On a Gauss-Kuzmin-Type Problem For a Generalized Gauss-Kuzmin Operator

Abstract

A generalized limit probability measure associated with a random system with complete connections for a generalized Gauss-Kuzmin operator, only for a special case, is defined, and its behaviour is investigated. As a consequence a specific version of Gauss-Kuzmin-type problem for the above generalized operator is obtained.

1. Introduction

Let 𝑌=𝐶([0,1]) be the Banach space of complex-valued continuous functions on [0,1] under the supremum norm, and let 𝑁^∗={1,2,…}, 𝑁={0,1,2,…}. Then for every 𝑓∈𝑌 and for every 𝛼≥1 the function 𝐺_𝛼𝑓 introduced by Fluch [1] and defined by𝐺_𝛼𝑓(𝑤)=_{𝑥∈𝑁∗}𝛼²()⋅()𝛼𝛼𝑥+𝑤𝛼𝑥+𝛼−1+𝑤⋅𝑓𝛼𝑥+𝛼−1+𝑤, (1.1) for all 𝑤∈[0,1], is called a generalized Gauss-Kuzmin operator.

The present paper arises as an attempt to determine a generalized limit probability measure, only for a special case, associated with a random system with complete connections for the above generalized Gauss-Kuzmin operator obtained in Ganatsiou [2], for every 𝛼>2. This will give us the possibility to obtain a specific variant of Gauss-Kuzmin-type problem for the above operator.

Our approach is given in the context of the theory of dependence with complete connections (see Iosifescu and Grigorescu [3]). For a more detailed study of the theory and applications of dependence with complete connections to the metrical problems and other interesting aspects of number theory we refer the reader to [4–9] and others.

The paper is organized as follows. In Section 2, we present all the necessary results regarding the ergodic behaviour of a random system with complete connections associated with the generalized Gauss-Kuzmin operator 𝐺_𝛼 obtained in [2], in order to make more comprehensible the presentation of the paper. In Section 3, we introduce the determination of a limit probability measure associated with the above random system with complete connections, only for a special case, for every 𝛼>2, which will give us the possibility to study in Section 4 a specific version of the associated Gauss-Kuzmin type problem.

2. Auxiliary Results

For every 𝛼≥1, we consider the function 𝜌_𝛼 defined by𝜌_𝛼(𝑤)=𝛼[]𝛼+𝑤,𝑤∈0,1, (2.1) and setg_𝑛=𝐺^𝑛_𝛼𝑓𝜌_𝛼,𝑛∈𝑁, (2.2) where 𝐺_𝛼^𝑛+1f=G_𝛼(𝐺^𝑛_𝛼f), for every 𝑛∈𝑁 and for every f∈Y.

Then we obtain the following statement which gives a relation deriving from an analogous of the Gauss- Kuzmin type equation.

Proposition 2.1. The function 𝑔_𝑛 satisfies 𝑔_𝑛+1(𝑤)=_{𝑥∈𝑁∗}𝛼⋅(𝛼+𝑤)(𝛼𝑥+𝑤)⋅(𝛼𝑥+𝛼+𝑤)⋅𝑔_𝑛𝛼𝛼𝑥+𝛼−1+𝑤, (2.3) for any 𝑛∈𝑁 and 𝑤∈[0,1].

Furthermore we obtain the following.

Proposition 2.2. For every 𝛼≥1, the function 𝑃_𝛼()=()𝑤,𝑥𝛼⋅𝛼+𝑤()⋅()[]𝛼𝑥+𝑤𝛼𝑥+𝛼+𝑤,𝑤∈0,1,𝑥∈N^∗, (2.4) defines a transition probability function from ([0,1],𝐵_[0,1]) to (𝑋,𝑃(𝑋)), where 𝑋=N^∗ and𝑃(𝑋) the power set of 𝑋.

Equation (2.3) and Proposition 2.2 lead to the consideration of a family of random systems with complete connections (RSCCs)(W,𝑊)(X,𝑋),𝑢_𝛼,P_𝛼,𝛼≥1, (2.5) where𝑊=[0,1],𝑊=𝐵_[0,1],𝑋=𝑁^∗(𝑋),𝑢,𝑋=𝑃_𝛼()=𝛼𝑤,𝑥𝛼𝑥+𝛼−1+𝑤,𝑃_𝛼()=()𝑤,𝑥𝛼⋅𝛼+𝑤(𝛼𝑥+𝑤)⋅(𝛼𝑥+𝛼+𝑤),𝑤∈𝑊,𝑥∈𝑋. (2.6) In the next, we consider the transition probability function 𝑄_𝛼, 𝛼≥1, of the Markov chain associated with the family of the RSCCs (2.5) and the corresponding Markov operator 𝑈_𝛼, 𝛼≥1, defined by 𝑈_𝛼𝑓(𝑤)=_{𝑥∈𝑁∗}𝛼⋅(𝛼+𝑤)()⋅()𝛼𝛼𝑥+𝑤𝛼𝑥+𝛼+𝑤⋅𝑓𝛼𝑥+𝛼−1+𝑤, (2.7) for all complex-valued measurable bounded functions 𝑓 on [0,1].

