Polynomial growth of the derivative for diffeomorphisms on tori

We consider area--preserving diffeomorphisms on tori with zero entropy. We classify ergodic area--preserving diffeomorphisms of the 3--torus for which the sequence $\{Df^n\}_{n\in{\Bbb N}}$ has polynomial growth. Roughly speaking, the main theorem says that every ergodic area--preserving $C^2$--diffeomorphism with polynomial uniform growth of the derivative is $C^2$--conjugate to a 2--steps skew product of the form \[\tor^3\ni(x_1,x_2,x_3)\mapsto (x_1+\alpha,\ep x_2+\beta(x_1),x_3+\gamma(x_1,x_2))\in\tor^3,\] where $\ep=\pm 1$. We also indicate why there is no 4--dimensional analogue of the above result. Random diffeomorphisms on the 2--torus are studied as well.


1.
Introduction. Let M be a compact Riemannian smooth manifold and let µ be a probability Borel measure on M having full topological support. Let f : (M, µ) → (M, µ) be a smooth measure-preserving diffeomorphism. An important question of smooth ergodic theory is the following: whether there is a relation between asymptotic properties of the sequence {Df n } n∈N and dynamical properties of the dynamical system f : (M, µ) → (M, µ). There are results describing a close relation in the case where M is the torus. For example, if f is homotopic to the identity, the coordinates of the rotation vector of f are rationally independent and the sequence {Df n } n∈N is uniformly bounded, then f is C 0 -conjugate to an ergodic rotation (see [8] p.181). Moreover, if {Df n } n∈N is bounded in the C r -norm (r ∈ N ∪ {∞}), then f and the ergodic rotation are C r -conjugated (see [8] p.182). On the other hand, if {Df n } n∈N has an "exponential growth", more precisely if f is an Anosov diffeomorphism, then f is C 0 -conjugate to an algebraic automorphism of the torus (see [11]).
A natural question is what can happen between the above extreme cases? The aim of this paper is to classify measure-preserving tori diffeomorphisms f for which the sequence {Df n } n∈N has polynomial growth. The first definition of polynomial growth of the derivative was proposed in [4]. In [4], the following result has been proved.
Proposition 1.1. Let f : T 2 → T 2 be an ergodic area-preserving C 2 -diffeomorphism. If the sequence {n −τ Df n } n∈N converges a.e. (τ > 0) to a nonzero function, then τ = 1 and f is algebraically (i.e. via a group automorphism) conjugate to the skew product of an irrational rotation on the circle and a circle cocycle with nonzero topological degree.
Moreover, the author in [5] showed that if f : T 2 → T 2 is an ergodic areapreserving C 3 -diffeomorphism for which the sequence {n −1 Df n } n∈N is C 0 -separated from 0 and ∞ and it is bounded in the C 2 -norm, then f is also algebraically conjugate to the skew product of an irrational rotation on the circle and a circle cocycle with nonzero topological degree.
We also recall the main result of [13] asserting that if f : T 2 → T 2 is a homotopic to the identity symplectic diffeomorphism with a fixed point, then f is equals the identity map or there exists c > 0 such that max( Df n ∞ , Df −n ∞ ) ≥ cn for any natural n (see [14] for some generalizations).
In the present paper some versions of Proposition 1.1 are discussed. In Section 2 we consider the random case. In Section 3 we classify area-preserving ergodic C 2diffeomorphisms of a polynomial uniform growth of the derivative on the 3-torus, i.e. diffeomorphisms for which the sequence {n −τ Df n } n∈N converges uniformly to a non-zero function. It is shown that if the limit function is of class C 1 , then τ is 1 or 2, and the diffeomorphism is C 2 -conjugate to a 2-step skew product. We indicate why there is no 4-dimensional analogue of Proposition 1.1 in Section 4.

2.
Random diffeomorphism on the 2-torus. Throughout this section we will consider smooth random dynamical systems over an abstract dynamical system (Ω, F, P, T ), where (Ω, F, P ) is a Lebesgue space and T : (Ω, F, P ) → (Ω, F, P ) is an ergodic measure-preserving automorphism. We will consider a compact Riemannian C ∞ -manifold M equipped with its Borel σ-algebra B as a phase space for smooth random diffeomorphisms. A measurable map f Z × Ω × M (n, ω, x) −→ f n ω x ∈ M satisfying for P -a.e. ω ∈ Ω the following conditions • f 0 ω = Id M , f m+n ω = f m T n ω • f n ω for all m, n ∈ Z, • f n ω : M → M is a smooth function for all n ∈ Z, is called a smooth random dynamical system (RDS). Of course, the smooth RDS is generated by the random diffeomorphism f ω = f 1 ω in the sense that for n < 0.

