Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 30, 2005, 361–383

DIFFERENTIABLE RIGIDITY AND SMOOTH CONJUGACY

Yunping Jiang

Queens College of CUNY, Department of Mathematics Flushing, NY 11367, U.S.A.; yunqc@forbin.qc.edu and CUNY Graduate School, Department of Mathematics 365 Fifth Ave., New York, NY 10016, U.S.A. and Chinese Academy of Sciences Academy of Mathematics and System Sciences, Beijing 100080, P. R. China

Abstract. We started a study of regularity of the conjugacy between two dynamical systems with singularities. We complete this study for quasi-hyperbolic maps. We prove that if the conjugacy between two quasi-hyperbolic maps is differentiable at one point with uniform bound, then it is piecewise smooth. Furthermore, if the exponents of these two maps at all power law critical points are also the same, then the conjugacy is piecewise diffeomorphic. The degree of the smoothness of the conjugacy can be also calculated.

1. Introduction

A remarkable result in geometry is the so-called Mostow’s rigidity theorem. This result assures that two closed hyperbolic 3-manifolds are isometrically equivalent if they are homeomorphically equivalent [Mo]. A closed hyperbolic 3-manifold can be viewed as the quotient space of a Kleinian group acting on the upper-half 3space. So a homeomorphic equivalence between two closed hyperbolic 3-manifolds can be lifted to a homeomorphism of the upper-half 3-space preserving group actions. The homeomorphism can be extended to the boundary of the upper-half 3-space as a boundary map. The boundary is the complex plane and the boundary map is a quasi-conformal homeomorphism. A quasi-conformal homeomorphism of the complex plane is absolutely continuous. Following this property plus the group action is that the boundary map has no invariant line field. Thus it is a Möbius transformation.

For closed hyperbolic Riemann surfaces, the situation is quite complicated. A closed hyperbolic Riemann surface can be viewed as a Fuchsian group acting on the upper-half plane too. A homeomorphic equivalence between two closed hyperbolic Riemann surfaces can be also lifted to a homeomorphism of the upperhalf plane preserving group actions. This homeomorphism can be extended to the boundary of the upper-half plane as a boundary map. In this case, the boundary is the real line and boundary map is a quasisymmetric homeomorphism. The complication comes from the fact that a quasisymmetric homeomorphism may not be absolutely continuous. However, this complication is a rich supply for Teichmüller theory. Actually if the boundary map is absolutely continuous, then following the idea in the proof of Mostow’s rigidity theorem, it is a Möbius transformation. Furthermore, Tukia [T] proved that the boundary map is, in general, uniformly complicated at every point. Actually if the boundary map is smooth at one point, then it is absolutely continuous, therefore, it is a Möbius transformation.

2000 Mathematics Subject Classification: Primary 37E99; Secondary 53C24, 52C25.

The research is partially supported by NSF grants and PSC-CUNY awards and Hundred Talents Program from Academia Sinica.

Shub and Sullivan [SS] introduced this kind of study for the conjugacy between two smooth expanding circle maps. The conjugacy in this case is always quasisymmetric (refer to [J4, Chapter 3]). They proved that if the conjugacy is absolutely continuous then it is smooth. The degree of the smoothness is the same as the degree of the smoothness for both maps.

Following Shub and Sullivan, we started a similar study for dynamical systems with singularies. In the paper [J3], we investigated the conjugacy between two generalized Ulam–von Neumann transformations. A generalized Ulam–von Neumann transformation is a certain interval map with one power law type singularity (see Section 2 for more details). We proved that the conjugacy between two generalized Ulam–von Neumann transformations is smooth if they have the same type power law singularity and their asymmetries are the same and their eigenvalues at all corresponding periodic points are the same. Later, the asymmetry is removed from the result in [J5], [J6]. Moreover, in [J5], [J6], we studied the smoothness of the conjugacy between two geometrically finite one-dimensional maps. A geometrically finite one-dimensional map is a certain one-dimensional map with finitely many power law singularities and with certain Markovian property (see Section 2 for more details). In our previous paper [J1], we have proved that the conjugacy between two geometrically finite one-dimensional maps is quasisymmetric always. In [J5], [J6], we defined a smooth invariant called the scaling function for a geometrically finite one-dimensional map. We further showed that the scaling function and the exponents of power law singularities are complete smooth invariants. That is, the conjugacy between two geometrically finite onedimensional maps are smooth if and only if they have the same scaling function and the exponents of the corresponding singularities are the same.

In this paper, we continue our study and make a more complete theory for quasi-hyperbolic maps which is a certain one-dimensional map with finitely many non-recurrent critical points (see Definition 2). We first study one point differentiable rigidity problem in dynamical systems. The problem concerns how the differentiability of the conjugacy between two dynamical systems at one point affects the global differentiability of the conjugacy. Our first theorem is a result similar to Tukia’s results [T] for Fuchsian groups. This result (Theorem 1) says that if the conjugacy between two quasi-hyperbolic maps is differentiable at one point with uniform bound then it is piecewise C^{1} . The exceptional points are those in the closure of post singular orbits. The asymptotic behavior of the derivative of the conjugacy near those points is taken care of in the second result. Our second theorem (Theorem 2) says that, in addition to the assumption in our first theorem, if exponents at all corresponding singular points are the same, then the conjugacy is a piecewise C^{1+β} diffeomorphism. Actually, from the proof of our second theorem, we know exactly how exponents at singular points affect the asymptotic behavior of the derivative of the conjugacy at those points. In addition, if both quasi-hyperbolic maps satisfy a certain α-Hölder condition, 0 < α ≤ 1, (see Definition 1 for more precise meaning about this) and if γ is the maximum number among all the exponents at all singularities for both maps, then β = α/γ in our second theorem.

Following our first and second theorems, we could get more results in the study of the rigidity problem in dynamical systems. For example, we show two stronger conditions than that in Theorem 1. The first condition is given in our third result (Lemma 6). We prove that if the conjugacy is differentiable at one point with non-zero derivative and if there is a neighborhood about this point such that the absolute value of the eigenvalue at every periodic point in this neighborhood is the same as that at the corresponding periodic point, then the conjugacy is differentiable at this point with uniform bound. This result combined with our first and second theorems gives us that if the conjugacy is differentiable at one point with non-zero derivative and if the absolute value of the eigenvalue at every periodic point in a small neighborhood of this point is the same as that at the corresponding periodic point, then the conjugacy is piecewise C^{1} . Furthermore, if all the exponents at the corresponding singular points are also the same, then the conjugacy is a piecewise C^{1+β} diffeomorphism for some fixed 0 < β ≤ 1 (Corollary 1).

The other condition is absolute continuity. We show that if the conjugacy is absolutely continuous on any small interval, then it is piecewise C^{1} . Furthermore, if all the exponents at the corresponding singular points are also the same, then the conjugacy is a piecewise C^{1+β} diffeomorphism for some fixed 0 < β ≤ 1 (Corollary 2).

The paper is organized as follows. In Section 2, we define a quasi-hyperbolic map (Definition 2). We also provide several examples of such a map. In Section 3, we study the nonlinearity of a quasi-hyperbolic map. In this section, we prove one of our key technical results, the C^{1+Hölder} –Koebe–Denjoy type distortion lemma (Lemma 2) for a quasi-hyperbolic map. By using this technical result, we prove in Section 4 our main results (Theorems 1 and 2). In the same section, we show two stronger conditions than those in Theorem 1 and prove two more rigidity results (Lemma 6, Corollary 1, and Corollary 2).