This gives us the possibility of obtain the following.

Proposition 2.3. The family of RSCCs (2.5) is with contraction. Moreover, its associated Markov operator 𝑈_𝛼 given by (2.7) is regular with respect to 𝐿([0,1]), the Banach space of all real-valued bounded Lipschitz functions on [0,1].

On the contrary the RSCC associated with a concrete piecewise fractional linear map (see Ganatsiou [10]) is not an RSCC with contraction since 𝑟₁=1 and its associated Markov chain is not compact and regular with respect to the set 𝐿([0,1]), even though there exists a point 𝑦^∗∈(0,1), such that lim_{𝑛→∞𝑛}𝑦−𝑦^∗=0, (2.8) for all 𝑦∈(0,1). This corrects the escape of [10] gives an RSCC associated with a concrete piecewise fractional linear map which is not uniformly ergodic (a special case of [4]).

By virtue of Proposition 2.3, it follows from [3, Theorem  3.4.5] that the family of RSCCs (2.5) is uniformly ergodic. Furthermore, Theorem  3.1.24 in [3] implies that, for every 𝛼≥1, there exists a unique probability measure 𝛾_𝛼 on 𝐵_[0,1], which is stationary for the kernel 𝑄_𝛼, such thatlim_𝑛→∞𝑈^𝑛_𝛼𝑓=¹₀𝑓𝑑𝛾_𝛼,𝑓∈𝐿([0,1]). (2.9) This means that𝛾_𝛼(𝐵)=¹₀𝑄_𝛼(𝑤,𝐵)𝛾_𝛼(𝑑𝑤), (2.10) where𝑄_𝛼(𝑤,𝐵)=_{𝑋∈𝐵𝑤}𝑃_𝛼(𝑤,𝑥), (2.11) with𝐵_𝑤=𝑥∈𝑁^∗∣𝑢_𝛼(𝑤,𝑥)∈B,forevery𝐵∈𝑊,𝑤∈[0,1]. (2.12) Moreover, for some c>0and0<𝜃<1, we have𝑈^𝑛_𝛼𝑓−¹₀𝑓𝑑𝛾_𝛼≤c⋅𝜃^𝑛⋅𝑓_𝐿, (2.13) for all 𝑛∈N^∗ and 𝑓∈𝐿([0,1]), where ‖⋅‖_𝐿 denotes the usual norm in 𝐿([0,1]), where𝑈^∞_𝛼𝑓=¹₀𝑓(𝑤)𝛾_𝛼(𝑑𝑤). (2.14) In general the form of the limit probability measure associated with the family of random systems with complete connections (2.5) cannot be determined but this is possible only for a special case as we prove in the following section.

For the proofs of the above results we refer the reader to Ganatsiou [2].

3. A Limit Probability Measure Associated with the Family of RSCCs

Now, we are able to determine a limit probability measure associated with the family of RSCCs (2.5) as is shown in the following.

Proposition 3.1. The probability measure 𝛾_𝛼 has the density 𝜌_𝛼(𝑤)=𝛼[]𝛼+𝑤,forevery𝑤∈0,1, (3.1) with constant 1/𝛼⋅log(1+𝛼⁻¹) only for the special case 𝑎⋅𝑢⁻¹+1−𝑎[𝑢⁻¹+𝑎⁻¹]<1, for every 𝑎>2, 0<𝑢≤1.

