsilon\mu_n - L_n^0\mu_n}{\varepsilon} \quad\text{is converging in }L^1,\tag{23}\] furthermore \[\label{eq:strong95derivative95noise} \lim_{\varepsilon\to0}\sup_{n\in\mathbb{Z}} \left\|\frac{L_n^\varepsilon\mu_n - L_n^0\mu_n}{\varepsilon}-g_n\right\|_{L^1}=0, \qquad \sup_{n\in\mathbb{Z}}\|g_n\|_{L^1}<\infty.\tag{24}\] (If \(q\) is \(C^1\) and 18 holds, one may identify \(g_n\) explicitly as \(g_n(y)=\int \mu_n(x)\,[-q'(y-f_n(x))\,\dot{f}_n(x)]\,dx\), and \(\sup_n\|g_n\|_{L^1}\le \|q'\|_{L^1}\sup_n\|\dot{f}_n\|_\infty\).)

Finally, the auxiliary continuity requirement in Assumption 2 holds because, for each fixed \(v\in L^1\), \[\|(L_n^\varepsilon-L_n^0)v\|_{L^1}\le \|v\|_{L^1}\,\|q(\cdot-f_n^\varepsilon(\cdot))-q(\cdot-f_n^0(\cdot))\|_{L^1(dy)} \to 0\] uniformly in \(n\) by 18 and 22 . ⁶

Therefore Assumption 2 holds with \(B_s=B_w=L^1\). Applying Theorem 5 we finally get

Proof. Mass preservation 20 holds. Since in this case \(B_w=L^1\) the assumption (A2) is satisfied with \(M_w=1\). Uniform exponential loss of memory on \(L^1_0\) follows from Theorem 9 applied to the strictly positive kernel 17 , yielding 21 . Uniform bounds for the equivariant family hold with \(M_s=1\) since \(\|\mu_n^\varepsilon\|_{L^1}=1\). Assumption 2 is ensured by 24 and the \(L^1\) continuity argument above. Therefore Theorem 5 applies and yields the claimed conclusions.  

7 Discussion and Conclusions↩︎

In this work, we have shown that it is possible to develop a response theory for time-dependent systems, posing rigorous foundations to the formal calculations and numerical results presented in [77]. Our framework is axiomatic. Our results are a first general step in the direction of providing a meaningful extension of response theory for nonautonomous systems, providing results that apply to random and deterministic sequential systems having a general time-dependence (which could be periodic or aperiodic) and in some sense extend previous findings on response theory for stochastic differential equations having time periodic coefficients [76].

Our strategy of proof revolves around requesting loss of memory for the reference system, which allows to define a unique equivariant measure supported on the pullback attractor, and defining a generalized transfer operator that acts on sequences of measures.

The required conditions a) uniform regularity of equivariant measures; b) differentiable perturbation; and c) exponential loss of memory; somewhat resemble and adapt to the nonautonomous case the typical conditions used to obtain the linear response in the autonomous case. We discuss two explicit examples where our framework applies, namely the case of sequential expanding maps and the composition of random maps with large enough noise. We believe that, in addition to the application examples presented in the article, the axiomatic framework also applies, for example, to the sequential composition of uniformly hyperbolic deterministic systems, by having the associated operators act on suitable anisotropic spaces (see [31]). Anisotropic Banach spaces are tailored to hyperbolic dynamics: they encode different regularity along expanding and contracting directions, and thus allow one to treat SRB-type equivariant states supported on (possibly fractal) pullback attractors within the same strong/weak operator framework used here. This answers one of the key challenges mentioned in the introduction.

We conclude by highlighting two important aspects that deserve attention in separate publications. So far we have studied under which conditions it is possible to establish a response theory and predict the impact of small perturbations to the dynamics via explicit response operators. In [16], [17] it is explained that in the case of autonomous dynamics, the breakdown of response theory is intimately associated with the closure of the transfer operator gap in suitably defined spaces (in that case, \(L^2\)) and with the appearance of criticality associated with tipping behavior [85], [86]. Indeed, the framework developed in this paper - or, more specifically, the study of the conditions under which the perturbative theory developed here fails - might instead pave the way for understanding comprehensively more complex critical behaviour like that associated with the so-called rate-induced [87], [88] and phase-induced [89] tipping, which are of great relevance in assessing, e.g., climate and ecosystems stability and resilience [90], [91].

