QUASISYMMETRIC PROPERTY FOR CONJUGACIES BETWEEN ANOSOV DIFFEOMORPHISMS OF THE TWO-TORUS

HUYI HU AND YUNPING JIANG

Abstract. We prove that the restrictions of the conjugacy between two Anosov diffeomorphisms of the two-torus to the stable and unstable manifolds are quasisymmetric homeomorphisms.

(This paper is dedicated to Professor Lo Yang for his 70^{th} birthday)

1. Introduction

The study of the quasisymmetric property for a conjugacy between two one-dimensional maps has led to solutions of many important problems in one-dimensional dynamical systems and in complex dynamical systems. We give a partial list of references in this direction [9, 10, 15, 11, 21, 5, 19, 18].

A quasisymmetric homeomorphism could be very singular, that is, it could map a set of positive Lebesgue measure to a set of zero Lebesgue measure or vice versa. Generally speaking, as a conjugacy between certain two onedimensional dynamical systems, a homeomorphism must be either totally singular or smooth (see, for examples, [9, 12, 13, 14, 15, 16]). However, a quasisymmetric homeomorphism has many important geometric properties. For example, a quasisymmetric homeomorphism of the real line can be extended to the whole complex plan as a quasiconformal homeomorphism (refer to [2]).

We would like to push the study of the quasisymmetric property into higher dimensional dynamical systems but with either one-dimensional stable manifolds or one-dimensional unstable manifolds. In [3], Cawley did a similar study and more emphasized on the geometric structure of the space of Anosov diffeomorphisms of the two-torus parametrized by potentials on stable and unstable manifolds. In this paper, we study the quasisymmetric property of a conjugacy between two Anosov diffeomorphisms of the two-torus when the conjugacy is restricted to stable and unstable manifolds. The main technique we use in this paper is the Markov partition method (see [22]) which has been used in the study of the quasisymmetric property of one-dimensional dynamical systems (see [9]).

2. Notations and the Main Theorem

Let 𝕋^{2} be the two-torus. Let f be an Anosov diffeomorphism of the twotorus. Suppose f is at least C^{1+α} for some 0 < α ≤ 1. By the definition, f is an Anosov diffeomorphism if there is a invariant splitting of the tangent bundle T 𝕋^{2} = E^{s} ⊕ E^{u}, where the subbundle E^{s} is contracted by f, and the subbundle E^{u} is expanded by f. That is, by considering the Lebesgue metric ∥ · ∥ on 𝕋^{2}, there are two constants 0 < μ < 1 and C_{0} > 0 such that for all n ≥ 0,

∥Df^{n}v∥ ≤ C_{0}μ^{n}∥v∥, ∀v ∈ E^{s},

and

∥Df^{−n}v∥ ≤ C_{0}μ^{n}∥v∥, ∀v ∈ E^{u}.

The only 2-dimensional smooth manifold that support an Anosov diffeomorphism is the two-torus.

The stable and unstable manifold theorem [6] says that for an Anosov diffeomorphism f, 𝕋^{2} can be foliated by two transversal C^{1+α} submanifolds W^{s} and W^{u} such that T W^{s} = E^{s} and T W^{u} = E^{u}. Here W^{s} and W^{u} are called the stable and unstable manifolds for f. For each x in 𝕋^{2}, the stable manifold W^{s}(x) and the unstable manifold W^{u}(x) passing x are

W^{s}(x) = {y ∈ 𝕋^{2} | d(f^{n}(x), f^{n}(y)) → 0, n → ∞}

and

W^{u}(x) = {y ∈ 𝕋^{2} | d(f^{−n}(x), f^{−n}(y)) → 0, n → ∞}.

Each W^{s}(x) or W^{u}(x) is a connecting C^{1+α} immersed submanifold.

Suppose f and g are two Anosov diffeomorphisms of the two-torus. We say that f and g are topologically conjugate if there is a homeomorphism h of the two-torus such that

h ∘ f = g ∘ h.

