FRITZ SCHWEIGER
A milestone in the history of metric number theory was Kuzmin’s proof of Gauss conjecture [2]. In an earlier attempt I tried to generalize this result for a suitable class of fibred systems (see [3]) with a convergence rate of orderσ(s). Hereσ(s) is the maximum of the diameters of cylinders of ranks. Unfortunately, some years later Berechet and Iosifescu detected a serious flaw in the proof. A subsequent pa- per could partially remedy the situation but the convergence rate in this corrected proof was lowered toσ(√
s) [5]. However, I am still convinced that the conver- gence rate is at leastσ(s) (but see [6] for a better rate for continued fractions).
In this note a new proof for this result is offered.
AMS 2010 Subject Classification: 11K55, 28D99.
Key words: ergodic theory, invariant measures.
1. INTRODUCTION
Let (B, T) be a fibred system [3, 4] with given measure λ. This notion includes well known dynamical systems like g-adic expansions and continued fractions. Let A be the transfer operator defined by the equation
Z
T−1E
fdλ= Z
E
(Af)dλ.
Then a Kuzmin theorem is a statement of the form
s→∞lim Asf =h Z
B
fdλ, where h is the density of the invariant measureµλ.
Many years ago for a certain class of fibred systems and for a class of functions the convergence result
Asf−h Z
B
fdλ=O(σ(s)) was claimed. Here
σ(s) := max
(k1,...,ks)diamB(k1, . . . , ks).
REV. ROUMAINE MATH. PURES APPL.,56(2011),3, 229–234
This result looked very natural but however the proof contained a serious gap which was detected many years later by A. Berechet. The proof could be restored only partially. In [5] essentially the result
Asf−h Z
B
fdλ=O(σ(√ s))
is proved. A Kuzmin theorem with exponential convergence rate is given in [1].
In this paper we suppose that the qualitative result
s→∞lim Asf =h Z
B
fdλ
is true. We show that for d-dimensional fibred systems the original claim can be proved. The class of fibred systems ([0,1]d, T) considered here is subject to the following conditions.
(A) lims→∞σ(s) = 0.
(B) The map T can be extended as a C1-map to everyB(k), the closure of B(k), andTsB(k1, . . . , ks) = [0,1]d=B for all cylinders.
(C) If
Z
E
ω(k1, . . . , ks)dλ= Z
T−sE∩B(k1,...,ks)
dλ then there is a constantC ≥1 such that
sup
x∈[0,1]d
ω(k1, . . . , ks;x)≤C inf
x∈[0,1]dω(k1, . . . , ks;x).
(D) There is a constantR≥1 such that
kV(k1, . . . , ks)x−V(k1, . . . , ks)yk ≤Rkx−yk.
(E) There is a constantL such that
|ω(k1, . . . , ks;x)−ω(k1, . . . , ks;y)| ≤Lλ(B(k1, . . . , ks))kx−yk.
Conditions (A)–(C) are sufficient to ensure the existence of a finite in- variant measure µ∼λand the ergodicity of T.
The class of functionsf : [0,1]d→Rsatisfies the conditions (α) 0< m0 ≤f ≤M0, m0 =m0(f), M0 =M0(f), (β) |f(x)−f(y)| ≤N0kx−yk, N0 =N0(f).
It is easy to see that the iterates Asf, belong to the same class with some uniform constants m1=m1(f), M1 =M1(f), N1 =N1(f).
Conditions (A)–(E) show that a Kuzmin theorem is valid in the form Asf−h
Z
B
fdλ=O(σ(√ s)).
We will apply this result to the Jacobians of the map but we will give a stronger convergence rate for continuous functions f ≥0 which satisfy a Lip- schitz condition (β). Instead of working with the transfer operator A it is more convenient to use the transfer operatorU which belongs to the invariant measure µ, namely
Z
E
(U f)dµ= Z
T−1E
fdµ.
Therefore we replace the Jacobians ω(k1, . . . , ks) by the equivalent quantities κ(k1, . . . , ks) which are defined by
Z
E
κ(k1, . . . , ks)dµ= Z
T−sE∩B(k1,...,ks)
dµ.
Then we claim the following result.
Theorem. We have Usf−
Z
B
fdµ=O(σ(s)).
2. KUZMIN’S THEOREM
This section is devoted to the proof of the announced theorem.
Proof. By the choice of the operatorU we see that U1 = 1,
which is equivalent to
X
(k1,...,ks)
κ(k1, . . . , ks) = 1.
We suppose
Z
B
fdµ= 1.
The proof will be divided into several steps.
