Ergodic Properties Of θ\theta-Expansions And A Gauss-Kuzmin-Type Problem

A generalization of the regular continued fractions was given by Chakraborty and Rao \cite{CR-2003}. For the transformation which generates this expansion and its invariant measure, the Perron-Frobenius operator is given and studied. For this expansion, we apply the method of Rockett and Sz\"usz \cite{RS-1992} and obtained the solution of its Gauss-Kuzmin-type problem.Comment: 19 page

On convergence rate in the Gauss-Kuzmin problem for θ\theta-expansions

After providing an overview of $\theta$-expansions introduced by Chakraborty and Rao, we focus on the Gauss-Kuzmin problem for this new transformation. Actually, we complete our study on these expansions by proving a two-dimensional Gauss-Kuzmin theorem. More exactly, we obtain such a theorem related to the natural extension of the associated measure-dynamical system. Finally, we derive explicit lower and upper bounds of the error term which provide interesting numerical calculations for the convergence rate involved.Comment: 21 page

A Generalization of the Gauss-Kuzmin-Wirsing constant

We generalize the result of Wirsing on Gauss transformation to the generalized tranformation $T_p(x)=\{\cfrac{p}{x}\}$ for any positive integer $p$. We give an estimate for the generalized Gauss-Kuzmin-Wirsing constant.Comment: 12 page

Limiting modular symbols and the Lyapunov spectrum

This paper consists of variations upon the theme of limiting modular symbols. Topics covered are: an expression of limiting modular symbols as Birkhoff averages on level sets of the Lyapunov exponent of the shift of the continued fraction, a vanishing theorem depending on the spectral properties of a generalized Gauss-Kuzmin operator, the construction of certain non-trivial homology classes associated to non-closed geodesics on modular curves, certain Selberg zeta functions and C^* algebras related to shift invariant sets.Comment: 25 pages LaTe

A Gauss-Kuzmin-L\'evy theorem for R\'enyi-type continued fractions

We consider an interval map which is a generalization of the R\'enyi transformation. For the continued fraction expansion arising from this transformation, we prove a result concerning the asymptotic behavior of the distribution functions of this map. More exactly, we use Sz\"usz's method to prove a Gauss-Kuzmin-L\'evy-type theorem.Comment: 11 page

On invariant Mobius measure and Gauss-Kuzmin face distribution

There exists and is unique up to multiplication by a constant function a form of the highest dimension on the manifold of n-dimensional continued fractions in the sense of Klein, such that the form is invariant under the natural action of the group of projective transformations PGL(n+1). A measure corresponding to the integral of such form is called a Mobius measure. In the present paper we deduce an explicit formulae to calculate invariant forms in special coordinates. These formulae allow to give answers to some statistical questions of theory of multidimensional continued fractions. As an example, we show in this work the results of approximate calculations of frequencies for certain two-dimensional faces of two-dimensional continued fractions

A dependence with complete connections approach to generalized R\'enyi continued fractions

We introduce and study in detail a special class of backward continued fractions that represents a generalization of R\'enyi continued fractions. We investigate the main metrical properties of the digits occurring in these expansions and we construct the natural extension for the transformation that generates the R\'enyi-type expansion. Also we define the associated random system with complete connections whose ergodic behavior allows us to prove a variant of Gauss-Kuzmin-type theorem.Comment: 18 page

Convergence rate for R\'enyi-type continued fraction expansions

This paper continues our investigation of Renyi-type continued fractions studied in \cite{Sebe&Lascu-2018}. A Wirsing-type approach to the Perron-Frobenius operator of the R\'enyi-type continued fraction transformation under its invariant measure allows us to study the optimality of the convergence rate. Actually, we obtain upper and lower bounds of the convergence rate which provide a near-optimal solution to the Gauss-Kuzmin-L\'evy problem.Comment: 14 page

On Gauss-Kuzmin Statistics and the Transfer Operator for a Multidimensional Continued Fraction Algorithm: the Triangle Map

The Gauss-Kuzmin statistics for the triangle map (a type of multidimensional continued fraction algorithm) are derived by examining the leading eigenfunction of the triangle map's transfer operator. The technical difficulty is finding the appropriate Banach space of functions. We also show that, by thinking of the triangle map's transfer operator as acting on a one-dimensional family of Hilbert spaces, the transfer can be thought of as a family of nuclear operators of trace class zero.Comment: 23 page

Dependence with complete connections and the Gauss-Kuzmin theorem for N-continued fractions

We consider a family $\{T_N:N \geq 1 \}$ of interval maps as generalizations of the Gauss transformation. For the continued fraction expansion arising from $T_N$, we solve its Gauss-Kuzmin-type problem by applying the theory of random systems with complete connections by Iosifescu.Comment: 19 page