2. Quasi-hyperbolic one-dimensional maps

Let M be the interval [0, 1] or the unit circle R/Z. Let f: M → M be a piecewise C^{1} map. A point c ∈ M is said to be singular if either f′(c) does not exist or f′(c) exists but f′(c) = 0. A singular point c is said to be power law if there is an interval (c − τ_{c}, c + τ_{c}), τ_{c} > 0, such that the restrictions of f to (c − τ_{c}, c) and to (c, c + τ_{c}) are C^{1} and such that there is a real number γ ≥ 1 such that the limits

lim_{x→c−} \frac{f′(x)}{|x − c|^{γ−1}} = B^{−} and lim_{x→c+} \frac{f′(x)}{|x − c|^{γ−1}} = B^{+}

exist and are non-zero. The number γ is called the exponent at c.

For a power law singular point c, let

r_{c,−}(x) = \frac{f′(x)}{|x − c|^{γ−1}} , x ∈ (c − τ_{c}, c)

and

r_{c,+}(x) = \frac{f′(x)}{|x − c|^{γ−1}} , x ∈ (c, c + τ_{c}).

A singular point c is called critical if γ > 1.

Let SP denote the set of all singular points and let CP denote the set of all critical points. Let PSO = ⋃_{i=1}^{∞} f^{i}(SP) be the set of post-singular orbits.

Remark 1. The exponent is C^{1} -invariant meaning that if f and g are C^{1} conjugated maps (i.e., there is a C^{1} diffeomorphism h such that h ∘ f = g ∘ h), then the exponents of f and g are the same at corresponding power law critical points.

Remark 2. In general, we can define the left and right exponents at a singular point as follows. A singular point c is said to be power law if there is an interval (c − τ_{c}, c + τ_{c}), τ_{c} > 0, such that f|(c − τ_{c}, c) and f|(c, c + τ_{c}) are C^{1} and there are two real numbers γ^{−} ≥ 1 and γ^{+} ≥ 1 such that the limits

lim_{x→c−} \frac{f′(x)}{|x − c|^{γ^{−}−1}} = B^{−} and lim_{x→c+} \frac{f′(x)}{|x − c|^{γ^{+}−1}} = B^{+}

exist and are non-zero. The numbers γ^{−} and γ^{+} are called the left and right exponents of f at c. For a power law critical point c, let

r_{c,−}(x) = \frac{f′(x)}{|x − c|^{γ^{−}−1}} , x ∈ (c − τ_{c}, c)

and

r_{c,+}(x) = \frac{f′(x)}{|x − c|^{γ^{+}−1}} , x ∈ (c, c + τ_{c}).

The left and right exponents are orientation-preserving C^{1} -invariant, meaning that if h is an orientation-preserving C^{1} diffeomorphism such that

h ∘ f = g ∘ h,

then the left and right exponents of f and g are the same at corresponding power law critical points. Under this definition of a power law singular point, our main results (Theorems 1 and 2) will be true too if we replace the conjugacy by the orientation-preserving conjugacy.

Our f in this paper always satisfies that

(1) SP is finite (could be empty) and

(2) every singular point in SP is of power law type.

Let τ > 0 be a real number. For every critical point c ∈ CP, let U_{c} = [c − τ, c + τ]. Define

U(τ) = ⋃_{c∈CP} U_{c} and V (τ) = \overline{M \ U(\frac{1}{2}τ)}.

Denote

U_{c,−} = U_{c} ∩ (c − τ, c) and U_{c,+} = U_{c} ∩ (c, c + τ).

Definition 1. We call f in this paper C^{1+α} for some 0 < α ≤ 1 if there is τ > 0 such that

(i) f on every component of M \ SP is C^{1} and the derivative is α-Hölder continuous, and

(ii) every r_{c,±} | U_{c,±} is α-Hölder continuous.

A sequence of intervals {I_{i}}_{i=0}^{n} is called a chain (with respect to f) if

(a) I_{i} ⊂ M \ SP for all 0 ≤ i ≤ n;

(b) f: I_{i} → I_{i+1} is a C^{1} -diffeomorphism for every 0 ≤ i ≤ n − 1; and

(c) either I_{i} ⊆ V (τ) for all 0 ≤ i ≤ n − 1 or the last interval I_{n} ⊆ U(τ) (but in the latter case, some I_{i}, 0 < i < n − 1, may not be contained in U(τ) or V (τ)).

Definition 2. We call f quasi-hyperbolic if

(1) f is C^{1+α} for some 0 < α ≤ 1;

(2) \overline{PSO} ∩ U(τ) = ∅ for some number τ > 0; and

(3) there are two constants C > 0 and 0 < μ < 1 such that for any chain {I_{i}}_{i=0}^{n}, |I_{0}| ≤ Cμ^{n}|I_{n}|. Let τ > 0 be a fixed small number in the rest of this paper. We first show several examples of f which are quasi-hyperbolic. A point

p ∈ M is called periodic of period k if f^{i}(p) ≠ p for all 0 < i < k but f^{k}(p) = p. When k = 1, we also call it fixed. For a periodic point p of period k, e_{p} = (f^{k})′(p) is called the eigenvalue of f at p. Then p is called attractive if |e_{p}| < 1; parabolic if |e_{p}| = 1; expanding if |e_{p}| > 1. The first example comes from a theorem (see [MS, Theorem 6.3, pp. 261–262]). A critical point c of a C^{2} map is called non-degenerate if f′(c) = 0 and f′′(c) ≠ 0.

Example 1. A C^{2} map f with only non-degenerate critical points such that PSO and SP are both finite and PSO ∩ SP = ∅ and all periodic points are expanding.

The Schwarzian derivative of a C^{3} map h is defined as

S(h) = \frac{h′′′}{h′} − \frac{3}{2}(\frac{h′′}{h′} )^{2}.

We say that h has negative Schwarzian derivative if S(h)(x) < 0 for all x. Singer (see [Si]) proved that if f is C^{3} and has negative Schwarzian derivative, then the immediate basin of every attractive or parabolic periodic orbit contains at least one critical orbit. Therefore, if f has negative Schwarzian derivative and if PSO ∩ SP = ∅ and if \overline{PSO} contains neither attractive nor parabolic periodic points, then all periodic points of f are expanding. The map f is said to be preperiodic if for every singular point c, f^{m}(c) is an expanding periodic point for some integer m ≥ 1. Then \overline{PSO} = PSO contains neither attractive nor parabolic periodic points. A special case of Example 1 is that

Example 2. A preperiodic C^{3} map f having negative Schwarzian derivative. Let SO = ⋃_{n=0}^{∞} f^{n}(SP) be the union of singular orbits of f. If SO is non-

empty and finite, let η_{1} = {I_{0}, …, I_{k−1}} be the set of the closures of intervals in M \ SO, then (f, η_{1}) has Markovian property. That means that

(i) I_{0}, …, I_{k−1} have pairwise disjoint interiors, (ii) the union ⋃_{i=0}^{k−1} I_{i} of all intervals in η_{1} is M, (iii) the restriction f: I → f(I) for every interval I in η_{1} is homeomorphic, and (iv) the image f(I) of every interval I in η_{1} is the union of some intervals in η_{1} .

We call η_{1} a Markov partition. Let g_{i} = (f | I_{i})^{−1} be the inverse of f: I_{i} → f(I_{i}) for each I_{i} ∈ η_{1} . A sequence w_{n} = i_{0} ⋯ i_{n−1} of 0’s, ⋯, (k − 1)’s is called admissible if the domain f(I_{i_{l}}) of g_{i_{l}} contains I_{i_{l+1}} for all 0 ≤ l < n − 1. For an admissible sequence w_{n} = i_{0} ⋯ i_{n−1} of 0’s, ⋯, (k − 1)’s, we can define g_{w_{n}} = g_{i_{0}} ∘ ⋯ g_{i_{n−1}} and I_{w_{n}} = g_{w_{n}}(f(I_{i_{n−1}})). Let η_{n} be the set of the intervals I_{w_{n}} for all admissible sequences of length n. It is also a Markov partition of M with respect to f. We call it the n^{th} -partition of M induced from (f, η_{1}). Let κ_{n} be the maximum of the lengths of intervals in η_{n}.

