A NEW PROOF OF KUZMIN S THEOREM

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1 A NW PROOF OF KUZMIN S THORM FRITZ SCHWIGR 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 order σ(s). Here σ(s) is the maximum of the diameters of cylinders of rank s. Unfortunately, some years later erechet and Iosifescu detected a serious flaw in the proof. A subsequent paper could partially remedy the situation but the convergence rate in this corrected proof was lowered to σ( s) [5]. However, I am still convinced that the convergence rate is at least σ(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 (, 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 fdλ = (Af)dλ. T 1 Then a Kuzmin theorem is a statement of the form lim s As f = h fdλ, where h is the density of the invariant measure µ λ. Many years ago for a certain class of fibred systems and for a class of functions the convergence result A s f h fdλ = O(σ(s)) was claimed. Here σ(s) := max diam (k 1,..., k s ). RV. ROUMAIN MATH. PURS APPL., 56 (2011), 3,

2 230 Fritz Schweiger 2 This result looked very natural but however the proof contained a serious gap which was detected many years later by A. erechet. The proof could be restored only partially. In [5] essentially the result A s f h 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 lim s As f = h 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) lim s σ(s) = 0. () The map T can be extended as a C 1 -map to every (k), the closure of (k), and T s (k 1,..., k s ) = [0, 1] d = for all cylinders. (C) If ω(k 1,..., k s )dλ = dλ then there is a constant C 1 such that T s sup ω(k 1,..., k s ; x) C inf ω(k 1,..., k s ; x). x [0,1] d x [0,1] d (D) There is a constant R 1 such that V (k 1,..., k s )x V (k 1,..., k s )y R x y. () There is a constant L such that ω(k 1,..., k s ; x) ω(k 1,..., k s ; y) Lλ((k 1,..., k s )) x y. Conditions (A) (C) are sufficient to ensure the existence of a finite invariant measure µ λ and the ergodicity of T. The class of functions f : [0, 1] d R satisfies the conditions (α) (β) 0 < m 0 f M 0, m 0 = m 0 (f), M 0 = M 0 (f), f(x) f(y) N 0 x y, N 0 = N 0 (f). It is easy to see that the iterates A s f, belong to the same class with some uniform constants m 1 = m 1 (f), M 1 = M 1 (f), N 1 = N 1 (f).

3 3 A new proof of kuzmin s theorem 231 Conditions (A) () show that a Kuzmin theorem is valid in the form A s f h 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 Lipschitz condition (β). Instead of working with the transfer operator A it is more convenient to use the transfer operator U which belongs to the invariant measure µ, namely (Uf)dµ = fdµ. T 1 Therefore we replace the Jacobians ω(k 1,..., k s ) by the equivalent quantities κ(k 1,..., k s ) which are defined by κ(k 1,..., k s )dµ = dµ. Then we claim the following result. Theorem. We have U s f T s fdµ = O(σ(s)). 2. KUZMIN S THORM This section is devoted to the proof of the announced theorem. Proof. y the choice of the operator U we see that which is equivalent to We suppose U1 = 1, κ(k 1,..., k s ) = 1. fdµ = 1. The proof will be divided into several steps. 1. There exist points ξ s (k 1,..., k s ) such that 1 = f(ξ s )µ((k 1,..., k s )).

4 232 Fritz Schweiger 4 2. We remark that the series f(v (k 1,..., k s )x)κ(k 1,..., k s ; x) = (U s f)(x) is absolutely and uniformly convergent. 3. Denote by α s and β s points from (k 1,..., k s ) such that for all points y (k 1,..., k s ) the inequality f(α s ) f(y) f(β s ) holds. Now suppose that for a given x the inequality f(β s )κ(k 1, k 2,..., k s ; x) < 1 holds. We denote by f(β s )κ(k 1, k 2,..., k s ; x) a finite partial sum. Then we apply the operator U and find f(β s )U t κ(k 1, k 2,..., k s ; x) < 1 for t 1. Since, as outlined in the introduction, we find lim U t κ(k 1,..., k s ) = µ((k 1,..., k s )) t f(β s )µ((k 1, k 2,..., k s )) 1. Since this is true for any partial sum, we finally obtain f(β s )µ((k 1, k 2,..., k s )) 1. We observe f(ξ s ) f(β s ) and see that f(ξ s ) = f(β s ), hence f(y) = f(β s ) for all points y (k 1,..., k s ). 4. In a similar way the inequality f(α s )U t κ(k 1, k 2,..., k s ; x) > 1 implies f(y) = f(α s ) for all points y (k 1,..., k s ). Note that, if f is piecewise constant on the cylinders (k 1,..., k s ), the function f is constant, i.e., f 1.

5 5 A new proof of kuzmin s theorem Therefore we can assume f(α s )κ(k 1, k 2,..., k s ; x) 1 f(β s )κ(k 1, k 2,..., k s ; x). We order the cylinders according to the size of µ((k 1, k 2,..., k s ), say. This gives an ordering for the points α s and β s which we write as α (1), α (2),... and β (1), β (2),.... We write κ (j) (x) for the corresponding κ(k 1,..., k s ; x). There are only finitely many points such that N f(β (j) )κ (j) (x) + j=1 j=n+1 f(α (j) )κ (j) (x) < 1. If the change appears at the cylinder (k 1,..., k s ) which corresponds to α (N+1) and β (N+1) then the intermediate value theorem shows the existence of a point z (k 1,..., k s ) such that f(β (j) )κ (j) (x) + f(z)κ (N+1) (x) + f(α (j) )κ (j) (x) = 1. j N j N+2 Hence we see that for every point x there is a selection of points η s (k 1,..., k s ) such that f(η s )κ(k 1, k 2,..., k s ; x) = Then we find f(v (k 1, k 2,..., k s )x)κ(k 1, k 2,..., k s ; x) 1 = = N 0 σ(s) f(v (k 1, k 2,..., k s )x)κ(k 1, k 2,..., k s ; x) f(η s )κ(k 1, k 2,..., k s ; x) κ(k 1, k 2,..., k s ; x) = N 0 σ(s). RFRNCS [1] A. erechet, A Kuzmin-type theorem with exponential convergence for a class of fibred systems. rgodic Theory Dynam. Systems 21 (2001),

6 234 Fritz Schweiger 6 [2] R. Kuzmin, Sur un problème de Gauss. In: Atti de Congresso Internaz. Mat. (ologna, 1928), Tom VI, Zanichelli, ologna, 1932, pp [3] F. Schweiger, rgodic Theory of Fibred Systems and Metric Number Theory. Oxford Univ. Press, Oxford [4] F. Schweiger, Multidimensional Continued Fractions. Oxford Univ. Press, Oxford [5] F. Schweiger, Kuzmin s theorem revisited. rgodic Theory Dynam. Systems 20 (2000), [6]. Wirsing, On the theorem of Gauss Kuzmin Lévy and a Frobenius type theorem for function spaces. Acta Arith. 24 (1974), Received 28 February 2011 Universität Salzburg Fachbereich Mathematik Hellbrunnerstraße 34 A-5020 Salzburg fritz.schweiger@sbg.ac.at