論文

We define two types of the α-Farey maps Fα and for, which were previously defined only for by Natsui (2004). Then, for each, we construct the natural extension maps on the plane and show that the natural extension of is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α-continued fractions does not vary by the choice of α,. This extends the result by Kraaikamp and Nakada (2000).

We show that the entropy of the α-continued fraction map w.r.t the absolutely continuous invariant probability measure is strictly less than that of the nearest integer continued fraction map when 0 < α <3−√5 2. This answers a question by C. Kraaikamp, T. A. Schmidt, and W. Steiner (2012). To prove this result we make use of the notion of the geodesic continued fractions introduced by A. F. Beardon, M. Hockman, and I. Short (2012).

We consider the nearest integer complex continued fraction map associated to the Euclidean field Q( p -d) for each d = 1; 2; 3; 7; 11. For each map, we see that there is an absolutely continuous ergodic invariant probability measure. We construct the natural extension of each map on a subset of C×C. Then the invariant measure for this extension is derived from the hyperbolic measure on H3 and the density function of the absolutely continuous invariant measure for the nearest integer map is given as its marginal. For each case, we have a fundamental domain with a fractal boundary for the lattice generated by the set of integers of the imaginary quadratic fields Q( p -d), d = 1; 2; 3; 7; 11, through the construction of the natural extension except for only one case.

For N∈ N≥ 2 and α∈ R such that 0<α≤N-1, we define Iα: = [α, α+ 1] and Iα-:=[α,α+1) and investigate the continued fraction map Tα:Iα→Iα-, which is defined as Tα(x):=Nx-d(x), where d: Iα→ N is defined by d(x):=⌊Nx-α⌋. For N∈ N≥ 7, for certain values of α, open intervals (a, b) ⊂ Iα exist such that for almost every x∈ Iα there is an n∈ N for which Tαn(x)∉(a,b) for all n≥ n. These gaps (a, b) are investigated using the square Υα:=Iα×Iα-, where the orbitsTαk(x),k=0,1,2,… of numbers x∈ Iα are represented as cobwebs. The squares Υα are the union of fundamental regions, which are related to the cylinder sets of the map Tα, according to the finitely many values of d in Tα. In this paper some clear conditions are found under which Iα is gapless. If Iα consists of at least five cylinder sets, it is always gapless. In the case of four cylinder sets there are usually no gaps, except for the rare cases that there is one, very wide gap. Gaplessness in the case of two or three cylinder sets depends on the position of the endpoints of Iα with regard to the fixed points of Iα under Tα.

We consider the complex continued fraction map T defined by R. Kaneiwa, I. Shiokawa, and J. Tamura (1975) associated with the Eisenstein field Q(−3). A significant aspect of their continued fraction map is that the real number part of this map T is exactly the simple continued fraction map (Gauss map). In this paper we characterize the set of strictly periodic expansions of continued fraction expansions associated to this map in terms of quadratic extensions of Q(−3) in connection with the closure of [Formula presented], where qn, n≥0, is the denominator of the nth principal convergent of the continued fraction expansion. Moreover, we show that the closure of [Formula presented] has positive Lebesgue measure on the complex plane C though it has infinitely many holes. This gives us that the construction of the natural extension of T on a subset of C2∖{diagonal} is equivalent to the geodesics over the hyperbolic space H3. Then the invariant measure for the natural extension map is given by the hyperbolic measure. Hence its projection to the complex plane is obviously the invariant measure for T.

We consider some types of complex continued fraction expansions and their associated maps of the Euclidean fields Q(−d), d=1,2,3,7 and 11 introduced by Hurwitz (1887) [7], Lakein (1973) [10] and Shiokawa et al. (1975) [20]. We show that there exists the Legendre constant for every type of continued fraction expansions introduced by them improving the proof by Ei et al. (2019) [4] in the case of the one by A. Hurwitz with Q(−1).

We study and compare natural generalizations of Euclid's algorithm for polynomials with coefficients in a finite field. This leads to gcd algorithms together with their associated continued fraction maps. The gcd algorithms act on triples of polynomials and rely on two-dimensional versions of the Brun, Jacobi–Perron and fully subtractive continued fraction maps, respectively. We first provide a unified framework for these algorithms and their associated continued fraction maps. We then analyse various costs for the gcd algorithms, including the number of iterations and two versions of the bit-complexity, corresponding to two representations of polynomials (the usual and the sparse one). We also study the associated two-dimensional continued fraction maps and prove the invariance and the ergodicity of the Haar measure. We deduce corresponding estimates for the costs of truncated trajectories under the action of these continued fraction maps, obtained thanks to their transfer operators, and we compare the two models (gcd algorithms and their associated continued fraction maps). Proving that the generating functions appear as dominant eigenvalues of the transfer operator allows indeed a fine comparison between the models.

We show the ergodicity of Tanaka–Ito type α-continued fraction maps and construct their natural extensions. We also discuss the relation between entropy and the size of the natural extension domain.

© Springer International Publishing AG, part of Springer Nature 2018. For given positive integers dj, 1 ⩽ j⩽ ∀ s, ∑j=1sdj even, we construct a piecewise rotation map of the circle with ∑j=1sdj+s discontinuous points such that its critical iterates generate translation surfaces with singularity orders dj, 1 ⩽ j⩽ s, and with any Rauzy class associated to this singularity orders. The construction of the piecewise rotation map is combinatorial, on the other hand, the construction of the translation surfaces is based on the idea by Cruz and da Rocha.

© 2018, Springer-Verlag GmbH Austria, part of Springer Nature. We consider the Hurwitz complex continued fraction map associated to the Gaussian field Q(i). We characterize the density function of the absolutely continuous invariant measure for the map associated to the Hurwitz continued fractions. For this reason, we construct a representation of its natural extension map (in the sense of an ergodic measure preserving map) on a subset of C× C. This subset is constructed by the closure of pairs of the n-th iteration of a complex number by the Hurwitz complex continued fraction map and -QnQn-1, where Qn is the denominator of the n-th convergent of the Hurwitz continued fractions. The absolutely continuous invariant measure for the natural extension map is induced from the invariant measure for Möbius transformations on the set of geodesics over three dimension upper-half space. Then the absolutely continues invariant measure for the Hurwitz continued fraction map is given by its marginal measure.

形式的べき級数に関する Farey map を Bruhat-Tits tree の中で表現した。また、Farey map に関するエルゴード理論的性質をいくつか証明した。

α-連分数の同値関係が α が減少するにつれて群作用の同値関係とどのように異なっていくかを示した。

有限体を係数をする多項式に対するユークリッドアルゴリズムに関して、fine bit の概念を導入し、コストに関する大数の法則を証明した。

非アルキメデス数体上のディオファンタス近似に対して、Kurzweil型定理を証明した。

interval exchange に関する renormalized Rauzy induction に関して Darling-Kac 集合が存在することを証明した。

Fully subtractive algorithm に関する Schweiger の予想を証明した。

非アルキメデス数体上の非斉次ディオファンタス近似の問題について、ある測度論的性質を証明した。

連分数変換において Legendre 定数と Lenstra 定数が一致することを証明した。

α-連分数に関するブルノ関数を定義し、その性質を議論した。特に、α=0の場合に semi-Brjuno数の概念を与え、Brjuno数との関係を論じた。