Definition 3. We call f geometrically finite if

(i) the set of singular orbits SO is non-empty and finite, (ii) no critical point is periodic, and (iii) there are constants C > 0 and 0 < μ < 1 such that κ_{n} ≤ Cμ^{n} for all n > 0.

Our next example of a quasi-hyperbolic map is a geometrically finite map (refer to [J4, Chapter 3] for more details about this). A special class of geometrically finite maps is generalized Ulam-von Neumann transformation, which is the first class we studied in this direction.

Definition 4. Suppose M = [−1, 1]. We call f a generalised Ulam–von Neumann transformation if (I) f is geometrically finite with only one singular point 0, (II) f(−1) = f(1) = −1 and f(0) = 1, (III) f | [−1, 0] is increasing and f | [0, 1] is decreasing.

One example of a generalized Ulam–von Neumann transformation is f(x) = 1 − 2|x|^{γ} for γ ≥ 1. Another one is f(x) = −1 + 2 cos(\frac{1}{2}πx). If f is a generalized Ulam–von Neumann transformation, let I_{0} = [−1, 0] and I_{1} = [0, 1]. We then have that f(I_{0}) = f(I_{1}) = M. Thus η_{0} = {I_{0}, I_{1}} is a Markov partition. The post-singular orbit PSO = ⋃_{i=1}^{∞} f^{i}(0) is {−1, 1}. Any two generalized Ulam–von Neumann transformations f and g are topologically conjugate by an orientationpreserving homeomorphism. The conjugacy is quasisymmetric (refer to [J1] or [J4, Chapter 3]). And according to our results (Theorems 1 and 2) in this paper, it is either totally singular or smooth on (−1, 1). If the conjugacy is smooth on (−1, 1) and if the exponents of f and g at 0 are the same, then the conjugacy is a diffeomorphism of [−1, 1].

Our last example is a circle expanding map which was first studied by Shub– Sullivan [SS]. Let M = R/Z be the unit circle. Then f is called a circle expanding map if there are constants C > 0 and μ > 1 such that

(f^{n})′(x) ≥ Cμ^{n}, x ∈ M, n ≥ 1.

An example of an expanding circle endomorphism is x ↦ dx (mod 1) , where d > 1 is an integer. If the topological degree of f is d, then f is topologically conjugate to x ↦ dx (mod 1) . A circle expanding map is quasi-hyperbolic without any singular point. The conjugacy is quasisymmetric (refer to [J4]) and, according to results in this paper, it is either a totally singular homeomorphism or a diffeomorphism of the circle.

3. Nonlinearity of a quasi-hyperbolic map

The study of nonlinearity is extremely important in dynamical systems. In one complex variable, Koebe’s distortion theorem (see [Bi]) represents a beautiful result showing how nonlinearity can be controlled for all schlicht functions (or called conformal maps) defined on the unit disk. In one-dimensional dynamics, the Denjoy distortion technique becomes an important tool to estimate the nonlinearity of a C^{2} diffeomorphism of the circle (see [D]). Koebe’s distortion theorem represents a kind of magic in mathematics because it shows that the nonlinearity can be controlled universally for all schlicht functions on the unit disk. By a detailed analysis of the reason behind this magic, we prove a Koebe–Denjoy type distortion lemma by combining ideas of Koebe and Denjoy. This result gives us a wonderful technique to control the nonlinearity of a quasi-hyperbolic map.

Take f a quasi-hyperbolic map. Let α, U = U(τ), and V = V (τ) be as in Definition 2. This notation is fixed in the rest of the paper. Dividing M into U and V is one of the key points in this section; the set V is away from all critical points CP and the set U is away from post-singular orbit PSO = ⋃_{n=1}^{∞} f^{n}(SP). In the set V we can use the Denjoy distortion technique to control the distortion of the iterates of f (see Lemma 1); in the set U we can prove a Koebe type distortion property (see Lemma 3) to control the distortion of the iterates of f.

A chain ℐ = {I_{i}}_{i=0}^{n} is said to be regulated if either I_{i} ⊆ V or I_{i} ⊆ U for all 0 ≤ i ≤ n. We use d( · , ·) to mean the distance between two points or two sets.

Lemma 1 (The first distortion lemma). There is a constant C > 0 such that for any regulated chain ℐ = {I_{i}}_{i=0}^{n} satisfying I_{i} ⊆ V for all 1 ≤ i ≤ n − 1 and for all x and y in I_{0} ,

|log(\frac{|(f^{n})′(x)|}{|(f^{n})′(y)|})| ≤ C|x_{n} − y_{n}|^{α},

where x_{n} = f^{n}(x) and y_{n} = f^{n}(y).

Proof. Let ξ be the set of intervals of V \ SP. Let

a = inf_{x∈V \SP} |f′(x)| > 0

and let

b = sup_{x≠y∈I, I∈ξ} \frac{|f′(x) − f′(y)|}{|x − y|^{α}} < ∞.

For any x and y in I_{0} , let x_{i} = f^{i}(x) and y_{i} = f^{i}(y) for 0 ≤ i ≤ n. Then

𝒜 = \frac{|(f^{n})′(x)|}{|(f^{n})′(y)|} = ∏_{i=0}^{n−1} \frac{|f′(x_{i})|}{|f′(y_{i})|}.

This implies that

| log 𝒜 | ≤ \frac{b}{a} ∑_{i=0}^{n−1} |x_{i} − y_{i}|^{α}.

From Definition 2, there are constants C > 0 and 0 < μ < 1 such that

|x_{i} − y_{i}| ≤ Cμ^{n−i}|x_{n} − y_{n}|, 0 ≤ i ≤ n − 1.

Therefore,

| log 𝒜 | ≤ \frac{bC}{a(1 − μ^{α})}|x_{n} − y_{n}|^{α} = B|x_{n} − y_{n}|^{α}, B = (bC)/(a(1 − μ^{α})).

Lemma 2 (C^{1+Hölder} –Denjoy–Koebe type distortion lemma). There are constants C, D > 0 such that for any regulated chain ℐ = {I_{i}}_{i=0}^{n} and for all x and y in I_{0} ,

|log(\frac{|(f^{n})′(x)|}{|(f^{n})′(y)|})| ≤ C|x_{n} − y_{n}|^{α/γ} + D \frac{|x_{n} − y_{n}|}{d(\{x_{n}, y_{n}\}, \overline{PSO})},

where x_{n} = f^{n}(x) and y_{n} = f^{n}(y).

Proof. For any x and y in I_{0} , let x_{i} = f^{i}(x) and y_{i} = f^{i}(y) for 0 ≤ i ≤ n. The ratio |(f^{n})′(x)|/|(f^{n})′(y)| equals the product

𝒜 = ∏_{i=0}^{n−1} \frac{|f′(x_{i})|}{|f′(y_{i})|}.

Let (x_{i}, y_{i}) mean the open interval in I_{i} bounded by x_{i} and y_{i}. We divide the set of intervals 𝒥 = {(x_{i}, y_{i})}_{i=0}^{n−1} into two subsets

𝒥_{1} = {(x_{i}, y_{i}) | (x_{i}, y_{i}) ⊆ V } and 𝒥_{2} = {(x_{i}, y_{i}) | (x_{i}, y_{i}) ⊆ U and (x_{i}, y_{i}) ⊄ V }.

Consider

∏_{x_{i},y_{i}∈𝒥_{1}} \frac{|f′(x_{i})|}{|f′(y_{i})|} and ∏_{x_{i},y_{i}∈𝒥_{2}} \frac{|f′(x_{i})|}{|f′(y_{i})|}.