Frank [4] and Manning [20] showed that every Anosov diffeomorphism f of the two-torus is topologically conjugate to a linear example; that is, to an automorphism defined by a hyperbolic element A of GL(2, ℤ) whose determinant has absolute value one. Thus every Anosov diffeomorphism f of the two-torus has a fixed point, which we always take it as 0. It is known that the conjugacy h between any two Anosov diffeomorphisms is Hölder continuous (this will also be a corollary of our main theorem in this paper).

There is another very important geometric concept for a homeomorphism of the real line called quasisymmetry in complex analysis. A homeomorphism H of the real line ℝ is called quasisymmetric if there is a constant M ≥ 1 such that

\frac{1}{M} ≤ |\frac{H(x + t) − H(x)}{H(x) − H(x − t)}| ≤ M, ∀x ∈ ℝ, ∀0 < t ≤ 1.

Suppose W_{f}^{s}(0) and W_{f}^{u}(0) and W_{g}^{s}(0) and W_{g}^{u}(0) are the stable and unstable manifolds for f and g. Since they are all connecting C^{1+α} submanifolds of the two-torus, we have C^{1+α} embeddings ρ_{s,f}, ρ_{u,f}, ρ_{s,g}, and ρ_{u,g} from ℝ onto W_{f}^{s}(0), W_{f}^{u}(0), W_{g}^{s}(0), and W_{g}^{u}(0), respectively. We assume that ρ_{s,f}, ρ_{u,f}, ρ_{s,g}, and ρ_{u,g} preserve the arc-length.

For the conjugacy h from f to g, define

H_{s} = ρ_{s,g}^{−1} ∘ h ∘ ρ_{s,f} and H_{u} = ρ_{u,g}^{−1} ∘ h ∘ ρ_{u,f}.

Then they are two homeomorphisms of the real lines. We say h|W_{f}^{s}(0) and h|W_{f}^{u}(0) are quasisymmetric if H_{s} and H_{u} are quasisymmetric. We will prove is the following:

Theorem 1. Suppose f and g are two conjugated Anosov diffeomorphisms of the two-torus and h is a conjugacy between f and g, that is, h ∘ f = g ∘ h. Then

h|W_{f}^{s}(0) : W_{f}^{s}(0) → W_{g}^{s}(0) and h|W_{f}^{u}(0) : W_{f}^{u}(0) → W_{g}^{u}(0)

are both quasisymmetric homeomorphisms.

It is known that a quasisymmetric homeomorphism of the real line is Hölder continuous [2]. So the Hölder continuity property of h, which is a known result for a long time, is a corollary of the above theorem.

Corollary 1. A conjugacy h between Anosov diffeomorphisms f and g of the two-torus is Hölder continuous.

3. Markov Partitions

Suppose f is an Anosov diffeomorphism of 𝕋^{2}. Then f has a local product structure, that is, for any ϵ > 0, there exists δ > 0 such that if d(x, y) ≤ δ, W_{ϵ}^{s}(x) ∩ W_{ϵ}^{y}(y) contains exact one point, denoted by [x, y], where W_{ϵ}^{s}(x) and W_{ϵ}^{u}(x) are the local stable and unstable manifold at x given by

W_{ϵ}^{s}(x) = {y ∈ 𝕋^{2} | d(f^{n}(x), f^{n}(y)) ≤ ϵ, ∀n ≥ 0},

and

W_{ϵ}^{u}(x) = {y ∈ 𝕋^{2} | d(f^{−n}(x), f^{−n}(y)) ≤ ϵ, ∀n ≥ 0}.

A set R whose diameter is less than δ is called a rectangle if x, y ∈ R implies [x, y] ∈ R. A rectangle R is proper if it is the closure of its interior. It is easy to check that if R is a rectangle, so is f(R), and if R and S are rectangles, so is R ∩ S, provided the diameters of the rectangles involved are all small.

For a rectangle R and a point x ∈ R, we denote W^{s}(x, R) = W_{ϵ}^{s}(x) ∩ R and W^{u}(x, R) = W_{ϵ}^{u}(x) ∩ R. Note that if R is connected, then both W^{s}(x, R) and W^{u}(x, R) are connected curves.