Consider the skew-product transformation T f : (Ω × M, F ⊗ B) → (Ω × M, F ⊗ B)
induced naturally by f as follows: Then T n f (ω, x) = (T n ω, f n ω x) for all n ∈ Z. We call a probability measure µ on Such measures can also be characterized in terms of their disintegrations µ ω , ω ∈ Ω by f ω µ ω = µ T ω P -a.e. A measure µ is said to be ergodic if T f : (Ω×M, F ⊗B, µ) → (Ω × M, F ⊗ B, µ) is ergodic. We say that µ has full support, if supp(µ ω ) = M for P -a.e. ω ∈ Ω.
In this section we will deal with almost everywhere diffentiable and C r -random dynamical systems with polynomial growth of the derivative. Suppose that f : RDS f is called µ-almost everywhere diffentiable if for every integer n and for µ-a.e. (ω, x) ∈ Ω × M there exists the derivative Df n ω (x) : for every n ∈ Z and P -a.e. ω ∈ Ω, where · n,ω,x is the operator norm in L(T x M, T f n ω x M ). In the paper we will discuss in details random diffeomorphisms on tori. Let d be a natural number. By T d we denote the d-dimensional torus {(z 1 , . . . , z d ) ∈ C d : |z 1 | = . . . = |z d | = 1} which most often will be treated as the quotient group R d /Z d ; λ ⊗d will denote Lebesgue measure on T d . We will identify functions on T d with Z d -periodic functions (i.e. periodic of period 1 in each coordinate) on R d . Let there exists a set A ∈ F ⊗ B such that µ(A) > 0 and g(x) = 0 for all x ∈ A. Moreover, if additionally Df n belongs to L 1 ((Ω × T d , µ), M d (R)) for all n ∈ N and the sequence {n −τ Df n } converges in L 1 ((Ω × T d , µ), M d (R)) then we say that f has τ -polynomial L 1 -growth of the derivative.
We now give an example of an ergodic RDS on T 2 with linear L 1 -growth of the derivative. Before we do it let us introduce a standard notation. Let τ : (X, B, µ) → (X, B, µ) be a measure-preserving ergodic automorphism of a standard Borel space and let G be a compact metric Abelian group. Then each measurable map ϕ : X → G determines a measurable cocycle over τ given by which will be identified with the function ϕ. We say that the cocycle ϕ is a coboundary if there exists a measurable map g : X → G such that ϕ = g − g • τ . We call the cocycle ϕ ergodic if the skew product is ergodic, where λ G is the Haar measure on G.
Lemma 2.1. The RDS f is ergodic and has linear L 1 -growth of the derivative.
The aim of this section is to classify C r -random dynamical systems on the 2torus that have polynomial (L 1 ) growth of the derivative and are ergodic with respect to an invariant measure having full support. We say that two random dynamical systems f and g on T d over (Ω, F, P, T ) are smoothly conjugate if there exists a smooth random diffeomorphism h : If additionally there exists a group automorphism A : T d → T d such that h ω = A for P -a.e. ω ∈ Ω, we say that f and g are algebraically conjugate. Given a smooth RDS f on T 2 over (Ω, F, P, T ) let us denote by ε : Ω → Z 2 the measurable cocycle over the automorphism T : Ω → Ω given by We will prove the following theorems.