As in the proof of Lemma 1, there are constants C_{1}, C_{2} > 0 such that

|log( ∏_{x_{i},y_{i}∈𝒥_{1}} \frac{|f′(x_{i})|}{|f′(y_{i})|})| ≤ C_{1} ∑_{x_{i},y_{i}∈𝒥_{1}} |x_{i} − y_{i}|^{α} ≤ C_{2} ∑_{x_{i},y_{i}∈𝒥_{2}} |x_{i} − y_{i}|^{α}.

The rightmost expression here is obtained from (3) of Definition 2.

Suppose x_{i} and y_{i} are in U_{j_{i}}(τ). Denote e_{i} = c_{j_{i}} as the critical point in U_{j_{i}}(τ) and κ_{i} = γ_{j_{i}} as the exponent at e_{i}. Let t_{i} = (κ_{i} − 1)/κ_{i}. To estimate the product ∏_{x_{i},y_{i}∈𝒥_{2}} |f′(x_{i})|/|f′(y_{i})|, we write it as the product of three factors:

ℬ = ∏_{x_{i},y_{i}∈𝒥_{2}} (\frac{|x_{i} − e_{i}|^{κ_{i}}}{|f(x_{i}) − f(e_{i})|} \frac{|f(y_{i}) − f(e_{i})|}{|y_{i} − e_{i}|^{κ_{i}}})^{t_{i}},

𝒞 = ∏_{x_{i},y_{i}∈𝒥_{2}} \frac{|y_{i} − e_{i}|^{κ_{i}−1}}{|f′(y_{i})|} \frac{|f′(x_{i})|}{|x_{i} − e_{i}|^{κ_{i}−1}} ,

and

𝒟 = ∏_{x_{i},y_{i}∈𝒥_{2}} (\frac{|f(x_{i}) − f(e_{i})|}{|f(y_{i}) − f(e_{i})|})^{t_{i}}.

From Definition 1 and as in the proof of Lemma 1, there is a constant C_{3} > 0 such that

| log ℬ|, | log 𝒞 | ≤ C_{3} ∑_{x_{i},y_{i}∈𝒥_{2}} |x_{i} − y_{i}|^{α}.

The estimation of 𝒟 is a key part in the proof. Let

\frac{f(x_{i}) − f(e_{i})}{f(y_{i}) − f(e_{i})} = 1 + \frac{f(x_{i}) − f(y_{i})}{f(y_{i}) − f(e_{i})}.

Then

𝒟 = exp(∑_{s=1}^{r−1} \frac{1}{t_{i_{s}}} log|1 + \frac{f(x_{i_{s}}) − f(y_{i_{s}})}{f(y_{i_{s}}) − f(e_{i_{s}})}|)

where 0 ≤ i_{1} < i_{2} < ⋯ < i_{r−1} < n. Let i_{r} = n. For each i_{s}, 1 ≤ s < r, consider the interval L_{s} bounded by y_{i_{s}} and e_{i_{s}} and the map h_{s} = f^{i_{s+1}−i_{s}}. Let R_{s} ⊆ L_{s} be the maximal interval containing y_{i_{s}} such that h_{s} on R_{s} is C^{1+α} and injective. One of the endpoints of R_{s} is y_{i_{s}} and the other is a preimage, denoted as e, of a singular point q ∈ SP under f^{r_{s}} for some 0 ≤ r_{s} < i_{s+1} − i_{s}. Let l_{s} = i_{s+1} − i_{s} − r_{s}. Then h_{s} on the minimal interval J_{s} containing x_{i_{s}} and R_{s} is C^{1+α} and injective and maps J_{s} onto an interval containing the points y_{i_{s+1}} , x_{i_{s+1}} and f^{l_{s}}(q). We enlarge every interval J of V into a closed interval J′ ⊃ J such that J′ ∩ CP = ∅ and such that the length of J′ ∩ U is greater than a constant a > 0. Let V′ = ⋃_{J∈V} J′ be the union of all these enlarged intervals and let U′ = M \ V′.

We consider each 1 ≤ s < r − 1 in two cases. The first is that f^{i}(J_{s}) ⊆ V′ for all 1 ≤ i < i_{s+1} − i_{s}. Then as in the proof of Lemma 1, there is a constant C_{4} > 0 such that

\frac{|f(x_{i_{s}}) − f(y_{i_{s}})|}{|f(y_{i_{s}}) − f(c_{i_{s}})|} ≤ C_{4} \frac{|x_{i_{s+1}} − y_{i_{s+1}}|}{|y_{i_{s+1}} − f^{l_{s}}(q)|} ≤ C_{4} \frac{|x_{i_{s+1}} − y_{i_{s+1}}|}{A}

where A > 0 is the distance between U′ and \overline{PSO}. The other is opposite to the first one. Let 1 ≤ k < i_{s+1} − i_{s} be the smallest integer such that f^{k}(J_{s}) ∩ U′ ≠ ∅. Since f^{i}(J_{s}) ⊆ V′ for all 1 ≤ i < k, we have

\frac{|f(x_{i_{s}}) − f(y_{i_{s}})|}{|f(y_{i_{s}}) − f(c_{i_{s}})|} ≤ C_{4} \frac{|x_{i_{s}+k} − y_{i_{s}+k}|}{|y_{i_{s}+k} − f^{k}(e)|}.

Since y_{i_{s}+k} is in V and f^{k}(e) is in U′

\frac{|f(x_{i_{s}}) − f(y_{i_{s}})|}{|f(y_{i_{s}}) − f(c_{k_{is}})|} ≤ C_{4} \frac{|x_{i_{s}+k} − y_{i_{s}+k}|}{B}

where B > 0 is the distance between V and U′. From Definition 2, there is a constant C_{5} > 0 such that

|x_{i_{s}+k} − y_{i_{s}+k}| ≤ C_{5}|x_{i_{s+1}} − y_{i_{s+1}}|.

We get

\frac{|f(x_{i_{s}}) − f(y_{i_{s}})|}{|f(y_{i_{s}}) − f(c_{i_{s}})|} ≤ C_{4}C_{5} \frac{|x_{i_{s+1}} − y_{i_{s+1}}|}{B}.

For s = r − 1, as in the proof of Lemma 1 and in the above argument, there is a constant C_{6} > 0 such that

\frac{|f(x_{i_{r−1}}) − f(y_{i_{r−1}})|}{|f(y_{i_{r−1}}) − f(c_{i_{r−1}})|} ≤ C_{6} \frac{|x_{n} − y_{n}|}{d(\{x_{n}, y_{n}\}, \overline{PSO})}.

Therefore, there is a constant C_{7} > 0 such that

|log 𝒟 | ≤ C_{7} ∑_{x_{i},y_{i}∈𝒥_{2}} |x_{i} − y_{i}| + C_{6} \frac{|x_{n} − y_{n}|}{d(\{x_{n}, y_{n}\}, \overline{PSO})}.

Combining all the estimates, we have constants C_{8} such that

|log(\frac{|(f^{n})′(x)|}{|(f^{n})′(y)|})| ≤ C_{8} ∑_{x_{i},y_{i}∈𝒥_{2}} |x_{i} − y_{i}|^{α} + C_{6} \frac{|x_{n} − y_{n}|}{d(\{x_{n}, y_{n}\}, \overline{PSO})}.

From Definition 2, there are constants C_{10} ≥ C_{9} > 0 and 0 < ν_{1} ≤ ν_{2} < 1 such that

|x_{i} − y_{i}| ≤ C_{9}ν_{1}^{i_{r−1}−i}|x_{i_{r−1}} − y_{i_{r−1}}| ≤ C_{10}ν_{2}^{n−1}|x_{n} − y_{n}|^{1/κ_{r−1}}.

