INFIMUM OF THE METRIC ENTROPY OF VOLUME PRESERVING ANOSOV SYSTEMS

. In this paper we continue our study [9] of the inﬁmum of the metric entropy of the SRB measure in the space of hyperbolic dynamical systems on a smooth Riemannian manifold of higher dimension. We restrict our study to the space of volume preserving Anosov diﬀeomorphisms and the space of volume preserving expanding endomorphisms. In our previous paper, we use the perturbation method at a hyperbolic periodic point. It raises the question whether the volume can be preserved. In this paper, we answer this question aﬃrmatively. We ﬁrst construct a smooth path starting from any point in the space of volume preserving Anosov diﬀeomorphisms such that the metric entropy tends to zero as the path approaches the boundary of the space. Similarly, we construct a smooth path starting from any point in the space of volume preserving expanding endomorphisms with a ﬁxed degree greater than one such that the metric entropy tends to zero as the path approaches the boundary of the space. Therefore, the inﬁmum of the metric entropy as a functional is zero in both spaces.

(Communicated by Rafael de la Llave) Abstract. In this paper we continue our study [9] of the infimum of the metric entropy of the SRB measure in the space of hyperbolic dynamical systems on a smooth Riemannian manifold of higher dimension. We restrict our study to the space of volume preserving Anosov diffeomorphisms and the space of volume preserving expanding endomorphisms. In our previous paper, we use the perturbation method at a hyperbolic periodic point. It raises the question whether the volume can be preserved. In this paper, we answer this question affirmatively. We first construct a smooth path starting from any point in the space of volume preserving Anosov diffeomorphisms such that the metric entropy tends to zero as the path approaches the boundary of the space. Similarly, we construct a smooth path starting from any point in the space of volume preserving expanding endomorphisms with a fixed degree greater than one such that the metric entropy tends to zero as the path approaches the boundary of the space. Therefore, the infimum of the metric entropy as a functional is zero in both spaces.

1.
Introduction. For a uniformly hyperbolic dynamical system f on a compact smooth Riemannian manifold M possessing an SRB measure µ f , it is well known that the metric entropy of f with respect to µ f is positive. The metric entropy changes in general when we perturb the dynamical system f in a neighborhood. Such change is in fact differentiable with respect to f under appropriately chosen differentiable structures [21]. In a recent paper [9], we have shown that while the entropy remains positive, it can be made as small as possible by making successive perturbations to f along a C 1 -path, still preserving the uniform hyperbolicity.
The question is raised whether there is some obstacle that might prevent the entropy from approaching zero while the dynamical system is perturbed along a path within a given subspace of uniformly hyperbolic dynamical systems. In this paper, we explore this question within the space of all volume preserving uniformly hyperbolic dynamical systems.
We will construct a C 1 -path in the space of all volume preserving uniformly hyperbolic dynamical systems and show that there is no obstacle to reduce the entropy while keeping the volume form preserved. The metric entropy along this path can be made as small as possible by making successive perturbations.
The metric entropy of an Anosov map f with respect to its SRB measure ν can be calculated by the formula according to the entropy formula given in [17]. Therefore, to have h µ (f ) arbitrarily small, we need to construct f such that |detDf | E u x (f ) | is sufficiently close to 1 on a set with volume sufficiently close to 1. The complement of this set has small volume and its contribution to the value of h ν is small even though |detDf | E u x (f ) | is large. This is the central idea to construct a path such that the metric entropy can be arbitrarily small. There are two possible approaches to realize this idea. One approach is to make the motion of f very slow in a small neighborhood of a fixed point, then the mass of the SRB measure with respect to f will mostly concentrate in this neighborhood. So the measure of this neighborhood respect to the SRB measure is close to 1 and |detDf | E u x (f ) | in this neighborhood is also close to 1. In the complement of this neighborhood, |detDf | E u x (f ) | may be large but the measure of this complement with respect to the SRB measure is small, so it contributes little to the value of h ν (f ). Such a map is close to some almost Anosov system that has infinite SRB measures ( [7]) and the limiting map along this path is this almost Anosov system. This is the approach we have used in the previous paper ( [9]). However, such a path cannot be made as a path of volume preserving Anosov systems. In this paper, we develop the other approach. We first reduce the speed of the motion of a Anosov system f in an entire neighborhood of a fixed point by embedding it into a flow. This forces the SRB measure again to concentrate in this neighborhood. However, we make sure that every Anosov system in the path has a smooth invariant measure. This enables us to apply Moser's Theorem to smoothly conjugate this path to another path consisting of all volume preserving Anosov systems. The smooth conjugacy will help spread this slow motion to nearly the entire manifold. Consequently, this Anosov system has a strong expansion and a strong contraction on a set with a small Lebesgue volume. The path {f t } 0<t≤1 we will construct is chosen in such a way that f t may converge to the identity point-wise, although the C 1 -norm of an Anosov system in the path may become increasingly large. We would like to emphasize the main difference of these two approaches: In the first approach, the path {f t } 0<t≤1 we constructed in [9] has a limiting map as an almost Anosov system (i.e., degenerate one); In the second approach, the path {f t } 0<t≤1 we construct may lead to the identity map where all Anosov properties will disappear in the limiting map (i.e., totally degenerate one).