Finally, a very attractive angle on response theory goes under the umbrella names of optimal response or linear request. It is a bottom-up effort, whereby one asks what is the most efficient way (i.e., how to choose the cheapest, in some norm, perturbation), to achieve a desired change in the statistical properties the system. A growing body of literature has been studying this statistical control problem for the case of autonomous dynamics [42], [55], [84], [92]–[97].

Given the clear theoretical as well as practical relevance of this approach to the problem of studying the response of a system to perturbations, it seems highly relevant to investigate whether these results can be extended to the case of non-autonomous reference dynamics.

8 Uniform exponential loss of memory for composition of operators satisfying a common Lasota Yorke inequality.↩︎

In this section, we show a relatively simple and general argument (see [98] or [99] for similar ones) that establishes exponential loss of memory for a sequential composition of operators which are nearby or slowly varying in a mixed topology. We first consider sequential compositions of operators which are all close, in the mixed strong–weak topology, to a fixed reference operator \(L_0\) with convergence to equilibrium. We then show that the same argument applies to slowly varying families, provided the convergence-to-equilibrium assumption is uniformly satisfied along the family.

Let \(B_{w}\) and \(B_{s}\) be normed vector spaces of signed measures on \(X\). Suppose \((B_{s},||~||_{s})\subseteq\) \((B_{w},||~||_{w})\) and \(||~||_{s}\geq ||~||_{w}\). Let us consider a sequence of bounded linear Markov operators \(\{L_{i}\}_{i\in \mathbb{N}}:B_{s}\rightarrow B_{s}.\) We will suppose furthermore that the following assumptions are satisfied by the \(L_{i}\):

• The operators \(L_{i }\) satisfy a common "one step" Lasota-Yorke inequality. There are constants \(B,\lambda _{1}\geq 0\) with \(\lambda _{1}<1\), \(B\geq 1\), such that for all \(f\in B_{s},\) \(\mu \in P_{w},\) \(i\in \mathbb{N}\)\[\left\{ \begin{array}{c} ||L_{i}f||_{w}\leq ||f||_{w} \\ ||L_{i}f||_{s}\leq \lambda _{1}||f||_{s}+B||f||_{w}.\end{array}\right. \label{1}\tag{25}\]

• There exists \(M\) large enogh, such that \(\lambda_1^{M}\leq \frac{1}{10(\frac{B}{1-\lambda_1}+1)}\) and

  \[||L_{0}^{M}(v)||_{w}\leq \frac{1-\lambda_1}{10B} ||v||_{s} \label{3}\tag{26}\] for each \(v\in V_s\), where\[V_{s}=\{\mu \in B_{s}|\mu (X)=0\}.\]

• The family of operators is near to \(L_0\), satisfying: \(\forall i\), \[||L_{i}-L_{0}||_{B_{s}\rightarrow B_{w}}\leq \frac{7(1-\lambda_1 )^2}{10M B({\frac{1}{1-\lambda_1}+B})}.\]

We remark that the assumption \((ML1)\) implies that the family of operators \(L_{i}\) is uniformly bounded when acting on \(B_{s}\) and on \(B_{w}.\) Furthermore, by the Markov property, \(\forall i,\) \(L_{i }(V_{s})\subseteq V_{s}\).

First, we state a Lasota-Yorke inequality for a sequential composition of operators satisfying \((ML1)\). The statement directly follows from the definitions.

The following lemma is an estimate for the distance of the sequential composition of operators from the iterations of \(L_0\).

Proof. By the assumptions we get\[||L_0g-L_{j}g||_{w}\leq \delta ||g||_{s}\]