A Markov partition for f is a set R = {R_{1}, ⋯, R_{n}} of proper connected rectangles satisfying:

(1) 𝕋^{2} = ∪_{i=1}^{n}R_{i};

(2) int(R_{i}) ∩ int(R_{j}) = ∅ for 1 ≤ i ≠ j ≤ n;

(3) fW^{s}(x, R_{i}) ⊂ W^{s}(f(x), R_{j}) if x ∈ R_{i} and f(x) ∈ R_{j};

(4) fW^{u}(x, R_{i}) ⊃ W^{u}(f(x), R_{j}) if x ∈ R_{i} and f(x) ∈ R_{j}.

Sinai proved that any Anosov diffeomorphism has a Markov partition of arbitrarily small diameter [22]. Since we only consider Anosov diffeomorphisms of the two-torus and since every such an Anosov diffeomorphism is topologically conjugate to a linear one, we can construct a canonical Markov partition for every f as follows. Note that diameters of rectangles in this canonical Markov partition may not be small.

Suppose A is a hyperbolic automorphism of 𝕋^{2} conjugating to f. We first construct a canonical Markov partition for A (refer to [1]). Note that A can be defined by a hyperbolic matrix whose absolute value of the determinant is 1. So the matrix has an eigenvalue whose absolute value is greater than 1 (called the unstable eigenvalue) and an eigenvalue whose absolute value is less than 1 (called the stable eigenvalue).

Suppose E^{s} and E^{u} are the stable and unstable eigenspaces of the matrix respectively. Then they are two transversal lines passing through the origin of ℝ^{2}. Suppose the unit square [0, 1)×[0, 1) is a copy of 𝕋^{2} on the plane. Project into the 𝕋^{2} a segment in E^{s} through the origin, and a segment in E^{u} through the origin. Extended these segments until they cut the 𝕋^{2} into parallelograms. The set of these parallelograms is our canonical Markov partition R_{A} for A. The reader may refer to [17, pp. 84-86] for more details and some pictures of a canonical Markov partition. Let h_{A} be the conjugacy from A to f, that is, h_{A} ∘ A = f ∘ h_{A}. Then R_{f} = h_{A}(R_{A}) is our canonical Markov partition for f.

4. Nested sequence of partitions on W^{s}(0) and W^{u}(0).

For a canonical Markov partition R = R_{f} = {R_{1}, ⋯, R_{n}} for f, we define

κ_{0}^{s} = {W^{s}(x, R_{i}) | x ∈ W^{s}(0), 1 ≤ i ≤ n}

and

κ_{0}^{u} = {W^{u}(x, R_{i}) | x ∈ W^{u}(0), 1 ≤ i ≤ n}.

So κ_{0}^{s} is a partition of W^{s}(0) into countably many segments W^{s}(x, R_{i}), x ∈ W^{s}(0), R_{i} ∈ R and κ_{0}^{u} is a partition of W^{u}(0) into countably many segments W^{u}(x, R_{i}), x ∈ W^{u}(0), R_{i} ∈ R. Then we define κ_{n}^{s} = f^{n}κ_{0}^{s} and κ_{n}^{u} = f^{−n}κ_{0}^{u} for any n ≥ 1. That is, κ_{n}^{s} consists of all segments l^{s} in W^{s}(0) such that f^{−n}(l) ∈ κ_{0}^{s} and κ_{n}^{u} consists of all segments l^{u} in W^{u}(0) such that f^{n}(l) ∈ κ_{0}^{u}. By the condition (3) and (4) we know that each element of κ_{n}^{s} or κ_{n}^{u} is a union of some elements of κ_{n+1}^{s} or κ_{n+1}^{u} respectively.

