Linear response due to singularities

It is well known that a family of tent-like maps with bounded derivatives has no linear response for typical deterministic perturbations changing the value of the turning point. In this note we prove the following result: if we consider a tent-like family with a cusp at the turning point, we recover the linear response. More precisely, let T ɛ be a family of such cusp maps generated by changing the value of the turning point of T 0 by a deterministic perturbation and let h ɛ be the corresponding invariant density. We prove that ε↦hε is differentiable in L 1 and provide a formula for its derivative.


Introduction
Let M be a compact manifold with a reference volume and T : M → M be a map, whose iterates determine the dynamics.A T -invariant measure µ; i.e.T * µ = µ, is said to be physical if there is a positive volume set B such that for any continuous observable f : for all x ∈ B. An important question is to study how such measures vary, in an appropriate topology, under suitable perturbations of T .Namely, consider a family {(M, T ε )} ε∈V , where V is a small neighbourhood of 0, T 0 := T and T ε → T 0 , as ε → 0, in some suitable topology.For many chaotic systems it turns out that T ε admits a unique physical measure µ ε .
The system (M, T 0 , µ 0 ) is called statistically stable if the map ε −→ µ ε is continuous, in an appropriate topology, at ε = 0. Quantitative statistical stability is provided by quantitative estimates on its modulus of continuity (see e.g.[1,27,22,21,35,11]).In the case where ε −→ µ ε is differentiable in some sense, the system is also said to admit linear response; i.e., the 'derivative' μ0 represents the first order term of the response of the system to the perturbation where the error term is understood in an appropriate topology.
Linear response results in the context of deterministic dynamics was first obtained in the case of uniformly hyperbolic systems [32] (see also [26]).Nowadays linear response results are known for systems outside the uniformly hyperbolic setting (see [14,16,19,23,27], the survey article [8] and references therein.)For systems with discontinuities or critical points the situation is quite complicated even in the presence of uniform expansion.Indeed, for suitable small perturbations of piecewise expanding maps there are several examples that lack statistical stability or linear response (see [27,31,9] and [3] for recent results in this direction).In the literature there are some results indicating that perturbations which are not changing the topological class 1 or tangent to the topological class, linear response is likely to occur, while for perturbations which are transversal to the topological class there is no linear response (see [14] for the case of piecewise expanding unimodal maps, [10] and [15] for smooth unimodal maps and [23] for results along this line in the case of rotations).
In this paper we study perturbations of one dimensional tent-like piecewise expanding maps with unbounded derivatives at the turning point (a cusp singularity).We prove linear response for a large class of deterministic perturbations also changing the image at the turning point and hence the topological class.The existence of such a linear response is due to the singularity.Indeed, unlike the usual tent maps with bounded derivative, which do not admit linear response for the same kind of perturbations ( [8,31]), linear response in the singular case is attained due to fact that the cusp has a 'regularizing effect' at the level of the action of the associated transfer operators.The present paper studies this regularization phenomenon due to a singularity, in the relatively simple case of tent-like maps.We believe however that this phenomenon might also appear in other important classes of dynamical systems and this motivates further studies in this direction.
Cusp like singularities and unbounded derivatives appear in several important systems including Lorenz-type maps [5,6] and billiard maps (see e.g.[13]).Unlike the case of Anosov flows [18] where linear and higher order response is proved, in the case of the classical Lorenz flow only statistical stability is known [2,5,6] although numerical evidence suggests that the Linear Response may hold in this case ( [33], [34]).By highlighting the role of singularities in studying linear response, we hope that the results of the present paper contribute to the understanding of linear response for Lorenz-like flows and billiard maps.
The paper is organised as follows.In Section 2 we introduce the class of systems we consider and state the main result, Theorem 1, of the paper.In Section 3 we prove Theorem 1 in a series of lemmas.In Section 4 a concrete family of maps satisfying the assumptions of Section 2. Concluding remarks are presented in Section 5.
Acknowledgements.The research of W. Bahsoun is supported by EPSRC grant EP/V053493/1.W. Bahsoun would like to thank the University of Pisa for its hospitality during his visit to S. Galatolo.The research of S.G. was partially supported by the research project "Stochastic properties of dynamical systems" (PRIN 2022NTKXCX) of the Italian Ministry of Education and Research and by a grant from MIUR (Dipartimenti di Eccellenza DM 11/05/2017, n. 262).The authors thank M. Ruziboev, D. Smania and V. Baladi for fruitful discussions and for suggesting useful references.The authors also thank anonymous referees for careful reading and useful comments.