hence the case \(n=1\) of \((\ref{2})\) is trivial. Let us now suppose inductively\[||L^{(j,j+n-2)}g-L_{0}^{n-1}g||_{w}\leq \delta (C_{n-1}||g||_{s}+(n-1)\frac{B}{1-\lambda _{1}}||g||_{w})\]then\[\begin{align} ||L_{j+n-1}L^{(j,j+n-2)}g-L_{0}^{n}g||_{w} &\leq &||L_{j+n-1}L^{(j,j+n-2)}g-L_{j+n-1}L_{0}^{n-1}g+L_{j+n-1}L_{0}^{n-1}g-L_{0}^{n}g||_{w} \\ &\leq &||L_{j+n-1}L^{(j,j+n-2)}g-L_{j+n-1}L_{0}^{n-1}g||_{w}+||L_{j+n-1}L_{0}^{n-1}g-L_{0}^{n}g||_{w} \\ &\leq &\delta (C_{n-1}||g||_{s}+(n-1)\frac{B}{1-\lambda _{1}}||g||_{w})+||[L_{j+n-1}-L_{0}](L_{0}^{n-1}g)||_{w} \\ &\leq &\delta (C_{n-1}||g||_{s}+(n-1)\frac{B}{1-\lambda _{1}}||g||_{w})+\delta ||L_{0}^{n-1}g||_{s} \\ &\leq &\delta (C_{n-1}||g||_{s}+(n-1)\frac{B}{1-\lambda _{1}}||g||_{w}) \\ &&+\delta (\lambda _{1}^{n-1}||g||_{s}+\frac{B}{1-\lambda _{1}}||g||_{w}) \\ &\leq &\delta \lbrack (C_{n-1}+\lambda _{1}^{n-1})||g||_{s})+n\frac{B}{1-\lambda _{1}}||g||_{w}]. \end{align}\]

The statement follows from the observation that, continuing the composition, \(C_{n}\) remains bounded by the sum of a geometric series.  

Proof. Remark that because of the Lasota-Yorke inequality, \(\forall j,i\geq 1,g\in B_s\) \[\label{sop24} ||L^{(j,j+i)}(g)||_s\leq (\frac{B}{1-\lambda_1}+1)||g||_s.\tag{27}\] Now by \((ML2)\) let us consider \(M\) such that \(\lambda_1^{M}\leq \frac{1}{10(\frac{B}{1-\lambda_1}+1)}\) and , \(M\) such that \(\forall i\geq M,g\in V_s\) \[||{L_0}^{M}g||_{w}\leq \frac{1-\lambda_1}{10B}||g||_s.\]

Since \[||L_{i}-L_0||_{s\rightarrow w}\leq \frac{7(1-\lambda_1 )^2}{10M B({\frac{1}{1-\lambda_1 }+B})}\] for all \(i\). By \((\ref{2})\), \(\forall j\geq M, i\geq M\) \[\begin{align} ||L^{(j,j+i-1)}g-{L_0 }^{i}g||_{w} & \leq \frac{7(1-\lambda_1 )^2}{10MB(\frac{1}{1-\lambda_1 }+B)}(\frac{1}{1-\lambda_1 }||g||_s+i\frac{B||g||_w}{(1-\lambda_1 )}) \\ & \leq \frac{7i(1-\lambda_1 )}{10MB}||g ||_s. \end{align}\] Hence \[\begin{align} ||L^{(j,j+M-1)}g||_w &\leq ||{L_0 }^{M}g||_w+ \frac{7M (1-\lambda_1 )}{10MB}||g ||_s \\ &\leq \frac{1-\lambda_1}{10B}||g||_s + \frac{7M(1-\lambda_1 )}{10MB}||g ||_s. \end{align}\]

Applying now the Lasota-Yorke inequality and the estimate 27 we get, for any \(j\geq M\) \[\label{222} \begin{align} ||L^{(j,j+2M-1)}g\Vert _{s} & \leq \lambda _{1}^{M}||L^{(j,j+M-1)}g||_s+\frac{B}{1-\lambda_1}\Vert L^{(j,j+M-1)}g\Vert _{w} \\ &\leq \frac{1}{10}||g||_s+\frac{B}{1-\lambda_1} \frac{1-\lambda_1}{10B}||g||_s+\frac{7BM (1-\lambda_1 )}{(1-\lambda_1)10MB}||g ||_s \\ &\leq\frac{9}{10}||g||_s \end{align}\tag{28}\] and \[||L^{(j,j+2kM-1)}g\Vert _{s}\leq(\frac{9}{10})^k||g||_s\] for each \(j\geq N_1\) and \(k\geq 1\), \(g\in V_s\) establishing the result.  

We now formalize a consequence of the proof of Lemma 10, which is useful when the reference operator is not fixed, but changes slowly along the sequence. The idea is that the proof of Lemma 10 is local in time: on a block of length \(M\), the sequence only has to be close to a single “frozen” operator, namely the operator at the beginning of the block. The assumptions we will take and the notion of "slow enough change" will be explicitly verifiable in concrete example.