5. Holonomy map

For any two segments l^{s} and l̃^{s} of κ_{0}^{s} in a same rectangle R ∈ R, a holonomy map θ^{s}(x) : l^{s} → l̃^{s} is defined by sliding along the unstable curves, that is, for any x ∈ l^{s}, θ^{s}(x) = [z, x], the only point contained in the intersection W^{s}(z) ∩ W^{u}(x), where z is any point in l̃^{s}. Similarly, for any two segments l^{u} and l̃^{u} of κ_{0}^{u} in a same rectangle R ∈ R, a holonomy map θ^{u}(y) : l^{u} → l̃^{u} is defined by sliding along the stable curves, that is, for any y ∈ l^{s}, θ^{u}(y) = [y, z], the only point contained in the intersection W^{s}(y) ∩ W^{u}(z), where z is any point in l̃^{u}. The proof of the following lemma can be founded in [6, 17] (also, refer to [7, Proposition 3.2]), using the facts that both stable and unstable foliations are codimension one.

Lemma 1. All holonomies are Lipschitz continuous with a uniform Lipschitz constant. More precisely, there is a constant C_{1} > 0 such that for any two segments l^{s} and l̃^{s} of κ_{0}^{s} in a same rectangle R ∈ R,

d(θ^{s}(x), θ^{s}(x′)) ≤ C_{1}d(x, x′), ∀ x, x′ ∈ l^{s},

and for any two segments l^{u} and l̃^{u} of κ_{0}^{s} in a same rectangle R ∈ R,

d(θ^{u}(y), θ^{u}(y′)) ≤ C_{1}d(y, y′), ∀ y, y′ ∈ l^{u}.

This lemma implies the following.

Lemma 2. There is a constant C_{1} > 1 such that

\frac{1}{C_{1}} ≤ \frac{|l^{s}|}{|m^{s}|}, \frac{|l^{u}|}{|m^{u}|} ≤ C_{1}

for all l^{s}, m^{s} ∈ κ_{0}^{s} and l^{u}, m^{u} ∈ κ_{0}^{u}, where | · | means the length of the segment.

Remark 1. Following the method used in one-dimensional dynamical systems (see [9]), Cawley [3] studied the quasisymmetric property of holonomies.

6. Distortions

For an Anosov diffeomorphism, we have that

Lemma 3 (Distortion). For any ϵ > 0, there is a constant C_{2} = C_{2}(ϵ) > 0 such that for any x, y ∈ W^{s}(0) with d^{s}(x, y) ≤ ϵ and n > 0,

\frac{1}{C_{2}} ≤ \frac{∥Df^{n}(y)|_{E_{y}^{s}}∥}{∥Df^{n}(x)|_{E_{x}^{s}}∥} ≤ C_{2}

and for any x, y ∈ W^{u}(0) with d^{u}(x, y) ≤ ϵ and n > 0,

\frac{1}{C_{2}} ≤ \frac{∥Df^{−n}(y)|_{E_{y}^{s}}∥}{∥Df^{−n}(x)|_{E_{x}^{s}}∥} ≤ C_{2},

where d^{s} and d^{u} are the distances along W^{s}(0) and W^{u}(0) respectively.

The proof of this lemma is the same as the proof of the naive distortion lemma in one-dimensional dynamical systems (see [9, Chapter 1]) and can be found in many books for hyperbolic dynamical systems, see, for example, [17].

7. Bounded nearby geometry

Definition 1. The nested sequences of partitions κ^{s} = {κ_{n}^{s}} or κ^{u} = {κ_{n}^{u}} are said to have bounded nearby geometry if there is a constant C > 0 such that for any two adjacent segments l^{s}, m^{s} ∈ κ_{n}^{s} or l^{u}, m^{u} ∈ κ_{n}^{u}, n ≥ 0,

\frac{1}{C} ≤ \frac{|l^{s}|}{|m^{s}|} ≤ C or \frac{1}{C} ≤ \frac{|l^{u}|}{|m^{u}|} ≤ C

respectively.

Theorem 2. Suppose f is a C^{1+α} Anosov diffeomorphism for some 0 < α ≤ 1. Then the nested sequences of partitions κ^{s} and κ^{u} have the bounded nearby geometry.

Proof. By Lemma 2, there is a constant C_{2} = C_{2}(2ϵ) > 0 such that

\frac{1}{C_{1}} ≤ \frac{|l^{s}|}{|m^{s}|}, \frac{|l^{u}|}{|m^{u}|} ≤ C_{1}

for any two adjacent segments l^{s}, m^{s} ∈ κ_{0}^{s} or l^{u}, m^{u} ∈ κ_{0}^{u}.