Therefore, we have constants C, D > 0 such that

|log(\frac{|(f^{n})′(x)|}{|(f^{n})′(y)|})| ≤ C|x_{n} − y_{n}|^{α/γ} + D \frac{|x_{n} − y_{n}|}{d(\{x_{n}, y_{n}\}, \overline{PSO})}.

A special case of Lemma 2 which is often used in this paper is

Lemma 3 (The second distortion lemma). Suppose U_{0} is an open set such that d(\overline{U_{0}}, \overline{PSO}) > 0. Then there is a constant C > 0 such that for any regulated chain {I_{i}}_{i=0}^{n} with I_{n} ⊆ U_{0} and for all x, y ∈ I_{0} ,

|log(\frac{|(f^{n})′(x)|}{|(f^{n})′(y)|})| ≤ C|x_{n} − y_{n}|^{α/γ},

where x_{n} = f^{n}(x) and y_{n} = f^{n}(y).

Remark 3. We also proved a similar result called the geometric distortion theorem for higher dimensional dynamical systems in [J2] (see also [J4, Chapter 2]). Its proof is also done by a detailed analysis of the reason behind Koebe’s distortion theorem in one complex variable and by combining ideas of Koebe and Denjoy. This result is a kind of a generalization of Koebe’s distortion theorem for a larger class of maps including many non-conformal maps. The reader who is interested in this result and its application to some higher dimensional dynamical systems may refer to [J4, Chapter 2].

4. One-point differentiable rigidity

We use Leb to mean the Lebesgue measure on M. Our map f (or g) in this section satisfies three more technical conditions:

(1) \overline{PSO} has measure zero, i.e., Leb(\overline{PSO}) = 0.

(2) The set \overline{PSO} is not an attractor. More precisely, there is an open neighborhood \overline{PSO} ⊂ W ≠ M such that for any point p ∈ M either f^{n} falls into \overline{PSO} eventually (i.e., \{f^{n}(p)\}_{n=N}^{∞} ⊆ \overline{PSO} for some N > 0) or it leaves W infinitely often (i.e., there is a subsequence {f^{∘n_{i}}(p)}_{i=1}^{∞} ⊆ M \ W).

(3) The map f is mixing, that is, for any intervals I, J ⊂ M, there is an integer n ≥ 0 such that f^{n}(J) ⊇ I.

The last two conditions are invariant under topological conjugacy. Condition (3) says that {f^{n}}_{n=0}^{∞} would not be decomposed into several dynamical systems.

Denote M_{0} = M \ \overline{PSO}. For any point p ∈ M, let

BO(p) = ⋃_{n=0}^{∞} f^{−n}(p)

be the set of all backward images of p. It is countable.

Suppose that f and g are two conjugated quasi-hyperbolic maps and that h is the conjugacy between f and g, i.e.,

h ∘ f = g ∘ h.

If h is differentiable at p ∈ M_{0} , then, from the last equation, h is differentiable at all points in BO(p).

Definition 5. We call h differentiable at p ∈ M_{0} with uniform bound if there are a small neighborhood Z of p and a constant C > 0 such that

C^{−1} ≤ |h′(q)| ≤ C, q ∈ BO(p) ∩ Z.
For x ∈ M, let
ω(x) = {y ∈ M | there is a subsequence f^{n_{i}}(x) → y as i → ∞}

be the set of all accumulation points of the forward orbit {f^{∘n}(x)}_{n=1}^{∞}. Then x is called self-recurrent if x ∈ ω(x).

Let Λ = ⋃_{n=0}^{∞} f^{−n}(\overline{PSO}). Then Leb(Λ) = 0. Let Ω be the set of all selfrecurrent points in M \Λ of f. We say a measurable set E in M has full Lebesgue measure if Leb(E) = Leb(M). A point p in a measurable set E_{0} is said to be a density point if

lim_{Leb(J)→0} \frac{Leb(E_{0} ∩ J)}{Leb(J)} = 1

where J runs over all intervals containing p.

Proposition 1. The set Ω has full Lebesgue measure in M.

Proof. Let Λ_{n} = ⋃_{i=0}^{n} f^{−i}(\overline{PSO}). Then Leb(Λ_{n}) = 0. Let Γ_{n} = M \ Λ_{n}. For any component J of Γ_{n}, let X(J) ⊆ J be the set of points in J such that {f^{∘n}(x)}_{n=1}^{∞} ∩ J ≠ ∅. We claim Leb(J \ X(J)) = 0. Let us first prove this claim.

Proof of the claim. We prove it by contradiction. Denote A = J \ X(J). Assume Leb(A) > 0. Almost every point in A \ Λ is a density point of A. Since Λ has zero measure, we can find a point u in A such that it is a density point of A and such that there is a subsequence {f^{∘n_{i}}(u)}_{i=1}^{∞} converging to a point q in M \ W where W is the neighborhood defined at the beginning of this section.

Let J_{0} be an open interval about q such that C_{1} = d(\overline{J_{0}}, \overline{PSO}) > 0. Assume that {f^{∘n_{i}}(u)}_{i=1}^{n} ⊂ J_{0} . Let J_{i} be the interval about u such that f^{∘n_{i}}: J_{i} → J_{0} is a C^{1+α} -diffeomorphism. Without loss of generality, we assume that {f^{∘k}(J_{i})}_{k=0}^{n_{i}} is a regulated chain for every i ≥ 1. From Definition 2, Leb(J_{i}) → 0 as i → ∞. Let A_{i} = f^{∘n_{i}}(A ∩ J_{i}) and A_{∞} = ⋂_{m=1}^{∞} ⋃_{i=m}^{∞} A_{i}. Then Lemma 3 implies that there is a constant C_{2} > 0 such that

C_{2}^{−1} ≤ \frac{|(f^{∘n_{i}})′(z)|}{|(f^{∘n_{i}})′(w)|} ≤ C_{2}

for all z, w ∈ J_{i} and all i ≥ 1. This and the fact that

lim_{i→∞} \frac{Leb(A ∩ J_{i})}{Leb(J_{i})} = 1

imply Leb(A_{∞}) = Leb(J_{0}) (refer to the proof of Theorem 1). By the construction, the forward post-orbit of every point in A_{∞} under iterates of f will not enter J. But on the other hand, because f is mixing the forward post-orbit of J_{0} under iterates of f must intersect with J. This is a contradiction. We proved the claim.

Note that Γ_{n} = M \ Λ_{n} gives a measurable partition of M and the lengths of intervals in this partition tend to zero as n goes to infinity. Thus, we have Ω = ⋂_{n=1}^{∞} ⋃_{J∈Γ_{n}} X(J). It is clear that

M \ Ω = Λ ∪ ⋃_{n=1}^{∞} ⋃_{J∈Γ_{n}}(J \ X(J)).

So

Leb(M \ Ω) ≤ Leb(Λ) + ∑_{n=1}^{∞} ∑_{J∈Γ_{n}} Leb(J \ X(J)) = 0.

In other words, Leb(Ω) = Leb(M).

We are ready to state one of our main results in this paper now.

Theorem 1. Suppose that f and g are conjugate quasi-hyperbolic maps and h is the conjugacy between f and g. If h is differentiable at one point p ∈ M_{0} with uniform bound, then h | M_{0} is C^{1} .

Proof. Suppose f and g are C^{1+α} for 0 < α ≤ 1. Let Z be an open interval about p and let C_{1} > 0 be a constant such that

d(\overline{Z}, \overline{PSO}), d(\overline{h(Z)}, \overline{h(PSO)}) > C_{1}

and such that

C_{1}^{−1} ≤ |h′(q)| ≤ C_{1}

for all q in BO(p) ∩ Z.