Family of maps and statement of the main result
For ε ∈ [0, δ), δ > 0, consider a family T ε : [0, 1] → [0, 1] of nonsingular maps, with respect to Lebesgue measure m on [0, 1].The transfer operator associated 1 A perturbation does not change the topological class if it changes a system to a system which is topologically conjugated to it with T ε , denoted by L Tε , is defined by duality as follows: for 4 ), we assume that T ε satisfies the following assumptions: (A1) T 0,ε := The power law divergence of the derivatives of T ε is uniform in ε: 1 we require that the transfer operators are close in a mixed norm when ε is small: A concrete family of maps satisfying assumptions (A1), ..., (A9) is presented in Section 4, see also Figure 1 for an example of a typical graph in such a family.Although it is known that L Tε admits a spectral gap when acting on the space of functions of generalized bounded variations [28], we study the action of L Tε on finer Banach spaces.Namely, the Sobolev spaces W i,1 , i = 1, 2. In particular, we show that L Tε admits a spectral gap on W i,1 , i = 1, 2. This will allow us to conclude that T ε admits an invariant density which is regular enough, in x.The latter is essential to derive the linear response formula that we are after in this work.The following result is the main result of the paper.
Also in this assumption we consider the weak norm to be L2 .The reason for this will become apparent in Section 3. with and the o is in the L 1 -topology.

Proof of Theorem 1
In this section we prove Theorem 1 in two steps.First, in subsection 3.1 we show that L Tε admits a uniform, in ε, spectral gap when acting on W 1,1 and W 2,1 .This implies, in particular, that T ε has a unique invariant density h ε ∈ W 2,1 .Then in subsection 3.2 we show that ε → h ε is differentiable in L 1 and obtain a formula for the derivative.
3.1.Uniform spectral gap on W 1,1 and on W 2,1 .In this section we prove that the transfer operators associated with our class of systems admit a uniform spectral gap when acting on suitable Sobolev spaces (Lemma 1).We obtain this as a consequence of classical results which are recalled below.
Lemma 1.Let (B, ∥ ∥ s ) be a Banach space, let ∥ ∥ w be a continuous semi-norm on B and Q a bounded linear operator on B, such that for any sequence x n with ∥x n ∥ s ≤ 1 it contains a Cauchy subsequence for ∥ ∥ w .
Assume there is λ ≥ 0 and C > 0 such that, for any Then the essential spectral radius of Q is bounded by λ.
We also recall the main result of [29], stating it in a simplified form suitable for our purposes.
Lemma 2. Let (B, ∥ ∥ s ) and ∥ ∥ w as above, and P ε : B → B with ε ≥ 0 be a family of bounded linear operators.Assume: there are C 1 > 0 and M ≥ 1 such that where τ (ε) → 0 monotonically and upper semicontinuously.Then there are ε 0 , a > 0 such that for all 0 ≤ ε ≤ ε 0 and f ∈ B 3.1.2.Uniform spectral gap for the associated transfer operators.Let We now state the main result of this section.
Proposition 1.There is δ 2 > 0 such that for any ε ∈ [0, δ 2 ) L Tε admits a unique invariant density h ε ∈ W 2,1 .Furthermore, L Tε has a uniform, in ε, spectral gap when acting on W 1,1 and W 2,1 .In particular, ∃ C > 0 such that for all ε ∈ [0, δ) To prove Proposition 1 we apply Lemma 1 and Lemma 2, we first prove few lemmas that will be used for this purpose and we start with an auxiliary lemma that verifies condition 3.3 in Lemma 2. First, consider the pointwise representation of the transfer operator associated with T ε Proof.By (3.7) for f ∈ W 1,1 , we have Notice that L Tε f ∈ W 1,1 .Indeed since f is bounded, by (3.7) lim x→aε L Tε f = 0. L Tε f is then continuous, with derivative almost everywhere.Furthermore, we note that in (3.8) when x → a ε then y → c and4 T ′′ By (3.9) we get where for each ε, λ Note that furthermore by assumptions (A4) and (A8) the quantities ∥ 1 The choice of L 2 as a weak space is due to the way the Hölder inequality is applied in 3.13, and to the fact that with the standing assumptions (A6), ..., (A8) imply ∥ By changing the weak space to another L p with p ≥ 2 one can allow different power law behavior for the singularity than the one in our assumptions and still get a Lasota Yorke inequality.Moreover, in such a setting, the compact embedding of the strong space into the weak one is still granted by the Rellich-Kondracov theorem.
Lemma 4.There exists M ≥ 0 such that for each ε ∈ [0, δ) where λ is the same as in Lemma 3.