1. There exist pointsξs∈B(k1, . . . , ks) such that
1 = X
(k1,...,ks)
f(ξs)µ(B(k1, . . . , ks)).
2. We remark that the series X
(k1,...,ks)
f(V(k1, . . . , ks)x)κ(k1, . . . , ks;x) = (Usf)(x) is absolutely and uniformly convergent.
3. Denote byαs andβs points from B(k1, ..., ks) such that for all points y ∈ B(k1, . . . , ks) the inequality f(αs) ≤ f(y) ≤ f(βs) holds. Now suppose that for a given x∈B the inequality
X
(k1,...,ks)
f(βs)κ(k1, k2, . . . , ks;x)<1 holds. We denote by
∗
X
(k1,...,ks)
f(βs)κ(k1, k2, . . . , ks;x) a finite partial sum. Then we apply the operator U and find
∗
X
(k1,...,ks)
f(βs)Utκ(k1, k2, . . . , ks;x)<1 for t≥1. Since, as outlined in the introduction,
t→∞lim Utκ(k1, . . . , ks) =µ(B(k1, . . . , ks)) we find
∗
X
(k1,...,ks)
f(βs)µ(B(k1, k2, . . . , ks))≤1.
Since this is true for any partial sum, we finally obtain X
(k1,...,ks)
f(βs)µ(B(k1, k2, . . . , ks))≤1.
We observe f(ξs)≤f(βs) and see that f(ξs) =f(βs), hencef(y) =f(βs) for all points y ∈B(k1, . . . , ks).
4. In a similar way the inequality X
(k1,...,ks)
f(αs)Utκ(k1, k2, . . . , ks;x)>1
implies f(y) = f(αs) for all points y ∈ B(k1, . . . , ks). Note that, if f is piecewise constant on the cylinders B(k1, . . . , ks), the function f is constant, i.e., f ≡1.
5. Therefore we can assume X
(k1,...,ks)
f(αs)κ(k1, k2, . . . , ks;x)≤1≤ X
(k1,...,ks)
f(βs)κ(k1, k2, . . . , ks;x).
We order the cylinders according to the size of µ(B(k1, k2, . . . , ks), say. This gives an ordering for the points αs andβs which we write asα(1), α(2), . . . and β(1), β(2), . . .. We write κ(j)(x) for the corresponding κ(k1, . . . , ks;x). There are only finitely many points such that
N
X
j=1
f(β(j))κ(j)(x) +
∞
X
j=N+1
f(α(j))κ(j)(x)<1.
If the change appears at the cylinder B(k1, . . . , ks) which corresponds to α(N+1) and β(N+1) then the intermediate value theorem shows the existence of a pointz∈B(k1, . . . , ks) such that
X
j≤N
f(β(j))κ(j)(x) +f(z)κ(N+1)(x) + X
j≥N+2
f(α(j))κ(j)(x) = 1.
Hence we see that for every point x ∈ B there is a selection of points ηs ∈ B(k1, . . . , ks) such that
X
(k1,...,ks)
f(ηs)κ(k1, k2, . . . , ks;x) = 1.
6. Then we find
X
(k1,...,ks)
f(V(k1, k2, . . . , ks)x)κ(k1, k2, . . . , ks;x)−1
=
=
X
(k1,...,ks)
f(V(k1, k2, . . . , ks)x)κ(k1, k2, . . . , ks;x)−
− X
(k1,...,ks)
f(ηs)κ(k1, k2, . . . , ks;x)
≤
≤N0σ(s) X
(k1,...,ks)
κ(k1, k2, . . . , ks;x) =N0σ(s).
REFERENCES
[1] A. Berechet, A Kuzmin-type theorem with exponential convergence for a class of fibred systems. Ergodic Theory Dynam. Systems21(2001), 673–688.
[2] R. Kuzmin,Sur un probl`eme de Gauss. In: Atti de Congresso Internaz. Mat. (Bologna, 1928), Tom VI, Zanichelli, Bologna, 1932, pp. 83–89.
[3] F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory. Oxford Univ. Press, Oxford 1995.
[4] F. Schweiger,Multidimensional Continued Fractions. Oxford Univ. Press, Oxford 2000.
[5] F. Schweiger, Kuzmin’s theorem revisited. Ergodic Theory Dynam. Systems 20(2000), 557–565.
[6] E. Wirsing, On the theorem of Gauss–Kuzmin–L´evy and a Frobenius–type theorem for function spaces.Acta Arith.24(1974), 507–528.
Received 28 February 2011 Universit¨at Salzburg Fachbereich Mathematik
Hellbrunnerstraße 34 A-5020 Salzburg fritz.schweiger@sbg.ac.at