We hence formalize below the slightly different assumptions to be taken in this case. We will assume again the common Lasota–Yorke inequality \((ML1)\). We replace the fixed convergence-to-equilibrium assumption \((ML2)\) by the following uniform "frozen" version.

• There exists \(M\geq 1\) such that \[\lambda_1^M \left( \frac{B}{1-\lambda_1}+1 \right) \leq \frac{1}{10},\] and, for every \(j\in\mathbb{N}\) and every \(v\in V_s\), \[\|L_j^M v\|_w \leq \frac{1-\lambda_1}{10B}\|v\|_s .\]

This means that each frozen operator \(L_j\) has an \(M\)-step convergence to equilibrium estimate, with constants independent of \(j\).

We also assume that the sequence is slowly varying on blocks of length \(M\).

• For every \(j\in\mathbb{N}\) and every \(\ell=0,\ldots,M-1\), \[\|L_{j+\ell}-L_j\|_{B_s\to B_w} \leq \delta_M,\] where \(\delta_M>0\) is small enough so that \[\delta_M \left( \frac{1}{1-\lambda_1} + \frac{MB}{1-\lambda_1} \right) \leq \frac{7(1-\lambda_1)}{10B}.\]

Equivalently, if one assumes the stepwise slow variation estimate \[\|L_{n+1}-L_n\|_{B_s\to B_w}\leq \eta \qquad \forall n,\] then \((ML3')\) follows whenever \(M\eta\leq \delta_M\).

By the same proof of Lemma 10 (see 28 ) one can obtain in this slightly different setting the following

directly implying

Showing that the argument of Lemma 10 does not require the whole sequence to remain close to one fixed operator \(L_0\). It is enough that, on each block of length \(M\), the sequence remains close to the frozen operator at the beginning of that block, and that all frozen operators satisfy the same \(M\)-step convergence-to-equilibrium estimate on \(V_s\).

8.1 Uniformly positive kernels imply exponential loss of memory in \(L^1\)↩︎

Let \((X,\mathcal{A})\) be a measurable space and let \(m\) be a reference probability measure on \(X\). We identify absolutely continuous probability measures \(\mu\ll m\) with their densities \(f=d\mu/dm\in L^1(m)\). A (time-dependent) Markov kernel is a family of transition probabilities \[K_n(x,dy),\qquad n\in\mathbb{Z},\] and the associated (annealed) transfer operator acting on \(L^1(m)\) is \[(L_n f)(y) := \int_X f(x)\,k_n(x,y)\,dm(x), \qquad\text{where } K_n(x,dy)=k_n(x,y)\,m(dy).\] Each \(L_n\) is positive and preserves mass: \(\int L_n f\,dm=\int f\,dm\).

Proof. Fix \(n\) and set \(\widetilde{K}_n(x,dy):=\frac{1}{1-\alpha}\bigl(K_n(x,dy)-\alpha m(dy)\bigr)\). Assumption 4 implies that \(\widetilde{K}_n(x,\cdot)\) is a Markov kernel (nonnegative and integrating to \(1\)), and we can write the convex decomposition \[\label{eq:kernel95split} K_n(x,dy) = \alpha\, m(dy) + (1-\alpha)\,\widetilde{K}_n(x,dy).\tag{29}\] Let \(\widetilde{L}_n\) be the transfer operator associated to \(\widetilde{K}_n\); then \[\label{eq:operator95split} L_n f = \alpha\Big(\int_X f\,dm\Big)\mathbf{1} + (1-\alpha)\widetilde{L}_n f,\tag{30}\] where \(\mathbf{1}\) denotes the constant function equal to \(1\) on \(X\).

Now let \(h\in L^1_0(m)\) (so \(\int h\,dm=0\)). By 30 , \[L_n h = (1-\alpha)\widetilde{L}_n h.\] Since \(\widetilde{L}_n\) is Markov, it is \(L^1\)-nonexpanding: \(\|\widetilde{L}_n h\|_{L^1}\le \|h\|_{L^1}\). Hence \[\|L_n h\|_{L^1} \le (1-\alpha)\|h\|_{L^1}.\] Iterating from \(m\) to \(n-1\) gives \[\|L_{n-1}\cdots L_m h\|_{L^1} \le (1-\alpha)^{n-m}\|h\|_{L^1}.\] Finally, for \(f,g\) with equal mass we have \(h=f-g\in L^1_0(m)\), yielding ?? .  

