A generic dimensional property of the invariant measures for circle diffeomorphisms

Given any Liouville number $\alpha$, it is shown that the nullity of the Hausdorff dimension of the invariant measure is generic in the space of the orientation preserving $C^\infty$ diffeomorphisms of the circle with rotation number $\alpha$.


Introduction
Denote by F the group of the orientation preserving C ∞ diffeomorphisms of the cirlce. For α ∈ R/Z, denote by F α the subspace of F consisting of all the diffeomorphisms whose rotation numbers are α, and by O α the subspace of F α of all the diffeomorphisms that are C ∞ conjugate to R α , the rotation by α.
In [Y1], J.-C. Yoccoz showed that O α = F α if α is a non-Liouville number. Before than that, M. R. Herman ([H], Chapt. XI) had obtained the converse by showing that for any Liouville number α the subspace O α is meager in F α .
For f ∈ F α , α irrational, denote by µ f the unique probability measure on S 1 which is invariant by f . The properties of µ f reflect the regularity of the conjugacy of f to R α . In [S], Victoria Sadovskaya improved the above result of M. R. Herman as follows. For d ∈ [0, 1] define where dim H (·) denotes the Hausdorff dimension of a measure. She showed that for any Liouville number α and any d ∈ [0, 1], the set S d α is nonempty. Notice that the Hausdorff dimension is an invariant of the equivalence classes of measures, and therefore dim H (µ f ) < 1 implies that µ f is singular w. r. t. the Lebesgue measure.
In [Y2], J.-C. Yoccoz showed the following theorem Theorem 1.1. For any irrational number the space O α is dense in F α in the C ∞ topology.
The proof of V. Sadovskaya is based on the method of fast approximation by conjugacy with estimate, developed in [FS], and if it is slightly modified it can be combined with the above theorem to show that for any Liouville number α and for any d ∈ [0, 1] the set S d α is C ∞ dense in F α . On the other hand M. R. Herman ([H],Prop I.8,p. 167) showed that the set S of the diffeomorphsim f ∈ F such that µ f is singular contains a G δ set in the C 1 topology in F .
These two results joined together does not imply immediately that S ∩ F α is a dense G δ set in the C r topology, as pointed out to the author by Mostapha Benhenda. The purpose of this paper is to settle down the situation. In fact we get a bit more.
Theorem 1. For any Liouville number α, the set S 0 α contains a countable intersection of C 0 open and C ∞ dense subsets of F α .

2.
1. An irrational number α is called a Liouville number if for any N ∈ N there is p/q ((p, q) = 1) such that |α − p/q| < 1/q N . We call α a lower Liouville number if the above p/q satisfies in addition that p/q < α. 1 For any lower Louville number α, N ∈ N and δ > 0 there is p/q such that |α − p/q| < δ/q N and p/q < α.

For a metric space
where B(x, r) denotes the open metric ball centered at x of radius r. The Hausdorff dimension of Z, denoted by dim H (Z), is defined by The lower box dimension of Z, denoted by dim B (Z), is defined by where N (ε, Z) denotes the minimal cardinality of ε-dense subsets of Z. Let X be a compact metric space, and µ a probability measure on X. The Hausdorff dimension dim H (µ) and the lower box dimension dim B (µ) of µ are defined respectively by It is well known that dim H (µ) ≤ dim B (µ).

2.3.
The proof of Theorem 1 is by the method of fast approximation by conjugacy with estimate. Let us prepare inequalities about the derivatives of circle diffeomorphisms which are necessary for the estimate.
For a C ∞ function ϕ on S 1 , we define as usual the C r norm ϕ r (0 ≤ r < ∞) by Since we include 1 in the definition of |f | r , we get the following inequality from the Faà di Bruno formula ( [H], p.42 or [S]).
The following inequality can be found as Lemma 5.6 of [FS] or as Lemma 3.2 of [S].
For q ∈ N, denote by π q : S 1 → S 1 the q-fold covering map. Simple computation shows: Lemma 2.3. Let h be a lift of k ∈ F by π q and assume Fix(h) = ∅. Then we have |h| r ≤ |k| r q r−1 .

2.4.
Here we prepare necessary facts about Moebius transformations on the circle. Let S 1 C = {z ∈ C | |z| = 1}, and Möb + (S 1 C ) the group of the orientation preserving Moebius transformations of C which leaves S 1 C invariant. We identify S 1 C with the circle S 1 = R/Z in a standard way. For k ∈ Möb + (S 1 C ), the diffeomorphism of S 1 corresponding to k is denoted byk. Define the expanding interval I(k) ofk by Then the inverse formula of the derivatives shows that The transformation k a is hyperbolic with an attractor z = 1 and a repellor z = −1.
Lemma 2.4. There is a constant C 3 (r) > 0 depending only on r such that for any 1/2 ≤ a < 1, Proof. First of all ρ(k a ) is proportional to the radius of the isometric circle of k a , {z ∈ C | |k ′ a (z)| = 1}, and the latter can easily be computed using the expression It follows that there is a constant c > 0 such that For k a , looked upon as a map from S 1 C to S 1 C , the real r-th derivative w. r. t. the angle coordinate is denoted by D r k a , while ϕ ′ denotes the complex derivative of a holomorphic map ϕ. It suffices to show for any r and z ∈ S 1 C , For r = 1, this follows immediately from (2.1) since Dk a = |k ′ a |. Now Dk a extends to a holomorphic function on a neighbourhood of S 1 C . Since arg(k ′ a (z)) = arg(k a (z)/z) and |k a (z)/z| = 1 for z ∈ S 1 C , we have It follows that where (Dk a ) ′ (z) = P 1 (z + a) 2 + Q 1 (az + 1) 2 , and P 1 and Q 1 are polynomials in z and a, showing (2.2) for r = 2. Now since Dk a is real valued on S 1 C , its derivative along the direction tangent to S 1 C is real. Therefore D 2 k a extends to a holomorphic function as This shows that showing (2.2) for r = 3.
The last argument for r = 3 can be applied for any r ≥ 4, completing the proof of the lemma.

