In this paper, we prove that almost every translation of $\mathbb{T}^2$ admits a symbolic coding which has linear complexity $2n+1$. The partitions are constructed with Rauzy fractals associated with sequences of substitutions, which are produced by a particular extended continued fraction algorithm in projective dimension $2$. More generally, in dimension $d\geq 1$, we study extended measured continued fraction algorithms, which associate to each direction a subshift generated by substitutions, called $S$-adic subshift. We give some conditions which imply the existence, for almost every direction, of a translation of the torus $\mathbb{T}^d$ and a nice generating partition, such that the associated coding is a conjugacy with the subshift.
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. Inspired by Rauzy's approach we construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any exponentially convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings of toral translations and bounded remainde…
We consider a symbolic coding of linear trajectories in the regular octagon with opposite sides identified (and more generally in regular 2n-gons). Each infinite trajectory gives a cutting sequence corresponding to the sequence of sides hit. We give an explicit characterization of these cutting sequences. The cutting sequences for the square are the well studied Sturmian sequences which can be analyzed in terms of the continued fraction expansion of the slope. We introduce an analogous continued fraction algorithm which we use to connect the cutting sequence of a trajectory with its slope. Our…
In this paper we discuss a coding and the associated symbolic dynamics for the geodesic flow on Hecke triangle surfaces. We construct an explicit cross section for which the first return map factors through a simple (explicit) map given in terms of the generating map of a particular continued fraction expansion closely related to the Hecke triangle groups. We also obtain explicit expressions for the associated first return times.
For a real number $0<λ<2$, we introduce a transformation $T_λ$ naturally associated to expansion in $λ$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of $T_λ$ provides an algorithm to expand any positive real number in $λ$-continued fraction. We prove the conjugacy between $T_λ$ and some $β$-shift, $β>1$. Some properties of the map $λ\mapstoβ(λ)$ are established: It is increasing and continuous from $]0, 2[$ onto $]1,\infty[$ but non-analytic.
We define families of aperiodic words associated to Lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots. An algorithm to construct symbolic words of satellite Lorenz knots is defined. We prove, subject to the validity of a previous conjecture, that Lorenz knots coded by some of these families of words are hyperbolic, by showing that they are neither satellites nor torus knots and making use of Thurston's theorem. Infinite families of hyperbolic Lorenz knots are generated in this way, to our knowledge, for the first time. The techniques used…
In this paper we give a geometric interpretation of the renormalization algorithm and of the continued fraction map that we introduced in arxiv:0905.0871 to give a characterization of symbolic sequences for linear flows in the regular octagon. We interpret this algorithm as renormalization on the Teichmueller disk of the octagon and explain the relation with Teichmueller geodesic flow. This connection is analogous to the classical relation between Sturmian sequences, continued fractions and geodesic flow on the modular surface. We use this connection to construct the natural extension and the …
This paper is part 1 of a series of papers culminating in the result that measure preserving diffeomorphisms of the disc or 2-torus are unclassifiable. It also addresses another classical problem: which abstract measure preserving systems are realizable as smooth diffeomorphisms of a compact manifold? The main result of this paper gives new symbolic representations of the Anosov-Katok diffeomorphisms.
Multidimensional Continued Fraction Algorithms are generalizations of the Euclid algorithm and find iteratively the gcd of two or more numbers. They are defined as linear applications on some subcone of $\mathbb{R}^d$. We consider multidimensional continued fraction algorithms that acts symmetrically on the positive cone $\mathbb{R}^d_+$ for $d=3$. We include well-known and old ones (Poincaré, Brun, Selmer, Fully Subtractive) and new ones (Arnoux-Rauzy-Poincaré, Reverse, Cassaigne). For each algorithm, one page (called cheat sheet) gathers a handful of informations most of them generated with …
In this article we study a renormalization scheme with which we find all semi-regular continued fractions of a number in a natural way. We define two maps, T_slow and T_fast: these maps are defined for (x,y) in [0,1], where x is the number for which a semi-regular continued fraction representation is developed by T_slow according to the parameter y. The set of all possible semi-regular continued fraction representations of x are bijectively constructed as the parameter y varies. The map T_fast is a "sped-up" version of the map T_slow, and we show that T_fast is ergodic with respect to a probab…
The decimal expansion real numbers, familiar to us all, has a dramatic generalization to representation of dynamical system orbits by symbolic sequences. The natural way to associate a symbolic sequence with an orbit is to track its history through a partition. But in order to get a useful symbolism, one needs to construct a partition with special properties. In this work we develop a general theory of representing dynamical systems by symbolic systems by means of so-called Markov partitions. We apply the results to one of the more tractable examples: namely hyperbolic automorphisms of the two…