Theorem 2.2. Let f be a C r -random dynamical system on T 2 over (Ω, F, P, T ) (r ≥ 1). Let µ be an f -invariant ergodic measure having full support on Ω × T 2 . Suppose that f has τ -polynomial growth of the derivative. Then τ ≥ 1 and f is algebraically conjugate to a random skew product of the form where F : Ω × T → T is a C r -random diffeomorphism of the circle. Moreover, there exist a random homeomorphism of the circle ξ : Ω × T → T and a measurable function α : Ω → T such that and consequently f is topologically conjugate to the random skew product • there exist a Lipschitz random diffeomorphism of the circle ξ : Ω × T → T with Dξ, Dξ −1 ∈ L ∞ (Ω × T, P ⊗ λ) and a measurable function α : For convenience of the reader the proofs of the above theorems are divided into a sequence of lemmas. Let f be a C r -random dynamical system on T d over (Ω, F, P, T ). Let µ be an f -invariant ergodic measure having full support on Ω×T d . Suppose that f has τ -polynomial growth of the derivative. Let g : By the Jewett-Krieger theorem, we can assume that Ω is a compact metric space, T : Ω → Ω is a uniquely ergodic homeomorphism and P is the unique T -invariant measure. Now choose a sequence {A k } k∈N of measurable subsets of A such that the functions g, Df : A k → M d (R) are continuous, all non-empty open subsets of A k (in the induced topology) have positive measure and µ(A k ) > 1 − 1/k for any natural k. Since the transformation Assume that (ω, x), (υ, y) ∈ B k . Then there exists an increasing sequence for any natural k, which proves the lemma.
Let us return to case d = 2. Suppose that A, B are non-zero real 2 × 2-matrixes such that A 2 = B 2 = AB = 0. Then (see Lemma 4 in [4]) there exist real numbers a, b = 0 and c such that It follows that g can be represented as where h : Ω × T 2 → R is a measurable function which is non-zero at µ-a.e. point and c ∈ R. We can omit the second case where because it reduces to case c = 0 after interchanging the coordinates, which is an algebraic isomorphism. Then by (2.4) we obtain for P -a.e. ω ∈ Ω and for all x ∈ T 2 , because µ has full support.
Consequently, the sequence n −τ Df n tends uniformly to zero.
for P -a.e. ω ∈ Ω. It follows that for i = 1, 2 there exists a C r+1 -random function Represent f as Therefore Since c is irrational, we conclude that a 11 (ω) − 1 = a 12 (ω) = 0, hence that u 1 (ω, ·) is 1 and c periodic, and finally u 1 (ω, ·) is a constant for µ-a.e. ω ∈ Ω. It is clear that the same conclusion can be obtained for u 2 , which completes the proof.
Lemma 2.6. If c is rational, then there exist a group automorphism A : Moreover, Proof. Let p and q be integers such that q > 0, gcd(p, q) = 1 and Thenμ is anf -invariant measure and for P -a.e. ω ∈ Ω and all x ∈ T 2 . Consequently, where F, ϕ : Ω × T → T are C r -random functions and It follows thatĥ ω depends only on the first coordinate.
Proof of Theorem 2.2. By Lemmas 2.5 and 2.6, to prove the first claim of the theorem it is enough to show that τ ≥ 1. Suppose that τ < 1.
and consequently 1 n It follows that the measurable cocycle Dϕ/ĥ : Ω × T → R over the skew product T F is recurrent (see [15]). Consequently, for ν-a.e. (ω, x) ∈ Ω × T there exists an increasing sequence of natural numbers contrary to (2.9). Now let us decompose ν ω = ν d ω + ν c ω , where ν d ω is the discrete and ν c ω is the continuous part of the measure ν ω . As this decomposition is measurable we can consider the measures ν d = Ω ν d ω dP (ω) and ν c = Ω ν c ω dP (ω) on Ω × T. It is easy to check that ν d and ν c are F -invariant. By the ergodicity of ν, either ν = ν d or ν = ν c .
3. Area-preserving diffeomorphisms of the 3-torus. In this section we give a classification of area-preserving ergodic diffeomorphisms of a polynomial uniform growth of the derivative on the 3-torus. A C 1 -diffeomorphism f : T 3 → T 3 has τpolynomial uniform growth of the derivative if the sequence {n −τ Df n } n∈N converges uniformly to a non-zero function. We first present a sequence of essential examples of such diffeomorphisms. We will consider 2-step skew products T α,β,γ,ε : T 3 → T 3 given by where α is irrational, ε = ±1 and β : T → T, γ : T 2 → T are of class C 1 . We will denote by d i (γ) the topological degree of γ with respect to the i-th coordinate for i = 1, 2. Here and subsequently, h xi stands for the partial derivative ∂h/∂x i .
The main result of this section is the following theorem.