The method in the proof of Theorem 1 is different from the method used in [9]. In [9], we construct a path in the space of all Anosov diffeomorphisms tending to the boundary of the space as an almost Anosov diffeomorphism. It is analogous to a path in the Teichmüller space of quasi-Fuchsian groups tending to the boundary of Teichmüller space as a Fuchsian group with a parabolic element. Thus, if we view this path as a path of Riemann surfaces up to homotopy, it tends to a Riemann surface with a node. However, in the proof of Theorem 1 in this paper, we construct a path without any concrete limiting point at one end. It is analogous to a path in the Teichmüller space of quasi-Fuchsian groups tending to the boundary of Teichmüller space as a totally degenerate Fuchsian group, that is, if we view this path as a path of Riemann surfaces up to homotopy, one can see no limiting object. To view the second statement more clearly, in Appendix B, we adapt the example from [12, Theorem 22] to construct a C 1 path of C r expanding maps {f t : 0 < t ≤ 1 6 } preserving the Lebesgue measure m such that as t → 0, h m (f t ) → 0 and f t (x) → x for all x except for one point at which the derivative of f tends to infinite.
We organize the paper as follows. In Section 2, we state our main results, Theorems 1 and 2. In Section 3, we prove our first main result, Theorem 1. We divide our proof into three main steps. In the first step, we linearize the given map in a neighborhood of a fixed point (in general, a periodic point). In the second step, we show that while we linearize the map, we can also preserve the volume form. We complete the proof in the third step by first embedding the linearized map into a flow and then using an idea of Katok [13] to slow down the flow to reduce the metric entropy while preserving the volume form. In Section 4, we prove our second main result for endomorphisms, Theorem 2. Since the proof is very similar to the proof of Theorem 1, we only give detailed proof for those parts that are different from the proof of Theorem 1. Finally in Appendix A, we state the Dacorogna-Moser Theorem that is used in proofs of theorems in the paper. A sketch of its proof is also provided since we need an additional statement on differentiable dependence on parameters. Appendix A also provides additional details omitted in the proofs of 1 and 2. Appendix B contains a one-dimensional example constructed explicitly where the C 1 path leads to the identity except for one point. 2. Statements of main results. Let M be a compact C ∞ Riemannian manifold of dimension n ≥ 2. Recall that a diffeomorphism f of M is called Anosov if the tangent bundle T M admits an exponential splitting T M = E s ⊗ E u where E s and E u are invariant under the differential operator df and there exist constants 0 < λ < 1 < µ and C > 0 such that Df n v ≤ Cλ n v and Df n u ≥ Cµ n u for all v ∈ E s x , u ∈ E u x , and x ∈ M . Let µ be the volume form (that is, Lebesgue measure) on M . Let Diff r (M ) denote the space of all C r , 1 < r ≤ ∞, diffeomorphisms of M . Using the C r -norm on the tangent bundle, we can equip Diff r (M ) with a differentiable Banach manifold structure. Let Diff r µ (M ) denote the space of all volume preserving C r -diffeomorphisms of M , that is, for any Borel set A ⊂ M , f ∈ Diff r µ (M ), µ(f (A)) = µ(A).
Let A r µ (M ) ⊂ Diff r µ (M ) denote the space of all volume preserving Anosov diffeomorphisms.
A C s , s ≥ 1, path in Diff r (M ) is a one parameter family {f t } 0<t≤1 such that t → f t is C s with respect to the usual Euclidean topology and the Banach manifold structure on Diff r (M ).