Proof. By virtue of uniqueness of 𝛾_𝛼 we have to show that it satisfies relation (2.10). Since the intervals [0,𝑢), 0<𝑢≤1 generate 𝐵_[0,1] it is sufficient to verify (2.10) only for 𝐵=[0,𝑢), 0<𝑢≤1.
Suppose that 𝐵=[0,𝑢). Then, for every 𝑤∈[0,1], we have𝐵_𝑤=𝑥∈𝑁^∗∣𝑢_𝛼(𝑤,𝑥)∈[0,𝑢)=𝑥∈𝑁^∗∣𝛼()=𝛼𝑥+𝛼−1+𝑤<𝑢𝑥∈𝑁^∗∣𝑥≥𝑢⁻¹−𝑤⋅𝛼⁻¹+𝛼⁻¹. (3.2) Hence by (2.11), we have that 𝑄_𝛼(𝑤,[0,𝑢))=𝛼+𝑤𝛼𝑢⁻¹−𝑤⋅𝛼⁻¹+𝛼⁻¹+𝑤, (3.3) where 𝑢⁻¹−𝑤⋅𝛼⁻¹+𝛼⁻¹=𝑢⁻¹+𝛼⁻¹,if0≤𝑤<𝛼⋅𝑢⁻¹+1−𝛼⋅𝑢⁻¹+𝛼⁻¹,𝑢⁻¹+𝛼⁻¹−1,if𝛼⋅𝑢⁻¹+1−𝛼⋅𝑢⁻¹+𝛼⁻¹<𝑤≤1. (3.4)
We consider the case 𝛼𝑢⁻¹+1−𝛼⋅[𝑢⁻¹+𝛼⁻¹]<1 or 𝑢⁻¹<[𝑢⁻¹+𝛼⁻¹], for every 𝛼>2,0<𝑢≤1. Consequently, we obtain that¹₀𝑄_𝛼(𝑤,[0,u))⋅𝜌_𝛼(𝑤)1𝑑𝑤=log1+𝛼⁻¹⋅¹₀𝑑𝑤𝛼⋅𝑢⁻¹−𝑤⋅𝛼⁻¹+𝛼⁻¹=1+𝑤log1+𝛼⁻¹⋅log𝛼⋅𝑢⁻¹+1−log𝛼⋅𝑢⁻¹𝑢+1−𝛼+log𝛼⋅⁻¹+𝛼⁻¹−𝛼+1−log𝛼⋅𝑢⁻¹+𝛼⁻¹. (3.5) In the next we put I=log𝛼⋅𝑢⁻¹+𝛼⁻¹−𝛼+1−log𝛼⋅𝑢⁻¹+𝛼⁻¹1=log1−𝑢⁻¹+𝛼⁻¹+1𝛼⋅𝑢⁻¹+𝛼⁻¹,II=log𝛼⋅𝑢⁻¹+1−log𝛼⋅𝑢⁻¹𝑢+1−𝛼=log1+𝛼𝑢−log1+𝛼−𝑢 (3.6) By taking the limit of 𝑢III=𝐼−log1+𝛼1−𝑢=log1−𝑢⁻¹+𝛼⁻¹+1𝛼⋅𝑢⁻¹+𝛼⁻¹𝑢−log1+𝛼−𝑢 (3.7) when 𝑢→1 we have that lim_𝑢→1𝑢log1+𝛼1−u=log𝛼,lim_𝑢→11log1−𝑢⁻¹+𝛼⁻¹+1𝛼⋅𝑢⁻¹+𝛼⁻¹1=log𝛼,forevery𝛼>2. (3.8) So part III tends to 0 when 𝑢→1. This means that lim_𝑢→1¹₀𝑄_𝛼(𝑤,[0,u))⋅𝜌_𝛼(𝑤)=1𝑑𝑤log1+𝛼⁻¹⋅lim_𝑢→1𝑢log1+𝛼 (3.9) which is equal to lim_𝑢→1^𝑢₀𝜌_𝛼(𝑤)𝑑𝑤=lim_𝑢→1^𝑢₀1𝛼⋅log1+𝛼⁻¹⋅𝛼=1𝛼+𝑤𝑑𝑤log1+𝛼⁻¹lim_𝑢→1()=1log𝛼+𝑢−log𝛼log1+𝛼⁻¹lim_𝑢→1log𝛼+𝑢𝛼=1log1+𝛼⁻¹lim_𝑢→1𝑢log1+𝛼 (3.10) and the proof is complete.

