Abstract
We show a general relation between fixed point stability of suitably perturbed transfer operators and convergence to equilibrium (a notion which is strictly related to decay of correlations). We apply this relation to deterministic perturbations of a large class of maps with indifferent fixed points. It turns out that the L 1 dependence of the a.c.i.m. on small suitable deterministic changes for these kind of maps is Hölder, with an exponent which is explicitly estimated.
Introduction
The statistical stability of dynamical systems is well understood in the uniformly hyperbolic case (see e.g. [19] and related references). In this case, quantitative estimations are available, proving the Lipschitz or even differentiable dependence of the physical measure under perturbations. For systems having a non uniformly expanding/hyperbolic behavior, much less is known. Qualitative results and some quantitative (providing precise information on the modulus of continuity) ones are known under different assumptions (see. e.g. the survey [3] , the papers [1] , [2] , [9] , [5] , [4] , [21] , [12] and [7] , for a recent survey, on differentiable dependence, where also some result beyond the uniformly expanding case are discussed). While these results do not give a quantitative information for the general case of deterministic perturbations of maps with indifferent fixed points, some very recent result give quantitative information (and even differentiability) for some deterministic perturbation of certain maps, leading to families of intermittent maps ( [8] , [10] , [16] ).
Introduction
In this note we show a general quantitative relation between speed of convergence to equilibrium and statistical stability. This relation gives nontrivial information for systems having different speed of convergence to equilibrium. We show the flexibility of this approach applying it to perturbations of a class of maps with indifferent fixed points having an absolutely continuous invariant probability measure. These are systems whose speed of convergence to equilibrium is bounded by a power law. We show that this implies the Hölder continuity of the absolutely continuous invariant measure under (suitable) deterministic perturbations of the system. The paper is structured as follows: in Section 2 we show a general result estimating quantitatively the stability of operator's fixed points under certain perturbations which is suitable to be applied to estimate the stability of physical measures. In Section 3 we apply this result to a class of maps with indifferent fixed points under a large set deterministic perturbations. We remark that our results applies to perturbations of actual intermittent maps and not only to perturbations of expanding maps leading to intermittent behavior after perturbation.
Introduction
Let us consider a dynamical system (X, T ) where X is a metric space and the space SM (X) of Borel measures with sign on X. The dynamics T naturally induces a function L : SM (X) → SM (X) which is linear and is called transfer
Introduction
for every measurable set B. If X is a manifold, the measure is absolutely continuous: dν = f dm (here m represents the Lebesgue measure) and T is nonsingular, the operator induces another operatorL :
Introduction
). By a small abuse of notation we will still indicate by L this operator.
Introduction
An invariant measure is a fixed point for the transfer operator. Let us now see a quantitative stability statement for these fixed points under suitable perturbations of the operator.
Introduction
Let us consider a certain system having a transfer operator L 0 for which we know the speed of convergence to equilibrium (see (1) below). Consider a "nearby" system L 1 having suitable properties which will be specified below: suppose there are two normed vector spaces of measures with sign B s ⊆ B w ⊆ SM (X) (the strong and weak space) with norms || || w ≤ || || s and suppose the operators L 0 and L 1 preserve the spaces:
Introduction
Let us consider
Introduction
the space of zero average measures in B s . The speed of convergence to equilibrium of a system will be measured by the speed of contraction to 0 of this space by the iterations of the transfer operator.
Introduction
Definition 2 Let φ(n) be a real sequence converging to zero. We say that the system has convergence to equilibrium with respect to norms || || w , || || s and speed
Introduction
Suppose f 0 ,f 1 ∈ B s are fixed probability measures of L 0 and L 1 . The following statement 1 gives a quantitative way to estimate the statistical stability of L 0 under perturbation if the speed of convergence to equilibrium is known.
Introduction
Theorem 3 Suppose we have estimations on the following aspects of the system:
Introduction
1. (speed of convergence to equilibrium) there is φ ∈ C 0 (R), φ(t) decreasing to 0 as t → ∞ such that L 0 has convergence to equilibrium with respect to norms || || w , || || s and speed φ. 