For every f ∈ Diff r µ (M ), µ is f -invariant. Hence we can define the metric entropy (measure-theoretic entropy) h µ (f ). Our first main result states that the infimum of Our second main result is for endomorphisms. Let End r (M ) be the space of C r , 1 < r ≤ ∞, endomorphisms of M equipped with a Banach manifold structure induced by the C r -norm in the tangent bundle of M . Let End r µ (M ) denote the space of all volume preserving endomorphisms f ∈ End r (M ), that is, for any Borel denote the space of all volume preserving expanding endomorphisms, that is, in addition to preserve the volume µ, it also satisfies where C > 0 and λ > 1 are two fixed constants. Our second main result states that the infimum of h µ (f ) over E r µ (M ) is zero.
Remark 1. The condition r > 2 is used in our proofs for the purpose of easy explanation and is not necessary, and can be replaced by the condition r > 1.

Remark 2.
The map f is topologically conjugate to f t for any t ∈ (0, 1] due to the structural stability of Anosov dynamical systems and expanding endomorphisms.
Hence, all maps f t , 0 < t ≤ 1, have the same topological entropy.
Remark 3. The path t ∈ (0, 1] → f t can be constructed to have higher order differentiability. For simplicity of the proofs, we have not included this part of discussion in this paper. 3. Proof of Theorem 1. We first give the strategy of the proof of our first main result. The proof has three steps. In the first step, we perturb the map f in its C 1 neighborhood to linearize f at a given hyperbolic periodic point. For the simplicity, we use a fixed point. In this perturbation, the resulting map may be no longer volume-preserving but still remaining as an Anosov diffeomorphism. In the second step, we use the Dacorogna-Moser theorem (Theorem A, we give a precise statement in Appendix A) to make the perturbed map volume-preserving within its C 1 neighborhood. We state the local version called the pasting lemma of the Dacorogna-Moser theorem as Lemma 2, whose proof can be found in [1]. In this step, we keep the perturbed map homotopic to f via a C 1 path. In the third step, we complete our proof by embedding the local linear map in a flow and slow down the flow to reduce the entropy. This is a technique also used by Katok in his work [13] in dimension two. We adapted his idea and show that it works for any dimension ≥ 2. We start our first step: linearization at a periodic point. In this step, we do not need f preserves the volume form.
Suppose f is a C r , r > 1, Anosov diffeomorphism on M . Then it always has a periodic point. For simplicity, we assume it has fixed point p. (Otherwise, we consider f n where n is the period of f at this periodic point.) With a fixed local chart φ : M → R n at p with φ(p) = 0, we can assume f is a local diffeomorphism on the Euclidean space R n with the fixed point p identified with the origin.
For a sufficiently small τ 0 > 0, there exists a one-parameter family of Anosov diffeomorphisms f τ , 0 < τ < τ 0 satisfying the following properties: (1) The C 1 distance between f and f τ is bounded by C 0 τ for a constant C 0 depending only on f .
are open neighborhoods of p with radii τ and τ 4 , respectively, and x → Df (0)x is the linear map induced by the derivative operator Proof. This lemma is similar to Lemma 3.1 in [9]. The small difference is that we now construct a smooth path f τ , τ ∈ (0, τ 0 ), such that the C 1 distance between f and f τ is bounded by C 0 τ . We construct f τ by perturbing f locally near the fixed point p.
Let η(t) be a C ∞ real function defined on [0, ∞) satisfying the following conditions. 1 4 ] and η(t) = 0, for all t ≥ 1. We define a map f τ by where |x| denotes the Euclidean norm of x. The map is well-defined as long as τ is We now get that f τ is C 1 close to f with the C 1 -distance bounded by C 0 τ and thus, lim

is linear and for
|x| ≥ τ , f τ = f . When 0 < τ < τ 0 for sufficiently small τ 0 , f τ clearly depends on τ smoothly since η(t) is C ∞ and f τ is also Anosov since f is Anosov.
Now we move to the second step: making f τ volume preserving. In this step, we assume that f preserves the volume form.