Let Ψ_{1} be the set of intervals of f^{−1}(Z) contained in Z. Inductively, let Ψ_{n} be the set of intervals of f^{−n}(Z) contained in Z \ (⋃_{I∈Ψ_{n−1}} I). Because f is mixing, there are infinitely many integers n such that Ψ_{n} is non-empty.

Suppose Ψ_{n} is non-empty. Then for any interval I ∈ Ψ_{n}, f^{n}: I → Z and g^{n}: h(I) → h(Z) are C^{1+α} -diffeomorphisms. Moreover we have that

\frac{|h(I)|}{|I|} = \frac{|(f^{n})′(x)|}{|(g^{n})′(h(y))|} \frac{|h(Z)|}{|Z|}

for some x, y ∈ I. Without loss of generality, we assume that {f^{k}(I)}_{k=0}^{n} and {g^{k}(h(J))}_{k=0}^{n} are regulated chains.

Let q ∈ I ⊆ Z be the preimage of p under f^{n}: I → Z. Then

\frac{(f^{n})′(q)}{(g^{n})′(h(q))} = \frac{h′(q)}{h′(p)}.

So we have that

C_{1}^{−2} ≤ \frac{|(f^{n})′(q)|}{|(g^{n})′(h(q))|} ≤ C_{1}^{2}.

Therefore,

C_{1}^{−2} \frac{|(f^{n})′(x)|}{|(f^{n})′(q)|} \frac{|(g^{n})′(h(q))|}{|(g^{n})′(h(y))|} \frac{|h(Z)|}{|Z|} ≤ \frac{|h(I)|}{|I|} ≤ C_{1}^{2} \frac{|(f^{n})′(x)|}{|(f^{n})′(q)|} \frac{|(g^{n})′(h(q))|}{|(g^{n})′(h(y))|} \frac{|h(Z)|}{|Z|}.

Now Lemma 3 implies that there is a constant C_{2} > 0 such that

C_{2}^{−1} ≤ \frac{|(f^{n})′(x)|}{|(f^{n})′(q)|} ≤ C_{2} and C_{2}^{−1} ≤ \frac{|(g^{n})′(h(q))|}{|(g^{n})′(h(y))|} ≤ C_{2}.

So we have a constant C_{3} > 0 such that

C_{3}^{−1} ≤ \frac{|h(I)|}{|I|} ≤ C_{3}.

Now we are going to prove that h | Z is bi-Lipschitz. Suppose x < y are in Z. Let Ψ_{1}(x, y) be the set of intervals of f^{−1}(Z) contained in [x, y]. Inductively, let Ψ_{n}(x, y) be the set of intervals of f^{−n}(Z) contained in [x, y] \ (⋃_{I∈Ψ_{n}} I). Then

⋃_{n=1}^{∞} ⋃_{I∈Ψ_{n}(x,y)} I

is the union of pairwise disjoint intervals and its closure is [x, y]. Let

A = [x, y] \ ( ⋃_{n=1}^{∞} ⋃_{I∈Ψ_{n}(x,y)} I).

Since every point z ≠ x, y in A is not self-recurrent, now Proposition 1 implies that Leb(A) = 0. Hence

Leb( ⋃_{n=1}^{∞} ⋃_{I∈Ψ_{n}(x,y)} I) = ∑_{n=1}^{∞} ∑_{I∈Ψ_{n}(x,y)} |I| = [x, y].

Similarly,

m( ⋃_{n=1}^{∞} ⋃_{I∈Ψ_{n}(x,y)} h(I)) = ∑_{n=1}^{∞} ∑_{I∈Ψ_{n}(x,y)} |h(I)| = h([x, y]).

The additive formula implies that

C_{3}^{−1} ≤ \frac{|h(x) − h(y)|}{|x − y|} ≤ C_{3}.

Therefore, h | Z is bi-Lipschitz.

Since h | Z is bi-Lipschitz, h′ exists a.e. in Z and is integrable (refer to [Br], [E]). Since (h | Z)′(x) is measurable, h | Z is a homeomorphism, and Leb(Λ) = 0, we can find a point p_{0} in Z \ Λ and a subset E_{0} containing p_{0} such that (refer to [Br], [E])

(1) h | Z is differentiable at every point in E_{0} ;

(2) p_{0} is a density point of E_{0} ;

(3) h′(p_{0}) ≠ 0; and

(4) the derivative h′ | E_{0} is continuous at p_{0} . From the beginning of this section, we know that there is a subsequence

{f^{n_{k}}(p_{0})}_{k=1}^{∞} ⊆ M \ W converging to a point q_{0} in M \ W. Let I_{0} = (a, b) be an open interval about q_{0} such that C_{4} = d(\overline{I_{0}}, \overline{PSO}) > 0. There is a sequence of intervals {I_{k}}_{k=1}^{∞} such that p_{0} ∈ I_{k} ⊆ Z and f^{n_{k}}: I_{k} → I_{0} is a C^{1+α} diffeomorphism. Without loss of generality, we may assume that {I_{l,k} = f^{l}(I_{k})}_{l=0}^{n_{k}} is a regulated chain for every k ≥ 1. From Definition 2, Leb(I_{k}) goes to zero as k tends to infinity. By Lemma 3, there is a constant C_{5} > 0, such that

|log(\frac{|(f^{n_{k}})′(w)|}{|(f^{n_{k}})′(z)|})| ≤ C_{5}

for any w and z in I_{k} and all k ≥ 1.

For any positive integer s, there is an integer N_{s} > 0 such that

\frac{Leb(E_{0} ∩ I_{k})}{Leb(I_{k})} ≥ 1 − \frac{1}{s}

for all k > N_{s}. Let E_{k} = f^{n_{k}}(E_{0} ∩ I_{k}). Then h is differentiable at every point in E_{k} and there is a constant C_{6} > 0 such that

\frac{Leb(E_{k} ∩ I_{0})}{Leb(I_{0})} ≥ 1 − \frac{C_{6}}{s}

for all k > N_{s} because {f^{n_{k}} | I_{k}}_{k=1}^{∞} have uniformly bounded distortion. Let E = ⋂_{s=1}^{∞} ⋃_{k>N_{s}} E_{k}. Then E has full measure in I_{0} and h is differentiable at every point in E with non-zero derivative.

Next, we are going to prove that h′ | E is uniformly continuous. For any x and y in E, let z_{k} and w_{k} be the preimages of x and y under the diffeomorphism f^{n_{k}}: I_{k} → I_{0} . Then z_{k} and w_{k} are in E_{0} . From h ∘ f = g ∘ h, we have that

h′(x) = \frac{(g^{n_{k}})′(h(z_{k}))}{(f^{n_{k}})′(z_{k})}h′(z_{k})

and

h′(y) = \frac{(g^{n_{k}})′(h(w_{k}))}{(f^{n_{k}})′(w_{k})}h′(w_{k}).

So

|log(\frac{h′(x)}{h′(y)})| ≤ |log|\frac{(g^{n_{k}})′(h(z_{k}))}{(g^{n_{k}})′(h(w_{k}))}| | + |log|\frac{(f^{n_{k}})′(w_{k})}{(f^{n_{k}})′(z_{k})}| | + |log(\frac{h′(z_{k})}{h′(w_{k})})|.

Applying Lemma 3 to both f and g, we can find a constant C_{7} > 0 such that

|log|\frac{(f^{n_{k}})′(w_{k})}{(f^{n_{k}})′(z_{k})}| | ≤ C_{7}|x − y|^{α}

and

|log|\frac{(g^{n_{k}})′(h(z_{k}))}{(g^{n_{k}})′(h(w_{k}))}| | ≤ C_{7}|h(x) − h(y)|^{α}

for all k ≥ 1. Therefore,

|log(\frac{h′(x)}{h′(y)})| ≤ C_{7}(|x − y|^{α} + |h(x) − h(y)|^{α}) + |log(\frac{h′(z_{k})}{h′(w_{k})})|

for all k ≥ 1. Since h′ | E_{0} is continuous at p_{0} , the last term in the last inequality tends to zero as k goes to infinity. Hence

|log(\frac{h′(x)}{h′(y)})| ≤ C_{7}(|x − y|^{α} + |h(x) − h(y)|^{α}).