Theorem 3.1. Let f : T 3 → T 3 be an area-preserving ergodic C 2 -diffeomorphism with τ -polynomial uniform growth of the derivative (τ > 0). Suppose that the limit function lim n→∞ n −τ Df n is of class C 1 . Then τ is 1 or 2, and f is C 2 -conjugate to a diffeomorphism of the form As in the previous section, the proof of the main theorem is divided into several lemmas. Suppose that f : T 3 → T 3 is an area-preserving ergodic diffeomorphism with τ -polynomial growth of the derivative. Assume that the limit of the sequence {n −τ Df n } n∈N , denoted by g : T 3 → M 3 (R), is of class C 1 . By Lemma 2.4, g(x) g(ȳ) = 0 and g(x) 2 = 0 for allx,ȳ ∈ T 3 . But the latter case can not occur because the square of the latter matrix is non-zero. It follows that there exists C ∈ GL 3 (R) such that Therefore we can find non-zero real 1 × 3-matrixesā 1 ,ā 2 such that A =ā T 1ā2 . As A 2 = 0 we haveā 1 ⊥ā 2 . Similarly, we can find non-zero real 1 × 3-matrixesb 1 ,b 2 such that B =b T 1b2 andb 1 ⊥b 2 . Letō ∈ R 3 be a non-zero vector orthogonal to bothā 1 andā 2 . As AB = BA = 0 we haveā 1 ⊥b 2 andā 2 ⊥b 1 . It follows that there exists a real matrix [d ij ] i,j=1,2 such that Then By the above lemma, there existsc ∈ R 3 such that for any two linearly independent vectorsā,b ∈ R 3 orthogonal toc there exist C 1 -functions h 1 , h 2 : for allx ∈ T 3 . We first treat the special case of Theorem 3.1 where the limit function g is constant.
Lemma 3.3. Let f : T 3 → T 3 be an area-preserving ergodic C 1 -diffeomorphism with τ -polynomial uniform growth of the derivative (τ > 0). Suppose that the limit function g = lim n→∞ n −τ Df n is constant. Then τ is 1 or 2, and f is algebraically conjugate to a diffeomorphism of the form Before we pass to the proof we introduce some notation. Let A ∈ GL 3 (R).

Denote by T 3
A the quotient group R 3 /(Z 3 A T ), which is a model of the 3-torus as well. Then the map A : T 3 → T 3 A , Ax =xA T establishes a smooth isomorphism between T 3 and T 3 A . Suppose that ξ : for allm ∈ Z 3 . Moreover, we can write and AN A −1 (resp.ξ) we will be called the A-linear (resp. the A-periodic) part of ξ. The name A-periodic is justified byξ(x +mA T ) =ξ(x) for allm ∈ Z 3 . Suppose that f : T 3 → T 3 is a smooth diffeomorphism with τ -polynomial uniform growth of the derivative and g : T 3 → M 3 (R) is the limit of the sequence {n −τ Df n } n∈N . Let us consider the diffeomorphismf : for allx,ȳ ∈ T 3 , and consequentlŷ for allx,ȳ ∈ T 3 A . Throughout this paper we denote by G(c) the subgroup of allm ∈ Z 3 such that m ⊥c. Of course, ifc ∈ R 3 \ {0}, then the rank of G(c) can be equal 0, 1 or 2. The reader can find further useful properties of the group G(c) in Appendix B.
Suppose that f : T 3 → T 3 is an area-preserving ergodic C 1 -diffeomorphism with τ -polynomial uniform growth of the derivative and the limit function g is constant. By Lemma 3.2, there exist mutually orthogonal vectorsā,c ∈ R 3 such that g =c Tā .
Lemma 3.4. Let f : T 3 → T 3 be an area-preserving C 1 -diffeomorphism. Suppose that f preserves orientation, has τ -polynomial uniform growth of the derivative and the limit function g = lim n→∞ n −τ Df n equalsc Tā , whereā ⊥c. Then the rank of G(ā) equals 2. Moreover, τ equals either 1 or 2.
Proof. Letb ∈ R 3 be a vector orthogonal to bothā andc such that det(A) = 1, where >From (3.13) we obtain Consequently, where β : R → R, γ : R 2 → R are C 1 -functions. Let N ∈ GL 3 (Z) stand for the linear part of f . Then the A-linear part off equals It follows that Suppose, contrary to our claim, that rank G(ā) < 2.