The map f τ in the first step may not preserve the volume form. Since τ is small, the perturbation is limited in one local coordinate chart U . Thus, we can assume f preserves the volume form Ω 0 = Ω 0 (x) = dx 1 ∧ · · · ∧ dx n , x ∈ U . Let Since both f and f τ are diffeomorphisms on a compact manifold, we must have We now need the following lemma, which is derived from the Dacorogna-Moser theorem in [5] as a local version. The theorem is also used to prove the pasting lemma in [1]. We will use this lemma twice in our proof: making both the map after the linearization of the map near the fixed point and the time-one map of the slowed-down flow volume preserving. A simplified version of Theorem 5 of [5] and its proof are provided in Appendix A. Lemma 2. Assume that Ω 0 and Ω 1 are two C r , r ≥ 1, volume forms in an open neighborhood U of the origin of R n . Assume that, where l 1 (τ ), l 2 (τ ) → 0 as τ → 0. Then, there exists a C r diffeomorphism T τ of U such that T τ maps the volume form Ω 1 to Ω 0 : T * τ Ω 1 = Ω 0 , i.e, detDT τ (x) = [a τ (x)] −1 and T τ is the identity outside the annulus A. Assume further that the relative density function a τ (x) and two radii l 1 (τ ) and l 2 (τ ) depend on τ differentiably. Then, T τ depends on τ differentiably. In particular, for each τ , there exists a constant κ such that The lemma can be obtained by applying Theorem A of the Appendix A to the open annulus A. Since the diffeomorphism constructed has support inside A, it can be extended to be a diffeomorphism of U .
Applying Lemma 2 to Ω 1 and Ω 0 defined for the map f τ , we have T τ such that This implies that map T τ f τ is volume preserving and linear inside the τ 4σ -neighborhood of the origin where σ denote the norm of Df (0). It is Anosov when τ is small.
By the mean value theorem, we conclude that the path τ ∈ (0, τ 0 ) → f τ is a C 1 -path in A r µ (M ). With a little abuse of notation, we again denote T τ f τ by f τ . In the third step, we complete the proof of Theorem 1. The main technique is to embed the linearization near the fixed point into a flow and then follow an idea of Katok in [13] to slow down the flow. The flow in [13] is of dimension two but we will show that the construction is completely parallel in the higher dimension case.
We start with a volume preserving Anosov diffeomorphism f that is linear in a r 2 -neighborhood of a fixed point p under a given system of coordinate charts. Since the perturbation of f will be restricted to this r 2 -neighborhood, we can assume that we are working in an r 2 -neighborhood of the Euclidean space with p = 0. Thus, Df p is an n × n hyperbolic matrix with |det(Df p )| = 1.
We now write Df p = EA, where A is a matrix whose eigenvalues are positive real numbers, and E is an isometry: a product of rotations and/or reflections restricted to each eigenspace of Df p (see Remark 5 in Appendix A). By choosing an appropriate basis, both E and A are block diagonal matrices: where E u and E s are both isometries and A u , A s are expanding and contracting linear maps, respectively. Note that both maps E and A preserve the stable and unstable subspaces of the linear map Df p .
Since the eigenvalues of A are all positive numbers, the linear map x → Ax can be embedded in a flow g t (x) [18] with g(x) = g 1 (x) = Ax. Let X (x) denote the vector field of the flow in B(p, r 2 ). Indeed, X (x) = Lx where L = ln(A). Note that L is also a block-diagonal matrix.
For each w ∈ [0, r 2 ], we define a vector field X w by Let g w be the time one map of the flow defined by the vector field X w . Since ψ w (s) = 1 for s ≥ s 1 , the choice of ψ 0 (s) implies g w (x) = g(x) for any x ∈ B(p, r 2 )\B(p, 7r2 8 ). Now we definef x ∈ B(p, r 2 ); f (x) x ∈ B(p, r 2 ).
For any two given differentiable volume forms ω 1 and ω 2 = ρ(x)ω 1 , ρ > 0, and a differentiable vector field X on M , the divergences of X with respect to the volume forms ω 1 and ω 2 satisfy the equation [11] div ω2 X = div ω1 X + grad ln ρ(x), X . (3. 2) It can be directly verified that the divergence of the vector field X w is zero with respect to the volume form defined by the measure ρ w dµ. Thus, the time-one map g w preserves the volume ρ w . So does the isometry E as the function ψ w (|x| n ) is invariant under E: ψ w (|x| n ) = ψ w (|Ex| n ).