For any n ≥ 1 and for any two adjacent segments l^{s} and m^{s} in κ_{n}^{s} or l^{u} and m^{u} in κ_{n}^{u}, f^{−n}(l^{s}) and f^{−n}(m^{s}) or f^{n}(l^{u}) and f^{n}(m^{u}) are two adjacent segments in κ_{0}^{s} or in κ_{0}^{u} respectively. Then we apply the distortion lemma, Lemma 3, to get the result. □

Lemma 4. For any c > 0, there exists k = k(c) > 0 such that for any n > 0, l^{s} ∈ κ_{n}^{s} and m^{s} ∈ κ_{n+k}^{s} with m^{s} ⊂ l^{s},

|m^{s}| ≤ c|l^{s}|,

and for any l^{u} ∈ κ_{n}^{u} and m^{u} ∈ κ_{n+k}^{u} with m^{u} ⊂ l^{u},

|m^{u}| ≤ c|l^{u}|.

Proof. By the above theorem we know that {|l̃^{s}| | l̃^{s} ∈ κ_{0}^{s}} and {|l̃^{u}| | l̃^{u} ∈ κ_{0}^{u}} are bounded above and below. Since f is uniformly contracting along the stable direction and f^{−1} is uniformly contracting along the unstable direction, we can take k > 0 such that for any l̃^{s} ∈ κ_{0}^{s} and m̃^{s} ∈ κ_{k}^{s} with m̃^{s} ⊂ l̃^{s},

|m̃^{s}| ≤ cC_{2}^{−1}|l̃^{s}|,

and for any l̃^{u} ∈ κ_{0}^{u} and m̃^{u} ∈ κ_{k}^{u} with m̃^{u} ⊂ l̃^{u},

|m̃^{u}| ≤ cC_{2}^{−1}|l̃^{u}|.

Note that if l^{s} ∈ κ_{n}^{s} and m^{s} ∈ κ_{n+k}^{s} with m^{s} ⊂ l^{s}, then f^{−n}(l^{s}) ∈ κ_{0}^{s} and f^{−n}(m^{s}) ∈ κ_{k}^{s} with f^{−n}(m^{s})^{s} ⊂ f^{−n}(l^{s}). Hence, we have

|f^{−n}(m^{s})| ≤ cC_{2}^{−1}|f^{−n}(l^{s})|,

and similarly if l^{u} ∈ κ_{n}^{u} and m^{u} ∈ κ_{n+k}^{u} with m^{u} ⊂ l^{u}, then

|f^{n}(m^{u})| ≤ cC_{2}^{−1}|f^{n}(l^{u})|,

Now we apply Lemma 3 to get |m^{s}| ≤ c|l^{s}| and |m^{u}| ≤ c|l^{u}|. □

8. Quasisymmetric property

Proof of Theorem 1. We adapted a technique in [9, 10] to prove from the bounded nearby geometry to the quasisymmetric property.

Suppose W_{f}^{s}(0) and W_{f}^{u}(0) and W_{g}^{s}(0) and W_{g}^{u}(0) are the stable and unstable manifolds for f and g at 0. Suppose ρ_{s,f} : ℝ → W_{f}^{s}(0), ρ_{u,f} : ℝ → W_{f}^{u}(0), ρ_{s,g} : ℝ → W_{g}^{s}(0), ρ_{u,g} : ℝ → W_{g}^{u}(0) are embedding maps preserving arc length.

We prove that H_{u} = ρ_{u,g}^{−1} ∘ h ∘ ρ_{u,f} : ℝ → ℝ is a quasisymmetric homeomorphism. The proof that H_{s} = ρ_{s,g}^{−1} ∘ h ∘ ρ_{s,f} : ℝ → ℝ is a quasisymmetric homeomorphism is the exactly same just by replacing u by s.

Let ξ_{n,f} = ρ_{u,f}^{−1}κ_{n,f}^{u} and ξ_{n,g} = ρ_{u,g}^{−1}κ_{n,g}^{u} for n ≥ 0. Then they are two sequences of nested partitions on the real line and H_{u}ξ_{n,f} = ξ_{n,g}.