Extreme value theorem for geodesic flow on the quotient of the theta group
HUN-REN Alfréd Rényi Institute of Mathematics（HUN-REN 阿尔弗雷德·雷尼数学研究所）；Department of Mathematical Sciences, Seoul National University（首尔大学数学科学系）；Research Institute of Mathematics, Seoul National University（首尔大学数学研究所）
We describe geometric algorithms that generalize the classical continued fraction algorithm for the torus to all translation surfaces in hyperelliptic components of translation surfaces. We show that these algorithms produce all saddle connections which are best approximations in a geometric sense, which generalizes the notion of best approximation for the classical continued fraction. In addition, they allow to list all systoles along a Teichmueller geodesic and all bispecial words which appear in the symbolic coding of linear flows. The elementary moves of the described algorithms provide a …
We study the arithmetic codings of hyperbolic automorphisms of the 2-torus, i.e. the continuous mappings acting from a certain symbolic space of sequences with a finite alphabet endowed with an appropriate structure of additive group onto the torus which preserve this structure and turn the two-sided shift into a given automorphism of the torus. This group is uniquely defined by an automorphism, and such an arithmetic coding is a homomorphism of that group onto the 2-torus. The necessary and sufficient condition of the existence of a bijective arithmetic coding is obtained; it is formulated in…
The fact that the euclidean algorithm eventually terminates is pervasive in mathematics. In the language of continued fractions, it can be stated by saying that the orbits of rational points under the Gauss map x-->{1/x} eventually reach zero. Analogues of this fact for Gauss maps defined over quadratic number fields have relevance in the theory of flows on translation surfaces, and have been established via powerful machinery, ultimately relying on the Veech dichotomy. In this paper, for each commensurability class of noncocompact triangle groups of quadratic invariant trace field, we constru…
We study the rotational behaviour on minimal sets of torus homeomorphisms and show that the associated rotation sets can be any type of line segments as well as non-convex and even plane-separating continua. This shows that restrictions holding for rotation sets on the whole torus are not valid on minimal sets. The proof uses a construction of rotational horseshoes by Kwapisz to transfer the problem to a symbolic level, where the desired rotational behaviour is implemented by means of suitable irregular Toeplitz sequences.
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 spectrum for $S$-adic shifts and tiling properties of associated Rauzy fractals are established under a generalized Pisot assumption together with a geometric coincidence condition. These general results extend the scope of the Pisot substitution conjecture to the $S$-adic framework. They are applied to families of $S$-adic shifts generated by Arnoux-Rauzy as we…
We consider a symbolic coding for geodesics on the family of Veech surfaces (translation surfaces rich with affine symmetries) recently discovered by Bouw and Moeller. These surfaces, as noticed by Hooper, can be realized by cutting and pasting a collection of semi-regular polygons. We characterize the set of symbolic sequences (cutting sequences) that arise by coding linear trajectories by the sequence of polygon sides crossed. We provide a full characterization for the closure of the set of cutting sequences, in the spirit of the classical characterization of Sturmian sequences and the recen…
We define a partition $\mathcal{P}_0$ and a $\mathbb{Z}^2$-rotation ($\mathbb{Z}^2$-action defined by rotations) on a 2-dimensional torus whose associated symbolic dynamical system is a minimal proper subshift of the Jeandel-Rao aperiodic Wang shift defined by 11 Wang tiles. We define another partition $\mathcal{P}_\mathcal{U}$ and a $\mathbb{Z}^2$-rotation on $\mathbb{T}^2$ whose associated symbolic dynamical system is equal to a minimal and aperiodic Wang shift defined by 19 Wang tiles. This proves that $\mathcal{P}_\mathcal{U}$ is a Markov partition for the $\mathbb{Z}^2$-rotation on $\math…
The Arnoux-Rauzy-Poincaré multidimensional continued fraction algorithm is obtained by combining the Arnoux-Rauzy and Poincaré algorithms. It is a generalized Euclidean algorithm. Its three-dimensional linear version consists in subtracting the sum of the two smallest entries to the largest if possible (Arnoux-Rauzy step), and otherwise, in subtracting the smallest entry to the median and the median to the largest (the Poincaré step), and by performing when possible Arnoux-Rauzy steps in priority. After renormalization it provides a piecewise fractional map of the standard $2$-simplex. We stud…
We extend the notion of Rauzy induction of interval exchange transformations to the case of toral $\mathbb{Z}^2$-rotation, i.e., $\mathbb{Z}^2$-action defined by rotations on a 2-torus. If $\mathcal{X}_{\mathcal{P},R}$ denotes the symbolic dynamical system corresponding to a partition $\mathcal{P}$ and $\mathbb{Z}^2$-action $R$ such that $R$ is Cartesian on a sub-domain $W$, we express the 2-dimensional configurations in $\mathcal{X}_{\mathcal{P},R}$ as the image under a $2$-dimensional morphism (up to a shift) of a configuration in $\mathcal{X}_{\widehat{\mathcal{P}}|_W,\widehat{R}|_W}$ where…