It means that the probability measure Since ψ 0 (0) = ψ (0) = 0, and ψ 0 is C ∞ , we know that the Taylor expansion of ψ 0 (s) has the form a 2 s 2 + o(s 2 ) near s = 0. It follows that Hence, for any fixed neighborhood U p, U ⊂ B(p, r 2 ), It is well known that by the entropy formula [17], if ν is absolutely continuous with respect to the Lebesgue measure, then the entropy for an Anosov map f is given by Since Df w is close to the identity map near p, we can find a small neighborhood B(p, δ) such that log |detDf w | E u x (fw) | < 2 for all |x| ≤ δ when w is sufficiently small. On the other hand, the function log |detDf w | E u x (fw) | is always bounded above. Thus, we can choose w small enough such that So we have h νw (f ) → 0 when w → 0. Finally, we use the Dacorogna-Moser theorem (Theorem A) again. There is a C ∞ diffeomorphism Ψ w that maps (M, ν w ) to (M, µ) satisfying Ψ * w ν w = µ. If we take f w = Ψ wfw Ψ −1 w , w ∈ (0, r 2 ]. then f w is a Anosov diffeomorphism and preserves µ. Since smooth conjugacy does not change the metric entropy, {f w } are the desired diffeomorphisms in Theorem 1. We have completed our proof of Theorem 1.

Remark 4. By (3.3) and (3.4)
we see thatf w has the absolutely continuous invariant measure ν w that is close to 1 for any neighborhood U of the fixed point p at which |detDf | E u x (f ) | is close to 1 if w is close to 0. Hencef w have small metric entropy with respect to ν w . On the other hand, since f w preserves the volume µ, we can see that the volume of Ψ w (U ) is close to 1, and |detDf | E u x (f ) | is close to 1 for x ∈ Ψ w (U ). The latter implies that Df | E u x (f ) is close to 1. Since |detDf | TxM | = 1, it implies Df | E u x (f ) is close to 1 as well.

4.
Proof of Theorem 2. The proof of Theorem 2 is similar to that of Theorem 1. So we only give a description of changes needed. Take a fixed point p of f . Let p 0 = p and p 1 , · · · , p k−1 be the other preimages of p under f : where k is the degree of the expanding map f . At each point p i , i = 0, 1, · · · , k − 1, take a local chart φ i : M → R n as in the first step in the proof of Theorem 1 such that φ i (p i ) = 0 ∈ R n , and y → Df | p y is the linear map induced by the derivative operator Df | p in R n . By taking a local chart at each p i , i = 0, 1, · · · , k − 1, we can assume that p i has a neighborhood U i that is a subset of the Euclidean space R n . We assume further that {U i } are pairwise disjoint and that each U i contains a ball B(p i , τ * ) of radius τ * > 0 about p i respectively. Take τ 0 ∈ (0, τ * ) such that f (B(p i , τ 0 )) ⊂ B(p, τ * ) for any i = 0, 1, · · · , k − 1.
On each B(p i , τ 0 ), we define f τ as in (3.1): Clearly, the map is well-defined as long as τ is sufficiently small. For x with |x| ≥ τ , f τ (x) = f (x). f τ is a C r map whenever f is C r (r ≥ 2). As in the first step in the proof of Theorem 1, we can also see that the C 1 norm of f τ − f is bounded by C 0 τ for some constant C 0 independent of τ , and therefore f τ is C 1 close to f with the C 1 -distance bounded by C 0 τ and thus, lim τ →0 f τ = f in C 1 -distance. For x ∈ U i with |x| ≤ τ 4 , f τ (x) = Df (0)x is linear; and f τ = f if |x| ≥ τ . When 0 < τ < τ 0 , f τ clearly depends on τ smoothly since η(t) is C ∞ and f τ is also uniformly expanding since f is. As in the second step in the proof of Theorem 1, On the annulus A = {x : τ 4 ≤ |x| ≤ τ }, choose a volume form Ω 0 and define |detDf τ (y)| −1 Ω 0 (y).

Then we have
A We can now apply Lemma 2 to make f τ volume preserving on M as well as linear in B(p, τ 4 ). Now we deform f τ by slowing down its dynamics near p. Denote again for simplicity f = f τ for a fixed τ ∈ (0, τ 0 ). We write Df p = EA as in the third step in the proof of Theorem 1, that is, A is a matrix whose eigenvalues are real numbers greater than 1, and E is an isometry: a product of rotations and/or reflections restricted to each eigenspace of Df p (see Remark 5 in Appendix A).
