On a Gauss-kuzmin Type Problem for Piecewise Fractional Linear Maps with Explicit Invariant Measure

A random system with complete connections associated with a piecewise fractional linear map with explicit invariant measure is defined and its ergodic behaviour is investigated. This allows us to obtain a variant of Gauss-Kuzmin type problem for the above linear map. 1. Introduction. It is known that a map T : [0, 1] → [0, 1] is called piecewise fractional linear if there is a partition D = {0 = k 0 < k 1 < ··· < k m = 1} such that for every j = 1, 2,...,m the map is defined by


T (x) =
A j + B j x C j + D j x , x ∈ k j−1 ,k j , (1.1) where A j D j − B j C j ≠ 0 and T [(k j−1 ,k j )] = (0, 1).According to Schweiger [9,10] such a map which satisfies the equalities T 0 = T 1 = 0 and T 0 < 1 is ergodic and admits an invariant measure.Its density h can be calculated explicitly and takes one of the following shapes: in the special case m = 2.But this is not true for m ≥ 3.
In order to verify his idea, Schweiger [11] introduced a map T : [0, 1] → [0, 1] defined by where N is a fixed positive integer.
It is obvious that is the nth iteration of T , n ∈ {1, 2,...}.As a consequence he proved that T is a piecewise fractional linear map with n = N +2 branches which admits an invariant measure with density The present paper arises as an attempt to find the limit and an estimation of the error µ(T −n > y) − α for a pregiven nonatomic measure µ on the σ -algebra B [0,1] of all the Borel subsets of [0, 1] associated with the above explicit piecewise fractional linear map T arising in case that m ≥ 3.That is to solve a variant of the Gauss-Kuzmin type problem for a piecewise fractional linear map with an explicit invariant measure.
Our approach is given in the context of the theory of dependence with complete connections (see [4]).For a more detailed study of the applications of dependence with complete connections to the metrical problems and other interesting aspects of number theory we refer the reader to [1,2,3,5,6,7,8] and others.
The paper is organized as follows.In Section 2, we define a random system with complete connections associated with the piecewise fractional linear map (1.3).In Section 3, we study the ergodic behaviour of the above random system.This gives us the possibility to obtain the asymptotic behaviour of the fact T −n > y , y ∈ [0, 1], as n → ∞, that is the associated Gauss-Kuzmin type problem.
In the next we need the following notation: is the σ -algebra of all Borel subsets of [0, 1], P (X) is the power set of X.
2. The random system with complete connections.Let µ be a nonatomic measure on the σ -algebra B [0,1] .Then we may define (2.1) Proposition 2.1 (the Gauss-Kuzmin type equation).The function V n , n ∈ N, satisfy the equation Proof.Taking into account the relations (1.3) and (1.4) we obtain that, for every y ∈ [0, 1], and the proof is complete.
Furthermore, suppose that V 0 exists and it is bounded (µ has bounded density).Then by induction we have that V n exists and it is bounded for any n ∈ ℵ * .If we derive the Gauss-Kuzmin type equation (2.2) we arrive at We denote and Then the relation (2.4) becomes (2.7) We can now prove the following proposition.
Proposition 2.2.The function defines a transition probability function from Proof.We must prove that we have that (2.11) By replacing j + 1 with m in the second sum we obtain that and the proof is complete.
Relation (2.7) and Proposition 2.2 lead to the definition of random system with complete connections (RSCC), (Y , ᐅ), (K, ), u, P , ( where (2.14) 3. The ergodic behaviour of the random system with complete connections.For any complex-valued function f defined on Y , we put If we denote by L(Y ) the set of all complex-valued measurable bounded Lipschitz functions defined on Y for which both |f | < ∞ and s(f ) < ∞, then it is clear that L(Y ) is a Banach space under the norm f L = |f |+s(f ).Furthermore, let Q denotes the transition probability function of the Markov chain associated with RSCC (2.13).Then the Markov operator U associated with RSCC (2.13) is given by for all f ∈ L(Y ).
In order to study the ergodic behaviour of the RSCC (2.13) we prove the following.

Proposition 3.1. RSCC (2.13) is a RSCC with contraction and its associated Markov operator given by (3.2) is regular with respect to L(Y ).
Proof.We have to prove that R 1 < ∞ and r 1 < 1, where r 1 and R 1 are defined in [4].To this end, we calculate the derivatives of u and P with respect to y.We have that dP (y, 0) dy It follows (cf.[4, pages 177-178]) that R 1 < ∞ and r 1 < 1, that is RSCC (2.13) is a RSCC with contraction.Now, in order to prove the regularity of the associated Markov operator U defined by (3.2) with respect to L(Y ) we have, according to Theorem 3.2.13 in [4], to find an element y * in [0, 1] such that lim n→∞ σ n (y) − y * = 0 ( for any y ∈ Y .Here σ n (y) denotes the support of Q n (y, •), where Q n is the kernel of U n .Let y ∈ Y .We define the iterative relation is strictly positive.So y n+1 ∈ σ 1 (y n ) and using [4, Lemma 3.2.14]we obtain by induction that y n ∈ σ n (y), for any n ∈ ℵ * .Since we obtain that U is regular, that is the desired result.By virtue of Proposition 3.1, it follows from [4, Theorem 3.4.5]that RSCC (2.13) is uniformly ergodic.Moreover, Theorem 3.1.24[4] implies that Q n (•, •) converges uniformly to a unique probability measure γ on ᐅ, which is stationary for the kernel Q, that is where with (3.13) Furthermore, there exist two positive constants q < 1 and c such that where In general, the form of γ cannot be determined but this is possible in our case as we prove in the following proposition.As a consequence we have and the proof is complete.

4.
A version of the Gauss-Kuzmin type theorem.Now, we are able to determine the limit of µ(T −n > y) as n → ∞ and give the rate of this convergence.
that is the solution of the associated Gauss-Kuzmin type theorem.