4. A Version of the Gauss-Kuzmin-Type Problem

Let 𝜇 be a nonatomic measure on the 𝜎-algebra 𝐵_[0,1]. Then we may define𝑉_𝑜(𝑤),𝑉=𝜇([0,𝑤])_𝑛(𝑤)=𝑉_𝑛𝑤,𝜇=^𝑤₀𝐺^𝑛_𝛼𝑓(𝑡)𝑑𝑡,𝑛∈𝑁^∗,𝑤∈[0,1]. (4.1) 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∈𝑁^∗ with𝑉^′_𝑛(𝑤)≡𝐺^𝑛_𝛼𝑓(𝑤)=G_𝛼G_𝛼^𝑛−1𝑓(𝑤),𝑓∈𝐿([0,1]),𝑛∈N^∗. (4.2) So ^𝑤₀𝑉^′_𝑛(𝑡)𝑑𝑡=^𝑤₀𝐺^𝑛_𝛼𝑓(𝑡)𝑑𝑡,𝑉_𝑛(𝑤)=^𝑤₀𝐺^𝑛_𝛼𝑓(𝑡)𝑑𝑡 (4.3) while 𝑔_𝑛(𝑤)=𝐺^𝑛_𝛼𝑓(𝑤)𝑝_𝛼(𝑤)≡𝑉^′_𝑛(𝑤)𝑝_𝛼(𝑤),𝑛∈𝑁. (4.4) Now, we are able to determine the limit lim_𝑛→∞𝑉_𝑛(1/𝑤) and to give the rate of this convergence, that is, a specific version of the associated Gauss-Kuzmin type problem.

Proposition 4.1. (i) If the density V^′₀ of 𝜇 is a Riemann integrable function, then lim_n→∞V_𝑛1𝑤=1log1+𝛼⁻¹⋅log𝛼𝑤+1𝛼𝑤,𝑤≥1,𝛼>2,𝑛∈𝑁^∗. (4.5)(ii) If the density V^′₀ of 𝜇 is an element of 𝐿([0,1)], then there exist two positive constants 𝑐 and 𝜃<1 such that lim_𝑛→∞𝑉_𝑛1𝑤=1+𝑞𝜃^𝑛⋅1log1+𝛼⁻¹⋅log𝛼𝑤+1𝛼𝑤, (4.6) for all 𝑤≥1, 𝑎>2, 𝑛∈𝑁^∗, where 𝑞=𝑞(𝜇,𝑛,𝑤)𝑤𝑖𝑡ℎ|𝑞|≤𝑐.

Proof. Let 𝑉^′₀∈𝐿([0,1]). Then 𝑔_o∈𝐿([0,1]), and by using relation (2.14) we have 𝑈^∞_𝛼𝑔₀≡lim_𝑛→∞𝑈^𝑛_𝛼𝑔₀=¹₀𝑔₀(𝑤)𝛾_𝛼(𝑑𝑤)=¹₀𝑉^′₀(𝑤)𝑑𝑤=1. (4.7) According to relation (2.13), there exist two positive constants 𝑐 and 𝜃<1 such that 𝑈^𝑛_𝛼𝑔₀=𝑈^∞_𝛼𝑔₀+𝑇^𝑛_𝛼𝑔₀,𝑛∈𝑁^∗,with𝑇^𝑛_𝛼𝑔₀≤c⋅𝜃^𝑛. (4.8) If we consider the Banach space 𝐶([0,1]) of all real continuous functions defined on [0,1] with the supremum norm, then since 𝐿([0,1]) is a dense subset of 𝐶([0,1]) we have lim_𝑛→∞𝑇ⁿ_𝛼𝑔₀=0,forevery𝑔_o∈𝐶([0,1]). (4.9) This means that it is valid for any measurable function 𝑔_𝑜 which is 𝛾_𝛼-almost surely continuous, that is, for any Riemann integrable function 𝑔_𝑜. Consequently we obtain lim_𝑛→∞𝑉_𝑛1𝑤=lim_𝑛→∞0^1/𝑤𝑈^𝑛_𝛼𝑔₀(𝑡)𝜌_𝛼(𝑡)𝑑𝑡=₀^1/𝑤𝜌_𝛼(𝑡)𝑑𝑡=₀^1/𝑤1log1+𝛼⁻¹⋅𝛼=1𝛼+t𝑑𝑡log1+𝛼⁻¹⋅log𝛼𝑤+1,𝛼𝑤 (4.10) that is the solution of the associated Gauss-Kuzmin type problem.

Remarks 4. (1) It is notable that for 𝛼=1 the RSCC associated with the generalized Gauss-Kuzmin operator is identical to that associated with the ordinary continued fraction expansion (see Iosifescu and Grogorescu [3]). Moreover the corresponding limit probability measure associated with the family of RSCCs (2.5) for 𝛼=1 is identical to the limit probability measure associated with the above random system with complete connections for the ordinary continued fraction expansion, that is, identical to the Gauss’s measure 𝛾 on 𝐵_[0,1] defined by 𝛾(𝐴)=1log2_𝐴𝑑𝑡𝑡+1,𝐴∈𝐵_[0,1]. (4.11)
(2) It is an open problem the determination of an analogous limit probability measure for the case 𝛼𝑢⁻¹+1−𝛼⋅[𝑢⁻¹+𝛼⁻¹]>1.