9 Evaluation of the constants for the perturbative estimate for expanding maps↩︎

In this section we show an explicit estimate for the mixed norm Lipschitz constant \(C(T_0)\) appearing in 11 . The results of this section combined with the ones presented in Section 5 can produce explicit examples of sequential expanding maps for which our linear response results hold.

Let \(T_0,T_1:\mathbb{S}^1\to\mathbb{S}^1\) be \(C^3\) expanding maps of the same degree \(n\ge 2\). Assume \(T_0\) is uniformly expanding: \[\label{eq:base95constants} \lambda_0:=\inf_{x\in\mathbb{S}^1}|T_0'(x)|>1,\qquad M_0:=\|T_0'\|_\infty<\infty,\qquad M_2:=\|T_0''\|_\infty<\infty.\tag{31}\] Assume \(T_1\) is \(C^2\)-close to \(T_0\): \[\label{eq:C295close95covering} \delta:=\|T_1-T_0\|_{C^2}\le \lambda_0 -1,\tag{32}\] so that \(\inf|T_1'|> 1\) and \(\|T_1'\|_\infty\le M_0+\delta\).

Since \(T_i\) is a covering of degree \(n\), there exist \(n\) global \(C^3\) inverse branches \(h_{i,j}:\mathbb{S}^1\to \mathbb{S}^1\) (\(j=1,\dots,n\)) such that \[T_i\circ h_{i,j}=\mathrm{id}_{\mathbb{S}^1},\qquad i\in\{0,1\}.\] The Perron–Frobenius operator \(L_i\) admits the representation \[\label{eq:PF95covering} (L_i f)(x)=\sum_{j=1}^n f(h_{i,j}(x))\,g_{i,j}(x), \qquad g_{i,j}(x):=\frac{1}{|T_i'(h_{i,j}(x))|}.\tag{33}\]

Proof. Fix \(x\in\mathbb{S}^1\) and set \(y_0=h_{0,j}(x)\), \(y_1=h_{1,j}(x)\). Then \(T_0(y_0)=x=T_1(y_1)\), hence \[|T_0(y_1)-T_0(y_0)|=|T_0(y_1)-T_1(y_1)|\le \|T_0-T_1\|_{C^0}.\] By the mean value theorem and \(\inf|T_0'|\ge \lambda_0\), we obtain \(\lambda_0|y_1-y_0|\le \|T_0-T_1\|_{C^0}\), which implies the claim.  

Proof. Fix \(x\in\mathbb{S}^1\) and write \(y_0=h_{0,j}(x)\), \(y_1=h_{1,j}(x)\). Then \[|g_{0,j}(x)-g_{1,j}(x)| \le \left|\frac{1}{|T_0'(y_0)|}-\frac{1}{|T_0'(y_1)|}\right| + \left|\frac{1}{|T_0'(y_1)|}-\frac{1}{|T_1'(y_1)|}\right|.\] For the first term, since \(u\mapsto 1/u\) is \(1/\lambda_0^2\)–Lipschitz on \([\lambda_0,\infty)\), \[\left|\frac{1}{|T_0'(y_0)|}-\frac{1}{|T_0'(y_1)|}\right| \le \frac{1}{\lambda_0^2}\,|T_0'(y_0)-T_0'(y_1)| \le \frac{M_2}{\lambda_0^2}\,|y_0-y_1|.\] Using Lemma 12 gives \[\left|\frac{1}{|T_0'(y_0)|}-\frac{1}{|T_0'(y_1)|}\right| \le \frac{M_2}{\lambda_0^3}\|T_0-T_1\|_{C^0}\le \frac{M_2}{\lambda_0^3}\delta.\] For the second term, by 32 we have \(|T_0'(y_1)|\ge \lambda_0\) and \(|T_1'(y_1)|\ge 1\), hence \[\left|\frac{1}{|T_0'(y_1)|}-\frac{1}{|T_1'(y_1)|}\right| = \frac{|T_1'(y_1)-T_0'(y_1)|}{|T_0'(y_1)T_1'(y_1)|} \le \frac{\|T_1'-T_0'\|_\infty}{\lambda_0} = \frac{1}{\lambda_0}\|T_0-T_1\|_{C^1} \le \frac{1}{\lambda_0}\delta.\] Combining the bounds yields the claim.  