□
We now show that the transfer operators are continuous in the L 2 norm.
Lemma 5.For each ε ∈ [0, δ) we have Proof.By (3.7) we have A similar estimate holds for ∥ψ 2,ε ∥ 2 L 2 .□ Proof of Proposition 1.Using Lemma 5, the Lasota Yorke inequalities proved in Lemma 3 and the fact that W 1,1 is compactly embedded in L 2 by the Rellich-Kondrakov theorem, applying Lemma 1, we get that the essential spectral radius ρ ess ≤ λ < 1.By this, any element of the spectrum with modulus strictly bigger than λ is an isolated eigenvalue.We now prove that spectral radius of L Tε , as an operator on W 1,1 , is 1.Since T • 1 = 1; 1 is in the spectrum of L Tε and in fact it is an eigenvalue (since ρ ess ≤ λ < 1).Moreover, L Tε cannot have eigenvalues ρ with |ρ| > 1 since L Tε is positive and it preserves integrals (see [12] properties of transfer operators).Therefore 1 is the spectral radius on L Tε as an operator on W 1,1 .
We now show that there are no other eigenvalues on the unit circle and that the eigenvalue 1 is simple.In the case of T 0 this is true since in (A5) we assume T 0 to be topologically mixing [35].Thus, L T0 has a spectral gap on W 1,1 .Since for any ε ∈ [0, δ), we proved in Lemma 3 a uniform (in ε) Lasota Yorke inequality for L Tε , by (A9) (2.1), the Keller-Liverani [29] spectral perturbation result implies that for sufficiently small 0 < ε ≤ δ 2 , the spectral projections corresponding to isolated eigenvalues of L Tε are in one to one correspondence with those of L T0 .Thus, 1 is also a simple eigenvalue for L Tε , for 0 < ε ≤ δ 2 , and L Tε as an operator on W 1,1 has no other eigenvalues on the unit circle.This in particular implies that T ε admits a unique invariant probability density h ε ∈ W 1,1 .
By Lemma 3, L Tε is bounded when acting on W 1,1 .Moreover, W 2,1 is compactly embedded in W 1,1 and thus Lemma 1 also implies the essential spectral radius of L Tε when acting on W 2,1 is smaller than λ.By the same reasoning as before we get that the spectral radius is 1.Since we have already proved that h ε is the unique invariant density in W 1,1 , it follows that h ε is the unique invariant probability density in W 2,1 and by the uniform Lasota Yorke inequality proved in Lemma 4 we have a uniform bound on its W 2,1 norm.
To prove (3.6) we apply Lemma 2 to L Tε when acting on W 1,1 0 .Considering ∥ ∥ L 2 as the weak norm.First notice that 1 is not in the spectrum of L T0 : W 1,1 0 → W 1,1 0 and by Lemma 1 the essential spectral radius of L Tε is bounded by λ.By Lemma 3 the operators L Tε indeed satisfy a uniform Lasota-Yorke Inequality.iterating this inequality and using (3.20) we get ).The application of Lemma 2 gives then directly (3.6).□ 3.2.Linear response derivation.Lemma 6. ε → h ε is differentiable, at ε = 0, in L 1 .In particular, with and the o is in the L 1 -topology.
Proof.We introduce the following notation: Since L Tε has a spectral gap on W 1,1 it eventually contracts exponentially on the subset of zero average functions W 1,1 0 and the following relation is well defined: Recall that h 0 ∈ W 2,1 .Therefore, the second part of assumption (A9) we have Thus, by (3.25), we have for some q ∈ W 1,1 0 , with the error o(ε) understood in W 1,1 .To obtain a formula for q, for x ∈ [0, a ε ), set g j,ε := T −1 j,ε (x) and consider To prove the statement for x ∈ [0, a ε ), we start from the relation T ε • g j,ε (x) = x and differentiate it with respect to ε and get ε .This provides the formula for q in (3.23).
To continue, recall that L Tε admits a uniform, in ε, spectral gap on W 1,1 .Therefore, G ε is uniformly bounded in L(W 1,1 0 , W 1,1 ) and we have where the above error is understood in W 1,1 .Moreover, by the stability result of [29], we get Using (3.26), (3.27) and (3.28) together with (3.24) we obtain in L 1 which proves differentiability of h ε and completes the proof of the theorem.□