Introduction
x , then
Introduction
Proof. The proof is a direct computation from the assumptions
Introduction
and we have the estimation for the modulus of continuity
Introduction
where the constant K 1 depends on M,C, C and not on the distance of the operators measured by ǫ.
Maps with indifferent fixed points
In this section we introduce a family of maps with indifferent fixed point; we then adapt some known results for these kind of maps to the form required to apply Theorem 3. 2. Each branch T i of T is increasing, weakly convex and can be extended to a C 1 function;
Maps with indifferent fixed points
3. T ′′ is continuous everywhere but in the points 0 and d;
Maps with indifferent fixed points
We say that T is in the set M 0 (α, c, C, d) if moreover T has an absolutely continuous invariant probability measure with density h such that.
Maps with indifferent fixed points
for some 0 < γ 0 < 1. We indicate with M 0 the set of maps that are in M 0 (α, c, C, d) for some value of the parameters.
Remark 6
We remark that the above conditions are verified for maps of the following kind:
Remark 6
, in particular this map is in M 0 (see [18] ).
Remark 7
If a map T is in M (α, c, C, d), the following condition is then satisfied: there is a constant C 3 ∈ (0, ∞) such that
Remark 7
For such a map, we can apply a useful estimation on the asymptotic behavior of f which can be found in [20] (Proposition 1.1, Theorem 1, Equation 3 ).
Remark 7
Proposition 9 Let us consider the transfer operator L T associated to T ∈ M (α, c, C, C 3 , d) and the following cone of decreasing functions
Remark 7
Now we estimate the speed of convergence to equilibrium. The following is proved in [18] Theorem 11 Let us consider
Remark 7
Similarly to what is done in [11] , Appendix B, for the decay of correlations, we show that by the uniform boundedness principle it is possible to see that a statement like (4) implies a statement on the convergence to equilibrium which is uniform also in f :
Remark 7
Proposition 12 Let us consider a system satisfying a non uniform convergence to equilibrium estimation, as in (4), then there is a constant
Remark 7
Proof. Let B L ∞ be the unit ball in L ∞ . For any (n, g) ∈ N×B L ∞ , let us consider the family of non-negative functions defined by
Remark 7
and
Remark 7
Moreover it follows from 4 that for each
Remark 7
By the uniform boundedness principle then (see [15] , Theorem 1.29, page 139 ) it follows sup
Remark 7
Which implies 5
Remark 13
The above proof generalizes to any speed of convergence to equilibrium. Moreover, since for any Lipschitz funcion f there isf ∈ C 1 such that ||f || C 1 ≤ ||f || Lip (where || || Lip denotes the Lipschitz norm) and ||f −f || 1 ≤ ǫ (g ∈ L ∞ , thus approximating f in L 1 is enough to approximate the integral of the product) then the above theorem also hold for Lipschitz functions f with the Lipschitz norm:
Application of Theorem 3 to maps with indifferent fixed points
To apply Theorem 3 we need some technical work to verify the required assumptions and provide the required estimations. This will be done in this section and in the following ones. Let us consider the following norm:
Application of Theorem 3 to maps with indifferent fixed points
We show now how Theorem 3 can be applied to our class of maps, considering || || α and || || 1 as strong and weak norms. The first step is to extend the result of Proposition 12 from Lipschitz observables to a class including functions with finite || || α norm.
Convergence to equilibrium for the || || α norm (item 1 of Theorem 3)
In the following proposition we show how a relation like Proposition 12 implies an estimation of the convergence to equilibrium, as Definition 2 with respect to || || 1 and the strong norm || || α
Proposition 14
If for a nonsingular function T and some γ > 0, C 2 ≥ 0,it holds that for each g ∈ L ∞ and f ∈ Lip[0, 1], with f dx = 0,
Proposition 14
then there is C 4 ≥ 0 such that
Proposition 14
where L is the transfer operator associated to T .