Proof. For each \(x\), the fundamental theorem of calculus gives \[|f(h_{0,j}(x))-f(h_{1,j}(x))| \le \int_{h_{1,j}(x)}^{h_{0,j}(x)} |f'(t)|\,dt.\] Integrate over \(x\in\mathbb{S}^1\) and apply Fubini: \[\int_{\mathbb{S}^1}|f(h_{0,j}(x))-f(h_{1,j}(x))|\,dx \le \int_{\mathbb{S}^1}|f'(t)|\,\bigl|\{x: t\in [h_{1,j}(x),h_{0,j}(x)]\}\bigr|\,dt.\] If \(t\) lies between \(h_{1,j}(x)\) and \(h_{0,j}(x)\), then \(x=T_1(h_{1,j}(x))\) lies within distance at most \(\|T_1'\|_\infty \|h_{0,j}-h_{1,j}\|_\infty\) of \(T_1(t)\).

For \(A_t:=\{x:\;t\in[\min(h_0(x),h_1(x)),\max(h_0(x),h_1(x))]\}\) we have \(A_t\subseteq T_1([t-\Delta,t+\Delta])\) where \(\Delta:=\|h_0-h_1\|_\infty\), hence \(m(A_t)\le 2\|T_1'\|_\infty\,\Delta\).

Substituting this into the previous inequality yields the first bound. The second bound follows from Lemma 12.  

Proof. From 33 , \[(L_0-L_1)f =\sum_{j=1}^n \bigl(f\circ h_{0,j}-f\circ h_{1,j}\bigr)\,g_{0,j} + \sum_{j=1}^n (f\circ h_{1,j})\,(g_{0,j}-g_{1,j}) =:A+B.\] For \(A\), use \(\|g_{0,j}\|_\infty\le 1/\lambda_0\) and Lemma 14: \[\|A\|_{L^1}\le \sum_{j=1}^n \|g_{0,j}\|_\infty\,\|f\circ h_{0,j}-f\circ h_{1,j}\|_{L^1} \le 2n\cdot \frac{1}{\lambda_0}\cdot \frac{M_0+\delta}{\lambda_0}\,\delta\,\|f'\|_{L^1}.\] Since \(\delta\le \lambda_0 -1\), this gives \[\|A\|_{L^1}\le 2n\cdot \frac{M_0+\lambda_0 -1}{\lambda_0^2}\,\delta\,\|f'\|_{L^1}.\] For \(B\), by change of variables \(x=T_1(y)\) on the image of \(h_{1,j}\), \[\|f\circ h_{1,j}\|_{L^1}=\int_{\mathbb{S}^1}|f(h_{1,j}(x))|\,dx =\int_{h_{1,j}(\mathbb{S}^1)} |f(y)|\,|T_1'(y)|\,dy \le \|T_1'\|_\infty\,\|f\|_{L^1}\le (M_0+\delta)\|f\|_{L^1}.\] Thus, using Lemma 13, \[\|B\|_{L^1} \le \sum_{j=1}^n \|f\circ h_{1,j}\|_{L^1}\,\|g_{0,j}-g_{1,j}\|_\infty \le n\,(M_0+\delta)\left(\frac{M_2}{\lambda_0^3}+\frac{1}{\lambda_0}\right)\delta\,\|f\|_{L^1}.\] Again using \(\delta\le\lambda_0-1\) and \(\|f\|_{W^{1,1}}=\|f\|_{L^1}+\|f'\|_{L^1}\) yields the stated bound.  

Acknowledgements↩︎

VL wishes to thank M. Branicki for stimulating conversations. VL acknowledges the partial support provided by the Horizon Europe Projects Past2Future (Grant No. 101184070) and ClimTIP (Grant No. 100018693), by the ARIA SCOP-PR01-P003—Advancing Tipping Point Early Warning AdvanTip project, by the European Space Agency Project PREDICT (Contract 4000146344/24/I-LR), and by the NNSFC International Collaboration Fund for Creative Research Teams (Grant No. W2541005).

SG acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Pisa, CUP I57G22000700001.

Data Availability Statement↩︎

No datasets were generated or analysed during the current study.

Conflict of interest↩︎

The authors declare that they have no conflict of interest.

References↩︎