4.
A family of maps satisfying assumptions (A1), ..., (A9) We provide an explicit example of a family of maps satisfying assumptions (A1), ..., (A9).Let ε ∈ [0, 1  10 ) and (4.1) . The graph of one member of this family is shown in Figure 1.In this example c = 1 2 and it is immediate to see that T ε satisfies (A1), (A2), (A3), (A5).Furthermore and , 1] and ε ∈ [0, 1  10 ).Thus, T ε satisfies (A4).Also T ′ ε satisfies (A6) with β = −7 8 .Computing the second derivative we get and thus the first part of (A7) is verified.Similarly one can proceed with the third derivative and verify the second part of (A7) and (A8).We now verify (A9).Let T 0,ε and T 1,ε be the branches of T ε , as in (A1).We have T i,ε = D ε • T i,0 , where D ε (x) = (1 − ε) (x).Let L Dε , L T0 denote the transfer operators associated with D ε and T 0 respectively.Note that have and for any x ∈ [0, 1], as ε → 0. Since we have already verified that T ε satisfies (A1), ..., (A8), by Lemma 4 we obtain that g := L T0 f ∈ W 1,1 when f ∈ W 1,1 .Furthermore there is an M > 0 such that ∥f ∥ W 1,1 ≤ 1 implies ∥g∥ W 1,1 ≤ M .Then, it is sufficient to prove that (4.2) sup as ε → 0. For brevity we will also denote L Dε g as We have Now changing the order of integration we get The estimate in (4.5) is explained in Figure 4 where the domains of integration are shown.We note that the left hand side integral in (4.5) is considered over the domain OAB in Figure 4 while the right hand side integral is considered over the larger domain OAC in the same figure.Therefore, The integration domains in (4.5).
and thus (4.2) is verified.Now for f ∈ W 2,1 we prove that the limit . By the same reasoning as above, using Lemma 4, and using the notation (4.3), we get this reduced to proving the existence of the limit lim ε→0 , where g = L T0 f as above and g ∈ W 2,1 .Now we have for each Since g ∈ W 2,1 ⊆ W 1,1 the convergence in L 1 of this limit can be proved as in (4.4).Now considering the derivative, since L Dε g(x) ∈ W 2,1 we have since g ′ ∈ W 1,1 we can repeat the same reasoning as in (4.5) and then get the convergence in L 1 of the derivative too, leading to the W 1,1 convergence of the limit considered in II.
Now we consider the limit related to I : We have Now changing the order of integration as in (4.5) we get By this where the last inequality holds since 0 ≤ t ≤ 1.On the other hand |xg ′ (x) − xh ′ (x)|dx ≤ ε 2 as before, and by uniform continuity of h ′ there is a function l(ε) such that l(ε) → 0 as ε → 0, such that |h ′ (t)−h ′ (x)| ≤ l(ε) and Err(ε) ≤ 2ε 2 + o(1) with ε 2 being arbitrary and o(1) → 0 as ε → 0, hence Err(ε) → 0 as ε → 0 and we established the L 1 convergence of I to the function x → xg ′ (x).Notice that we only used g ∈ W 1,1 in this proof.But for g ∈ W 2,1 then g ′ ∈ W 1,1 .This will be used in the next step, as now we are going to show the convergence of the same limit for the derivative.Indeed, we have which can be treated as before noting the similarity between the third line of (4.7) and I and the similarity between the fourth line of (4.7) and II.This concludes the verification of assumption (A9) for the family of maps in (4.1).

Concluding remarks
We presented a class of tent-like maps with singulartities where the presence of a cusp with a certain power law behavior induces a regularization at the level of the transfer operator associated to the map.This leads to a spectral gap of the transfer operator when acting on suitable Sobolev spaces.Unlike perturbations of tent maps in the bounded derivative case, we have shown that this induces linear response of the invariant density under deterministic perturbations of the system that changes the image of the critical point.
It would be interesting to explore how far regularization effect of singularities can play a role in obtaining linear response for more general systems with singularities, and perhaps with discontinuities.To implement a suitable generalization of the approach of this paper in more general settings, further technical work is needed (see for instance Remark 1 for a comment on the suitable choice of the weak space).