This means that h′ | E is uniformly continuous. So it can be extended to a continuous function ϕ on I_{0} . Because h | I_{0} is absolutely continuous and E has full measure,

h(x) = h(a) + ∫_{a}^{x} h′(x) dx = h(a) + ∫_{a}^{x} ϕ(x) dx

on I_{0} . This implies that h | I_{0} is actually C^{1} .

Now for any x ∈ M_{0} , let J ⊂ M_{0} be an open interval about x. By the mixing condition, there are an integer n > 0 and an open interval J_{0} ⊂ I_{0} such that f^{n}: J_{0} → J is a C^{1} diffeomorphism. By the equation h ∘ f = g ∘ h, we have that h | J is C^{1} . Therefore, h | M_{0} is C^{1} .

Consider the conjugacy h(x) = (2/π) arcsin(x) between f(x) = 1 − 2x^{2} and g(x) = 1 − 2|x| on [−1, 1]. The maps h and h^{−1} are both C^{1} on (−1, 1). But h′ is not uniformly continuous because the exponents of f and g at 0 are different. Note that the exponent at a singular point is invariant under C^{1} conjugacy. Furthermore, we have the following result to enhance Theorem 1.

Theorem 2. Suppose f and g and h are the same as those in Theorem 1 . If h is differentiable at one point in M_{0} with uniform bound and all the exponents of f and g at the corresponding singular points are the same, then h restricted to the closure of every interval of M_{0} is a C^{1+β} diffeomorphism for some 0 < β ≤ 1.

We prove this theorem by two lemmas. Let f and g be the maps and h be the conjugacy in Theorem 2. Suppose f and g are both C^{1+α}. Let U = U(τ) and V = V (τ) be the sets satisfied by both f and g. Let I_{0} and q_{0} be the interval and the point found in the proof of Theorem 1. Denote CP = {c_{1}, …, c_{d}} as the set of critical points and Γ = {γ_{1}, …, γ_{d}} as the corresponding exponents. Let γ = max{γ_{i} | 1 ≤ i ≤ d}. Let U_{i}(τ) = [c_{i} − τ, c_{i} + τ] for c_{i} ∈ CP.

Remark 4. Under the above assumption, β = α/γ in Theorem 2.

Lemma 4. The map h | U_{i}(τ) is C^{1+α}.

Proof. We use the same notation as in the proof of Theorem 1. Denote J = U_{i}(τ) = [d, e]. Since f has mixing condition and J ∩ \overline{PSO} = ∅, there is a preimage J_{k} ⊂ I_{0} of J under f^{n_{k}} such that J_{k} tends to q_{0} as k → ∞ and such that f^{n_{k}}: J_{k} → J is a C^{1} diffeomorphism where {n_{k}}_{k=1}^{∞} is a subset of the positive integers. Let F_{k} be the inverse of f^{n_{k}}: J_{k} → J. From the equation h ∘ f = g ∘ h, we have that h | J = g^{n_{k}} ∘ h ∘ F_{k}. So h | J is C^{1} and

h′(x) = \frac{(g^{n_{k}})′(h(z_{k}))}{(f^{n_{k}})′(z_{k})}h′(z_{k}) ≠ 0

for any x in J where z_{k} = F_{k}(x).

Without loss of generality, we assume that {J_{i,k} = f^{n_{k}−i}(J_{k})}_{i=0}^{n_{k}} are all regulated chains for all k > 0. For any x and y in J, let z_{k} and w_{k} be the preimage of x and y under the diffeomorphism f^{n_{k}}: J_{k} → J. Since

\frac{h′(x)}{h′(y)} = \frac{(g^{n_{k}})′(h(z_{k}))}{(f^{n_{k}})′(z_{k})} · \frac{(f^{n_{k}})′(w_{k})}{(g^{n_{k}})′(h(w_{k}))} · \frac{h′(z_{k})}{h′(w_{k})},

from Lemma 3, there is a constant C > 0 such that

|log(\frac{h′(x)}{h′(y)})| ≤ |log|\frac{(g^{n_{k}})′(h(z_{k}))}{(g^{n_{k}})′(h(w_{k}))}| | + |log|\frac{(f^{n_{k}})′(w_{k})}{(f^{n_{k}})′(z_{k})}| | + |log(\frac{h′(z_{k})}{h′(w_{k})})|

≤ C(|x − y|^{α} + |h(x) − h(y)|^{α}) + |log(\frac{h′(z_{k})}{h′(w_{k})})|.

Because h′(z_{k}), h′(w_{k}) → h′(q_{0}) as k → ∞, we have

|log(\frac{h′(x)}{h′(y)})| ≤ C(|x − y|^{α} + |h(x) − h(y)|^{α}).

This implies that h | J is actually C^{1+α}.

Lemma 5. The restriction of h to the closure of every interval J of V \\overline{PSO} is C^{1+(α/γ)} .

Proof. We always use C to denote a positive constant (although it may be different in different formulas). Since f has mixing condition, we can find a subsequence {n_{k}}_{k=1}^{∞} of the positive integers and intervals J_{k} ⊂ I_{0} such that J_{k} tends to q_{0} as k → ∞ and such that f^{n_{k}}: J_{k} → J is a C^{1} diffeomorphism. Let F_{k} be the inverse of f^{n_{k}}: J_{k} → J. From the equation h ∘ f = g ∘ h, we have that h | J = g^{n_{k}} ∘ h ∘ F_{k}. So h | J is C^{1} and

h′(x) = \frac{(g^{n_{k}})′(h(z_{k}))}{(f^{n_{k}})′(z_{k})}h′(z_{k}) ≠ 0

for any x in J where z_{k} = F_{k}(x).

Let J_{i,k} = f^{n_{k}−i}(J_{k}) for 0 ≤ i ≤ n_{k}. Without loss of generality, we assume that {J_{i,k}}_{i=0}^{n_{k}} are regulated chains for all k > 0. For any x and y in J, let z_{k} and w_{k} be the preimage of x and y under the diffeomorphism f^{n_{k}}: J_{k} → J. From the equation h ∘ f = g ∘ h, we have

\frac{h′(x)}{h′(y)} = \frac{(g^{n_{k}})′(h(z_{k}))}{(f^{n_{k}})′(w_{k})} · \frac{(f^{n_{k}})′(w_{k})}{(g^{n_{k}})′(h(w_{k}))} · \frac{h′(z_{k})}{h′(w_{k})}.

Thus,

|log(\frac{h′(x)}{h′(y)})| ≤ |log(\frac{|(g^{n_{k}})′(h(z_{k}))|}{|(g^{n_{k}})′(h(w_{k}))|})| + |log(\frac{|(f^{n_{k}})′(w_{k})|}{|(f^{n_{k}})′(z_{k})|})| + |log(\frac{h′(z_{k})}{h′(w_{k})})|.