The G δ set
In the rest of the paper we choose an arbitrary Liouville number α and fix it once and for all. Let us assume that α is a lower Liouville number (See 2.1.), the other case being dealt with similarly. In this section we define a G δ set B of F α in the C 0 topology, and show that any f ∈ B satisfies that dim H (µ f ) = 0. Notice that by the lower Liouville property, for p/q well approximating α, the iterate R q α has rotation number qα − p, a very small positive number.
Definition 3.1. For any n ∈ N, we define B n ⊂ F α to be the subset consisting of those f which satisfy the following condition.
There exist integers q n = q n (f ) > n, l n = l n (f ) > 0 and points c i = c n i (f ), with c qn = c 0 and d qn = q 0 , satisfying the following properties.
Clearly B n is C 0 open in F α , and therefore their intersection B = ∩ n B n is a G δ -set.
The following lemma follows from the flexibility of Definition 3.1, e. g. (3.1.c).
Since n is an arbitrary integer ≥ m, we have shown that dim B (C m ) = 0, as is required.

proof
The purpose of this section is to show: The proposition asserts that R α belongs to the C ∞ closure of B, which implies that B is C ∞ dense in F α , by virtue of Theorem 1.1 and Lemma 3.2. This, together with the fact that B is a G δ set of F α in the C 0 topology, completes the proof of Theorem 1.
Our overall strategy of the proof of Proposition 4.1 is as follows. Since α is lower Liouville we can choose a sequence of rationals α n = p n /q n well approximating α so that α n ր α. We shall construct a diffeomorphism h n ∈ F commuting with R αn , and set H n = h 1 h 2 · · · h n , f n = H n R αn+1 H −1 n . We are going to show that f n 's converge to f ∈ B. The commutativity condition above is quite useful when we estimates the norm of the functions Now a concrete construction gets started. Choose α 1 = p 1 /q 1 so that (A) 0 < α − α 1 < 2 −(r+1) , and let f 0 = R α1 .
Finally set h n to be the lift ofk an by the q n covering with Fix(h n ) = ∅,, where 1/2 ≤ a n < 1 is chosen such that Notice that h n R αn h −1 n = R αn and that h ±1 n − id 0 ≤ 2 −1 q −1 n . Lemma 4.2. We have d n+r (f n−1 , f n ) < 2 −(n+r+1) for n ≥ 1.
Proof. The proof is a routine calculation using the lemmata in Sect. 2 and condition (C). Just notice that f n−1 = H n R αn H −1 n , while f n = H n R αn+1 H −1 n , and that 0 < α n+1 − α n < α − α n .
Corollary 4.3. The limit f = lim n→∞ f n is a C ∞ diffeomorphism and d r (f, R α ) ≤ 2 −r .
Proof. The latter assertion is obtained from (A) and the following estimate.
Lemma 4.4. There exists a homeomorphism H of S 1 such that d 0 (H n , H) → 0. In what follows, we fix n ∈ N once and for all and will show that f ∈ B n . First of all let us study the dynamics of h n in details. Recall that h n is a lift ofk an by the q n -fold covering. So h n has q n repelling fixed points and q n attracting fixed points.

On the other hand by (D)
The expanding interval I(k n ) ofk n (See 2.4.) is centered at 1/2 and has length 2q −(n+1) n by (B), and I(k −1 n ) is the interval centered at 0 of the same length. Recall the dynamics ofk −1 an :k −1 an (I(k −1 an )) = S 1 \ Int I(k an ). Let [c ′ i , d ′ i ], 1 ≤ i ≤ q n be the lift of I(k −1 an ), located in this order in S 1 . Their ] are lifts of I(k an ), and has the same length 2q In fact, (4.2) can be shown by where we have used (E) in the last inequality. From (4.1), (4.2) and (4.3), we obtain that On the other hand we have n . Now the rotation number of R qn α is q n α − p n , a very small positive number. Let us estimate how long the orbit by R qn α of c ′′ i stays in the interval (c ′′ i , d ′′ i ). Let m n be the largest integer such that Then we have by (4.5) m n = ⌊(d ′′ i − c ′′ i )(q n α − p n ) −1 ⌋ ≥ 2 −1 q −1 n (q n α − p n ) −1 − 1. Next estimate how quickly the orbit of d ′′ i exits [d ′′ i , c ′′ i+1 ]. Let l n be the smallest positive integer such that R lnqn ]. Then it follows from (4.4) that l n = ⌊(c ′′ i+1 − d ′′ i )(q n α − p n ) −1 ⌋ + 1 ≤ 2 −(n+2) q −1 n (q n α − p n ) −1 + 1. By (C) the number q −1 n (q n α − p n ) −1 is sufficiently big, and we have (4.6) m n ≥ 2 n l n . Now consider f = HR α H −1 . Let c i = Hc ′′ i and d i = Hd ′′ i . Then m n is the largest integer such that f kqn c i ∈ (c i , d i ) if 1 ≤ k ≤ m n and l n the smallest positive integer such that f lnqn d i ∈ [d i , c i+1 ]. Thus (4.6) implies (3.1.d) and (3.1.e).