Since the eigenvalues of A are positive, the linear map x → Ax can be embedded in a flow g t (x) [18]. The local linear vector field X (x) is given by (ln A)x and the flow's time one map is given by x → g(x) = g 1 (x) = Ax on B(p, τ 4 ). In this case the divergence of the vector field is divX = tr ln A > 0. We construct a family of functions {ψ w (s)} and a vector field X w (x) as in the third step in the proof of Theorem 1 and let g w = g 1 w be the time one map of X w (x). Then we takef w = Eg w on B(p, r 2 ), andf w = f on the complement of B(p, r 2 ). By the same reasons,f w is a C r map for any w ∈ [0, r 2 ] and it depends on w in C r topology when w ∈ (0, r 2 ]. Furthermore,f w is expanding except Df 0 (p) = id.
For w ∈ (0, r 2 ], let ρ w (x) = [ψ w (|x| n )] −1 if x ∈ B(p, r 2 ), and ρ w (x) = 1 otherwise. Note that volume preserving of f implies that We now verify the mapf w preserves the volume ρ w . We only need to show that for any point x ∈ B(p, r 2 ), where the determinant is calculated with respect to the volume ρ w dµ.
Normalize the volume, we have that the probability measure The rest of the proof is identical to that of Theorem 1. Hence we have lim Appendix A. Dacorogna-Moser theorem and some remarks. In this part we provide a statement and a sketch of proof of a version of Moser's Theorem, given by Dacorogna and Moser in 1990, which directly gives Lemma 2 in Section 3. Also, we include here three remarks, two of which provide more details for the decomposition Df p = EA, and equation (3.2), used in Step 3 in the proof of Theorem 1, and the other gives explanation how to carry out the perturbation arguments along a periodic orbit instead of a fixed point.
In the following theorem, the differentiable dependence of the diffeomorphism on parameters is not explicitly mentioned in the original paper. Nevertheless, it can be derived from the proof of the theorem. We provide here a slightly modified version of this theorem with a differentiable dependence statement. A sketch of the proof is also provided. We refer readers interested in additional details to the original papers [5,19].
Theorem (Dacorogna and Moser [5]). Let r ≥ 1 be an integer and A be a bounded connected open set of R n with C r boundary ∂A. Let f (x), g(x) > 0 be two C r functions over the closureĀ = A ∪ ∂A. Assume that supp(f − g) ⊂ A and Then, the following statements hold true.
(1) There exists a diffeomorphism ϕ = ϕ(f, g) for every open subset E ⊂ A, where id is the identity map. This implies for all x ∈Ā.
Lemma 2 follows when we identify f (x)dx and g(x)dx with volume forms Ω 0 and Ω 1 = a τ (x)Ω 0 . Thus, T τ = ϕ is the desired diffeomorphism. Since ϕ = id when f (x) = g(x), the inequality T τ − id C r ≤ κ 1 − a τ (x) C r follows from the last statement of the theorem.
Details of the proof of part (1) of the theorem can be found in [19] and [5]. The results in [19] and [5] are more general. They include the case when supp(f − g) ⊂ A. For Lemma 2, we only need the theorem in the case when supp(f − g) ⊂ A. We provide a sketch of the proof appeared in [19,5] and explain why the statement (2) is true. We use essentially the same notation from [19,5] for objects involved in the proof.
Step one. The first step is to reduce the case for arbitrarily open set A to that when A is C k diffeomorphic to a unit cubeQ. This is based on the following statement.
j=0 is an open cover of a manifold M , including the case when M = A is an open set of R n , and g is a C r function on M satisfying the condition M gdx = 0. Then, there exist C r functions g j , j = 0, 1, 2, · · · , m such that g = m j=0 g j , supp(g j ) ∈ U j , and M g j dx = 0, j = 0, 1, 2, · · · , m.
Moreover, for each j, the map g → g j is linear and g j C r ≤ c g C r , where constant c depends on M and the covering {U j } m j=1 only. Indeed, let {φ j } m j=0 be a C r partition of unity for the open covering {U j } m j=0 . For each k = 1, 2, · · · , m, let ρ(k) < k denote the smallest integer such that U k ∩U ρ(k) = ∅ and for each j = 1, 2, · · · , m, let η k be a C r function such that supp(η k ) ⊂ U k ∩ U ρ(k) and η k dx = 1. Then, g j can be defined by where α jk = 1, ifj = k; −1, ifj = ρ(k); 0, otherwise and λ 1 , λ 2 , · · · , λ m is the solution to the nonsingular linear system m k=1 α jk λ k = gφ j dx, j = 1, 2, · · · , m.