Let n = n_{k} and let m = m(n_{k}) > 0 be the smallest integer such that J_{m,k} ⊆ U_{j}(τ) ⊆ U for some 1 ≤ j ≤ d. Let x_{i} = f^{n_{k}−i}(z_{k}) and y_{i} = f^{n_{k}−i}(w_{k}) for all 0 ≤ i ≤ n_{k}. Then

|log(\frac{h′(x)}{h′(y)})| ≤ | ∑_{i=1}^{m−1}(log |f′(y_{i})| − log |f′(x_{i})|)|

+ | ∑_{i=1}^{m−1}(log|g′(h(x_{i}))| − log|g′(h(y_{i}))|)|

+ |log(\frac{|g′(h(x_{m}))|}{|f′(x_{m})|} · \frac{|f′(y_{m})|}{|g′(h(y_{m}))|})|

+ | ∑_{i=m+1}^{n}(log |f′(y_{i})| − log |f′(x_{i})|)|

+ | ∑_{i=m+1}^{n}(log|g′(h(x_{i}))| − log|g′(h(y_{i}))|)|

+ |log(\frac{h′(z_{k})}{h′(w_{k})})|.

The last term tends to zero as k goes to infinity. We estimate the first five terms. From Lemma 1, there is a constant C > 0 such that

| ∑_{i=1}^{m−1}(log |f′(y_{i})| − log |f′(x_{i})|)| ≤ C|x − y|^{α}

380 Yunping Jiang and

| ∑_{i=1}^{m−1}(log|g′(h(x_{i}))| − log|g′(h(y_{i}))|)| ≤ C|h(x) − h(y)|^{α}.

From Lemma 3, there are constants C > 0 such that

\begin{aligned}| ∑_{i=m+1}^{n}(log |f′(y_{i})| − log |f′(x_{i})|)| & ≤ C|x_{m} − y_{m}|^{α} \\ & ≤ C|x_{m−1} − y_{m−1}|^{α/γ_{j}} ≤ C|x − y|^{α/γ_{j}}. \end{aligned}

Similarly,

| ∑_{i=m+1}^{n}(log|g′(h(x_{i}))| − log|g′(h(y_{i}))|)| ≤ C|h(x) − h(y)|^{α/γ_{j}}.

Now we consider

𝒮 = \frac{|g′(h(x_{m}))|}{|f′(x_{m})|} · \frac{|f′(y_{m})|}{|g′(h(y_{m}))|}.

Define

𝒮 = 𝒮_{1} · 𝒮_{2} · 𝒮_{3}

where

𝒮_{1} = \frac{|g′(h(x_{m}))|}{|h(x_{m}) − h(c_{j})|^{γ_{j}−1}} · \frac{|h(y_{m}) − h(c_{j})|^{γ_{j}−1}}{|g′(h(y_{m}))|} ,
𝒮_{2} = \frac{|x_{m} − c_{j}|^{γ_{j}−1}}{|f′(x_{m})|} · \frac{|f′(y_{m})|}{|y_{m} − c_{j}|^{γ_{j}−1}} ,

and

𝒮_{3} = (\frac{|h(x_{m}) − h(c_{j})|}{|x_{m} − c_{j}|})^{γ_{j}−1} · (\frac{|y_{m} − c_{j}|}{|h(y_{m}) − h(c_{j})|})^{γ_{j}−1}.

Now Lemma 4 implies that

|log S_{3}| ≤ C|x_{m} − y_{m}|^{α} ≤ C|x_{m−1} − y_{m−1}|^{α/γ_{j}} ≤ C|x − y|^{α/γ_{j}}.

From Definition 1,

|log S_{2}| ≤ C|x_{m} − y_{m}|^{α} ≤ C|x_{m−1} − y_{m−1}|^{α/γ_{j}} ≤ C|x − y|^{α/γ_{j}}

and |log S_{1}| ≤ C|h(x_{m}) − h(y_{m})|^{α} ≤ C|h(x_{m−1}) − h(y_{m−1})|^{α/γ_{j}} ≤ C|h(x) − h(y)|^{α/γ_{j}}. Thus, as k goes to infinity, we have that

|log(\frac{h′(x)}{h′(y)})| ≤ C(|x − y|^{α/γ} + |h(x) − h(y)|^{α/γ}).

This implies that h′ | J is actually C^{α/γ}. So is h′ | J̄. We have that h | J̄ is C^{1+α/γ}.

Proof of Theorem 2. Both Lemmas 4 and 5 for h and h^{−1} complete the proof.

We can use eigenvalues of f and g at corresponding periodic points to verify the condition, differentiable at one point with uniform bound, in Theorems 1 and 2.

Lemma 6. Suppose f and g and h are those in Theorem 1. If h is differentiable at a point p in M_{0} with non-zero derivative and if there is an open interval Y about p such that the absolute values of the eigenvalues of f and g at periodic points in Y and at corresponding periodic points in h(Y) are the same, then h is differentiable at p with uniform bound.

Proof. Let BO(p) be the backward orbit of p. Assume that C_{1} = d(\overline{Y} , \overline{PSO}) > 0. From Definition 2, there is an integer N ≥ 0 and an open interval Z about p such that every component of f^{−n}(Y) which intersects with Z is contained in Y for n > N.

For any q ∈ BO(p) ∩ Z, we have f^{n}(q) = p for some n ≥ 0. Since there are only a finite number of points in ⋃_{i=0}^{N} f^{−i}(p), we need only consider n > N. There is an open interval q ∈ J_{q} such that f^{n}: J_{q} → Y is a C^{1} -diffeomorphism where J_{q} ⊆ Y . Therefore, there is a fixed point r of f^{n} in J_{q}. By the assumption, we have that

|(f^{n})′(r)| = |(g^{n})′(h(r))|.

From the equation h ∘ f = g ∘ h, we also have that

h′(q) = \frac{(f^{n})′(q)}{(g^{n})′(h(q))}h′(p).

These imply that

|h′(q)| = \frac{|(f^{n})′(q)|}{|(f^{n})′(r)|} \frac{|(g^{n})′(h(r))|}{|(g^{n})′(h(q))|}|h′(p)|.

Without loss of generality, we assume that {f^{k}(J_{q})}_{k=0}^{n} is a regulated chain. Now applying Lemma 3, there is a constant C_{2} > 0 such that

C_{2}^{−1} ≤ \frac{|(f^{n})′(q)|}{|(f^{n})′(r)|} ≤ C_{2}

and

C_{2}^{−1} ≤ \frac{|(g^{n})′(h(r))|}{|(g^{n})′(h(q))|} ≤ C_{2}.

But h′(p) ≠ 0. We get a constant C_{3} > 0 such that

C_{3}^{−1} ≤ |h′(q)| ≤ C_{3}

for all q ∈ BO(p) ∩ Z.

The above lemma combined with Theorems 1 and 2 gives us the following result.

Corollary 1. Suppose f and g and h are those in Theorem 1. If h is differentiable at one point p in M_{0} with non-zero derivative and if the absolute values of the eigenvalues of f and g at periodic points in a small neighborhood Y of p and at corresponding periodic points in h(Y) are the same, then h | M_{0} is C^{1} . Furthermore, if all the exponents of f and g at the corresponding singular points are also the same, then h restricted on the closure of every interval of M_{0} is a C^{1+β} diffeomorphism for some fixed 0 < β ≤ 1.

The following rigidity result can now be obtained from Theorems 1, 2, and Corollary 1.

Corollary 2. Suppose f and g and h are those in Theorem 1. If there is a small interval Y of M such that h | Y is absolutely continuous, then h | M_{0} is C^{1} . Furthermore, if all the exponents of f and g at the corresponding singular points are also the same, then h restricted on the closure of every interval of M_{0} is C^{1+β} for some fixed 0 < β ≤ 1.

Proof. Since h | Y is absolutely continuous, it is differentiable a.e. on Y . Moreover, since h is a homeomorphism, there is a positive measure set in Z such that h is differentiable with non-zero derivatives. An absolutely continuous map preserves the absolute values of all the eigenvalues of f and g at periodic points in Y ∩ M_{0} and at corresponding periodic points in h(Y ∩ M_{0}). Therefore, the corollary follows from Corollary 1.

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Received 18 August 2004