Let Ω be the set of all endpoints of intervals I ∈ ξ_{n,f}, n = 0, 1 ⋯, ∞. It is a dense subset in ℝ.

For x ∈ Ω. Consider the interval [x − t, x]. There is a largest integer n ≥ 0 such that there is an interval I = [a, x] ∈ ξ_{n,f} satisfying [x − t, x] ⊆ I. Suppose J = [b, x] ∈ ξ_{n+1,f}. Then J ⊆ [x − t, x]. Let J′ = [x, c] ∈ ξ_{n+1,f}. From Theorem 2 for f, there is a constant C_{1} > 0 such that

C_{1}^{−1} ≤ \frac{|J′|}{|J|} ≤ C_{1}.

If |J′| > t, we have |J′| ≤ C_{1}|J| ≤ C_{1}t. Take k = k(C_{1}^{−1}) as in Lemma 4, and let J′_{k} = [x, c_{k}] ∈ ξ_{n+k,f}. Then J′_{k} ⊂ J′ and by the lemma we have |J′_{k}| ≤ C_{1}^{−1}|J′| ≤ t. This implies that J′_{k} ⊆ [x, x + t]. So we have

\frac{|H(J′_{k})|}{|H(I)|} ≤ \frac{|H(x + t) − H(x)|}{|H(x) − H(x − t)|} ≤ \frac{|H(J′)|}{|H(J)|},

where H(I) ∈ ξ_{n,g}, H(J), H(J′) ∈ ξ_{n+1,g}, and H(J′_{k}) ∈ ξ_{n+k+1,g}. Now from Theorem 2 for g, we have a constant C > 0 such that

C^{−1} ≤ \frac{|H(J′_{k})|}{|H(I)|} ≤ \frac{|H(x + t) − H(x)|}{|H(x) − H(x − t)|} ≤ \frac{|H(J′)|}{|H(J)|} ≤ C.

If |J′| ≤ t, we have |J′| ≥ C_{1}^{−1}t. We take the same k = k(C_{1}^{−1}) as above, and let J′_{−k} = [x, c_{−k}] ∈ ξ_{n−k,f}. Then J′_{−k} ⊃ J′ and by Lemma 4 we have |J′| ≤ C_{1}^{−1}|J′_{−k}| and therefore |J′_{−k}| ≥ C_{1}|J′| ≥ t. This implies that J′_{−k} ⊇ [x, x + t]. So we have

\frac{|H(J′)|}{|H(I)|} ≤ \frac{|H(x + t) − H(x)|}{|H(x) − H(x − t)|} ≤ \frac{|H(J′_{−k})|}{|H(J)|},

where H(I) ∈ ξ_{n,g}, H(J), H(J′) ∈ ξ_{n+1,g}, and H(J′_{−k}) ∈ ξ_{n−k+1,g}. Now from Theorem 2 for g, we have a constant C > 0, such that

C^{−1} ≤ \frac{|H(J′)|}{|H(I)|} ≤ \frac{|H(x + t) − H(x)|}{|H(x) − H(x − t)|} ≤ \frac{|H(J′_{−k})|}{|H(J)|} ≤ C.

For any x ∈ ℝ, since Ω is dense in [0, 1], we have a sequence x_{n} ∈ Ω such that x_{n} → x as n → ∞. For any t > 0, we have that

C^{−1} ≤ \frac{|H(x_{n} + t) − H(x_{n})|}{|H(x_{n}) − H(x_{n} − t)|} ≤ C.

Since H is continuous on ℝ, we get that

C^{−1} ≤ \frac{|H(x + t) − H(x)|}{|H(x) − H(x − t)|} ≤ C.

We proved the theorem. □

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Mathematics Department, Michigan State University, East Lansing, MI 48824, USA

E-mail address: hu@math.msu.edu

Department of Mathematics, the Graduate Center and Queens College of CUNY, Flushing, NY 11367

E-mail address: yunping.jiang@qc.cuny.edu