We may regard each set U j in the lemma as the unit cube Q, and construct the diffeomorphism ϕ on cube that satisfies the requirements of the theorem.
Step two. The construction is based on the implicit function theorem and induction on the dimension of the unit cube. We give an outline of this construction in the case when the dimension of the manifold is 2 to illustrate the C ∞ dependence of ϕ on the functions f and g. Assume Then choose a number δ > 0 sufficiently small and a 'cut-off' function ζ(t) ∈ C ∞ (0, 1) satisfying the following conditions: for all x 2 ∈ (0, 1]. Then define φ 2 (x 1 , x 2 ) = (x 1 , x 2 + ζ(x 1 )u(x 2 )), and take g 1 (x) = g(φ 2 (x))det(Dφ 2 (x)) ∈ C r (Q), where x = (x 1 , x 2 ).
It can be proved that ψ 2 (x 1 , x 2 ) is unique and of C r , supp(ψ 2 − id) ⊂ Q, and 1]. Further, since the function G is affine in f and g, the function u(x 2 ) depends on f and g smoothly, that is, the map (f, g) → u(x 2 ) is C ∞ .
Let ϕ(x) = φ 2 • φ 1 , which is the desired diffeomorphism that depends on f and g smoothly.
Remark 5. Given a nonsingular real matrix A, there exists a real matrix P such that P −1 AP consists of real Jordan blocks with each block corresponding to an eigenspace. Let B = P −1 AP . For each Jordan block B i , if the eigenvalue λ i is real, we multiply the block by matrix E i = I if λ > 0 and by E i = −I if λ < 0. If the eigenvalue is a complex number λ i = α + iβ, then we have matrix α β −β α on the diagonal, and let E i = 1 α 2 + β 2 α −β β α , which is a rotation in the eigenspace. In each case we have E −1 i = E i , B i = E i E i B i , and E i B i has positive real eigenvalues. We now have B = EE −1 B such that E −1 B has positive eigenvalues and E is either the identity, a reflection or a rotation. Since P −1 AP = B = EE −1 B, A = P BP −1 = P EP −1 P E −1 BP −1 . The fact |det(P BP −1 )| = 1 implies that P BP −1 preserves volume. Hence P E −1 BP −1 has positive real eigenvalues and preserves volume.
Remark 6. For a C 1 vector field X on a Riemannian manifold M with a metric < ·, · >, by Divergence Theorem, we only need to show that for any C ∞ test function g(x) with its support in a small neighborhood of x, the following equation holds: M g(div ω1 X+ < grad ln ρ(x), X >)dω 2 = M g(div ω2 X)dω 2 , where two volume forms satisfy the relation dω 2 = ρ(x)dω 1 .
Remark 7. The same perturbation argument can be carried out along a periodic orbit of a volume preserving Anosov diffeomorphism f . Let {p 1 , p 2 , · · · , p k } be any periodic orbit of f . Let g be the original Riemannian metric that defines the volume form preserved by f . Let g l be a smooth (Lyapunov) metric under which the exponential splitting of the tangent bundle E s ⊕ E u satisfies the conditions Df v s ≤ λ l v s and Df v u ≥ µ l v u for some λ l < 1 and µ l > 1 and any nonzero vectors v s ∈ E s , v u ∈ E u . We first perturb f using Moser's theorem so that it preserves the volume form induced by g l . We then choose an open neighborhood O(p i ) sufficiently small around each point of the periodic orbit. To obtain the perturbation, we first identify the consecutive points p i ≡ p i+1 , i = 1, 2, · · · , k; k + 1 ≡ 1 of the orbits, and treat the neighborhoods O(p i ) and O(p i+1 ) as the neighborhoods of the origin in a same Euclidean space. We further require that the identifications are chosen in a way that the eigenspaces of Df n at p i are invariant spaces of Df at p i . With the identification, for f : O(p i ) → O(p i+1 ) we still have Df pi = EA, where E and A are matrices with the same properties as that in the proof of Theorem 1. Hence we can apply the process of perturbation used in the proof of Theorem 1 to each of these k maps, we obtain mapf w | O(pi) : O(p i ) → O(p i+1 ). Now we go back to treat p i and p i+1 as different points in the manifold, and the mapf w | O(pi) : O(p i ) → O(p i+1 ) can be extended tof w : M → M by requiringf w = f outside O(p i ) for i = 1, · · · , k. Use this construction we can get that the entropy with respect to its SRB measure approaches zero as w → 0.