Proposition 14
Proof. Suppose ||g|| ∞ ≤ 1. There is no loss of generality in supposing
Proposition 14
Since we have convergence to equilibrium for functions of zero average we condiderf
Proposition 14
Since f q (x) is Lipschitz and ||f q || lip ≤ ||f || α q −α−1 , by 6
Proposition 14
and then
Applying Theorem 3
We can now apply Theorem 3, for suitable perturbations, obtaining that the physical invariant measure varies continuously, with a Holder continuity modulus. In this we will consider the norm || || α as a strong norm and the L 1 norm as a weak one.
Applying Theorem 3
We remark that the transfer operators are L 1 contractions then the constant at item 3 of Theorem 3 satisfiesC = 1. By Theorem 3 it follows:
Applying Theorem 3
Proposition 15 Let us suppose that T 0 ∈ M 0 (α, c, C, d) at beginning of Section 3, let L 0 be its transfer operator and f 0 its absolutely continuous invariant measure. Let T 1 ∈ M (α, c, C, d), let L 1 be its transfer operator and f 1 its absolutely continuous invariant measure. If T 0 and T 1 are such that
Applying Theorem 3
and L 1 is a small perturbation of L 0 , in the following sense:
Applying Theorem 3
then for each 0 < γ < 1 α −1 there is K 2 , depending on M,C = 1 (see Remark 4) and T 0 , but not on ǫ such that
Applying Theorem 3
We now show how assumptions (7) and (8) can be naturally supposed for deterministic perturbations in our class of maps, and we get an explicit estimation of M .
Bound on the strong norm of the invariant measures (item 2 of Theorem 3)
In this section we show how it is possible to have an estimation of the || || α norm of the absolutely continuous invariant measures of maps in the set M (α, c, C, d).
Bound on the strong norm of the invariant measures (item 2 of Theorem 3)
To estimate the || || α norm of this invariant density, we need to have an estimation on its derivative. In [18] the following is proved:
Bound on the strong norm of the invariant measures (item 2 of Theorem 3)
Then the cone
Bound on the strong norm of the invariant measures (item 2 of Theorem 3)
is invariant for the transfer operator.
Bound on the strong norm of the invariant measures (item 2 of Theorem 3)
The result follows by remarking that for a,
Bound on the strong norm of the invariant measures (item 2 of Theorem 3)
We need to have an explicit estimation of a, b in function of T . Since
Bound on the strong norm of the invariant measures (item 2 of Theorem 3)
.
Bound on the strong norm of the invariant measures (item 2 of Theorem 3)
Now let us find a sufficient condition for which the two summands are less or equal than ). We will obtain the conditions 9, 10, 11.
Bound on the strong norm of the invariant measures (item 2 of Theorem 3)
For the first summand
Bound on the strong norm of the invariant measures (item 2 of Theorem 3)
|DyT | 2 ≤ a is a sufficient condition. For the second summand
Bound on the strong norm of the invariant measures (item 2 of Theorem 3)
By Remark 10 we have the following Proposition 17 Let T ∈ M (α, c, C, C 3 , d), suppose T satisfies the assumptions of Lemma 16. Then the unique a.c.i.m h is such that
Bound on the strong norm of the invariant measures (item 2 of Theorem 3)
Proof. The condition ||h|| α ≤ A * follows from h(x) ≤ A * x −α (see Remark 10) . By Lemma 16 we have
Mixed norm for the difference of the operators for deterministic perturbations (item 4 of Theorem 3)
In this section we see a family of perturbations of maps with an indifferent fixed point, for which the difference ||(L δ − L)f || 1 can be estimated for each f such that ||f || α ≤ 1, allowing the application of Proposition 15. These are examples of perturbations for which we can have all the required estimations and apply our main results. Essentially we allow perturbations wich are small in a weighted L ∞ norm for the map and in the L ∞ norm for its derivative.
Proposition 18 Let us consider two maps
1,2 be the inverse branches of the two maps. Suppose that T 1 is a small perturbation of T 0 in the following sense:
Proposition 18 Let us consider two maps
then there is C 6 ≥ 0 not depending on
Proposition 18 Let us consider two maps
Proof. Writing the transfer operators explicitly:
Proposition 18 Let us consider two maps
by Lagrange theorem, where 

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