Proof of Lemma 3.3.
First, notice that f 2 preserves area and orientation, and n −τ Df 2n tends uniformly to 2 τcTā . By Lemma 3.4, rank G(ā) = 2. It follows that a = am ∈ aZ 3 , by Lemma B.1 (see Appendix B). Now choosen,k ∈ Z 3 such that the determinant of where ϕ : T × T 2 → T 2 is an area-preserving random diffeomorphism over the rotation by an irrational number α. Then Suppose thatnc T andkc T are rationally independent. Then by Lemma 2.5, Otherwise, by Lemma 2.6, there exist a group automorphism B : T 2 → T 2 and where ε = det Df , which proves the claim.

From (3.23) we obtain
where [ s 1 s 2 ] = [ l 1 l 2 ] K . If (s 1 , s 2 ) and (l 1 , l 2 ) are linearly independent, thenξ is constant. It follows that H is constant which reduces the problem to Lemma 3.3.
Otherwise, there exists a real number s such that (s 1 , s 2 ) = s(l 1 , l 2 ) and for any real x. Since f preserves area det DF (x) = ε = ±1 for allx ∈ T 3 . It follows that for any real x. SinceF 1 ,F 2 are 1-periodic, we have l 1 DF 1 (x) + l 2 DF 2 (x) = 0 and det K = ε. Therefore the function l 1F1 + l 2F2 is constant. Let us choose real numbers r 1 , r 2 such that the determinant of the matrix As ∂f n 1 /∂x 1 = s n and ∂f n 2 /∂x 2 = (ε/s) n we obtain s = ±1, becausef has polynomial uniform growth of the derivative. Moreover, Indeed, suppose, contrary to our claim, that s = −1. Consider the smooth function κ : T 3 → C given by κ(x) = e 2πixm T . Then κ • f 2 = κ. Since κ is smooth, we conclude that it is constant, by the ergodicity of f . Consequently,m = 0, which is impossible. Now choosen,k ∈ Z 3 such that the determinant of A :=  m n k   equals 1. Let us consider the diffeomorphismf : T 3 → T 3 given byf : Our claim now follows by the same arguments as in the proof of Lemma 3.3.
Case 1c. Suppose that rank G(c) = 2. Then we can assume thatā,b,c ∈ Z 3 andā,b generates G(c). Set q = det A ∈ N. Then the A-linear part off (which is equal K = A N A −1 ) belongs to M 3 (q −1 Z). Moreover, the functionsF : Consider the diffeomorphismf : for allm ∈ Z 2 . Therefore,f can also be treated as a diffeomorphism of the torus

24)
Dξ : R 2 → R is Z 2 -periodic and non-zero at each point. Moreover,f : T 3 → T 3 has τ -polynomial uniform growth of the derivative. More precisely, uniformly. Let us denote by ϕ t the Hamiltonian C 2 -flow on T 2 defined by the Hamiltonian equation .
Since ϕ t has no fixed point and T 2ξx1 (x)dx/ T 2ξx2 (x)dx = d 1 /d 2 is irrational, it follows that ϕ t is C 2 -conjugate to the special flow constructed over the rotation by an irrational number a and under a positive C 2 -function b : T → R, (see for instance [2,Ch. 16]) i.e. there exists an area-preserving C 2 -diffeomorphism ρ : R 2 → R 2 and a matrix N ∈ GL 2 (Z) such that where σ t (x 1 , x 2 ) = (x 1 , x 2 + t) and for an integer k. Then the quotient flow σ t a,b of the action σ t modulo the relation ∼ is the special flow constructed over the rotation by a and under the function b. Moreover, ρ : T 2 → M conjugates flows ϕ t and σ t a,b . Let is constant for each (x 1 , x 2 ) ∈ R 2 . It follows that the functionξ • ρ −1 : R 2 → R depends only on the first coordinate. Moreover, .
By the ergodicity off , Ξ and alsoξ is constant, which is impossible. By Theorem A.1 (see Appendix A), there is an area-preserving C 2 -diffeomorphism ψ : T 2 → T 2 such that is a skew product andξ • ψ(x 1 , x 2 ) = kx 1 + c, where k ∈ N and c ∈ R. Therefore Let us consider the area-preserving C 2 -isomorphism ρ :