Related papers: Ergodicity for $p$-adic continued fraction algorit…
We give continued fraction algorithms for a particular class of Fuchsian triangle groups. In particular, we give an explicit form of each such group that is a subgroup of the Hilbert modular group of its trace field and provide an interval…
The Arnoux-Rauzy-Poincar\'e multidimensional continued fraction algorithm is obtained by combining the Arnoux-Rauzy and Poincar\'e algorithms. It is a generalized Euclidean algorithm. Its three-dimensional linear version consists in…
We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers a so that either (a, a^2) or (a,…
This paper is devoted to the problem of ergodicity of $p$-adic dynamical systems. Our aim is to present criteria of ergodicity in terms of coordinate functions corresponding to digits in the canonical expansion of $p$-adic numbers. The…
The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to $p$-adic numbers where it presents many differences with…
We propose and study a generalized continued fraction algorithm that can be executed in an arbitrary imaginary quadratic field, the novelty being a non-restriction to the five Euclidean cases. Many hallmark properties of classical continued…
In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…
A continued fractions based verification of the Hurwitz values for the Hecke triangle groups is given, completing a program of Lehner's. Ergodic theory shows that Diophantine approximation by mediant convergents of the Rosen continued…
By using a jump transformation associated to the Romik map, we define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. Among…
It has been a long standing problem to find good symbolic codings for translations on the $d$-dimensional torus that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties.…
We introduce a family of adic transformations on diagrams that are nonstationary and nonsimple. This family includes some previously studied adic transformations. We relate the dimension group of each these diagrams to the dynamical system…
This paper studies geometric and spectral properties of $S$-adic shifts and their relation to continued fraction algorithms. These shifts are symbolic dynamical systems obtained by iterating infinitely many substitutions. Pure discrete…
This work is devoted to the study of families of infinite parabolic iterated function systems (PIFS) on a closed interval parametrized by vectors in $\mathbb{R}^d$ with overlaps. We show that the Hausdorff dimension and absolute continuity…
A real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse…
We consider the geometric generalization of ordinary continued fraction to the multidimensional case introduced by F. Klein in 1895. A multidimensional periodic continued fraction is the union of sails with some special group acting freely…
For any 1-lipschitz ergodic map $F:\; \mathbb{Z}^{k}_{p} \mapsto \mathbb{Z}^{k}_{p},\;k>1\in\mathbb{N},$ there are 1-lipschitz ergodic map $G:\; \mathbb{Z}_{p} \mapsto \mathbb{Z}_{p}$ and two bijection $H_k$, $T_{k,\;P}$ that $$G = H_{k}…
The Fourier coefficients of the Siegel-Eisenstein series are p-adically continued for all primes p, as meromorphic functions, using the reciprocal of a product of L-functions. A construction of p-adic meromorphic families of such series is…
J. Hurwitz introduced an algorithm that generates a continued fraction expansion for complex numbers $\alpha \in \mathbb{C}$, where the partial quotients belong to $(1+i)\mathbb{Z}[i]$. J. Hurwitz's work also provides a result analogous to…
We show that the ergodicity of an aperiodic automorphism of a Lebesgue space is equivalent to the continuity of a certain map on a metric Boolean algebra. A related characterization is also presented for periodic and totally ergodic…
We prove the convergence and ergodicity of a wide class of real and higher-dimensional continued fraction algorithms, including folded and $\alpha$-type variants of complex, quaternionic, octonionic, and Heisenberg continued fractions,…