arXiv subjects

Stefano Isola

Publications and source records attributed to Stefano Isola.

At least 19 recordsLinked to original sources

arXiv2026

Words and numbers: a dynamical systems perspective

Along with some known and less known results, we discuss new insights relating combinatorics of words and the ordering of the rationals from a dynamical systems point of view, somehow continuing along the path started in [BI]. We obtain in particular a set of results that structure and enrich the correspondence between the Stern-Brocot (SB) ordering of rational numbers and the corresponding ordering of Farey-Christoffel (FC) words, a class of words that, since their appearance in literature at the end of the 18th century, have revealed numerous relationships with other fields of mathematics. Among the results obtained here is the construction of substitution rules that act on the FC words in a parallel way to the maps on the positive reals that generate the permuted SB tree both vertically and horizontally. We further show that these rules naturally induce a map of the space of (infinite) Sturmian sequences into itself. Finally, a complete correspondence is obtained between the vertical and horizontal motions on the SB tree and the geodesic motions along scattering geodesics and the horocyclic motion along Ford circles in the upper half-plane, respectively.

math.DS↗

arXiv2023

On integer partitions and continued fraction type algorithms

We show that the additive-slow-Farey version of the traditional continued fractions algorithm has a natural interpretation as a method for producing integer partitions of a positive number $n$ into two smaller numbers, with multiplicity. We provide a complete description of how such integer partitions occur and of the conjugation for the corresponding Young shapes via the dynamics of the classical Farey tree. We use the dynamics of the Farey map to get a new formula for $p(2,n)$, the number of ways for partitioning $n$ into two smaller positive integers, with multiplicity. We then do the analogue using the additive-slow-Farey version of the Triangle map (a type of multi-dimensional continued fraction algorithm), giving us a method for producing integer partitions of a positive number $n$ into three smaller numbers, with multiplicity. However different aspects of this generalisations remain unclear.

math.NT↗

arXiv2022

A Poincaré map for the horocycle flow on $PSL(2,\mathbb{Z})\backslash \mathbb{H}$ and the Stern-Brocot tree

We construct a Poincaré map $\mathcal{P}_h$ for the positive horocycle flow on the modular surface $PSL(2,\mathbb{Z})\backslash \mathbb{H}$, and begin a systematic study of its dynamical properties. In particular we give a complete characterisation of the periodic orbits of $\mathcal{P}_h$, and show that they are equidistributed with respect to the invariant measure of $\mathcal{P}_h$ and that they can be organised in a tree by using the Stern-Brocot tree of rational numbers. In addition we introduce a time-reparameterisation of $\mathcal{P}_h$ which gives an insight into the dynamics of the non-periodic orbits. This paper constitutes a first step in the study of the dynamical properties of the horocycle flow by purely dynamical methods.

math.DS↗

arXiv2022

The simplest erasing substitution

In this work, we begin the study of a new class of dynamical systems determined by interval maps generated by the symbolic action of erasing substitution rules. We do this by discussing in some detail the geometric, analytical, dynamical and arithmetic properties of a particular example, which has the virtue of being arguably the simplest and that at the same time produces interesting properties and new challenging problems.

math.DS↗

arXiv2019

Series expansions for Maass forms on the full modular group from the Farey transfer operators

We deepen the study of the relations previously established by Mayer, Lewis and Zagier, and the authors, among the eigenfunctions of the transfer operators of the Gauss and the Farey maps, the solutions of the Lewis-Zagier three-term functional equation and the Maass forms on the modular surface $PSL(2,\Z)\backslash \HH$. In particular we introduce an "inverse" of the integral transform studied by Lewis and Zagier, and use it to obtain new series expansions for the Maass cusp forms and the non-holomorphic Eisenstein series restricted to the imaginary axis. As corollaries we obtain further information on the Fourier coefficients of the forms, including a new series expansion for the divisor function.

math.NT↗

arXiv2017

Diffuse Behaviour of Ergodic Sums Over Rotations

For a rotation by an irrational $α$ on the circle and a BV function $φ$, we study the variance of the ergodic sums $S_L φ(x) := \sum_{j=0}^{L -1} \, φ(x + jα)$. When $α$ is not of constant type, we construct sequences $(L_N)$ such that, at some scale, the ergodic sums $S_{L_N} φ$ satisfy an ASIP. Explicit non-degenerate examples are given, with an application to the rectangular periodic billiard in the plane.

math.DS↗

arXiv2015

On the generic triangle group

We introduce the concept of a generic Euclidean triangle $τ$ and study the group $G_τ$ generated by the reflection across the edges of $τ$. In particular, we prove that the subgroup $T_τ$ of all translations in $G_τ$ is free abelian of infinite rank, while the index 2 subgroup $H_τ$ of all orientation preserving transformations in $G_τ$ is free metabelian of rank 2, with $T_τ$ as the commutator subgroup. As a consequence, the group $G_τ$ cannot be finitely presented and we provide explicit minimal infinite presentations of both $H_τ$ and $G_τ$. This answers in the affirmative the problem of the existence of a minimal presentation for the free metabelian group of rank 2. Moreover, we discuss some examples of non-trivial relations in $T_τ$ holding for given non-generic triangles $τ$.

math.MG↗

arXiv2013

Continued fractions with SL(2, Z)-branches: combinatorics and entropy

We study the dynamics of a family K_alpha of discontinuous interval maps whose (infinitely many) branches are Moebius transformations in SL(2, Z), and which arise as the critical-line case of the family of (a, b)-continued fractions. We provide an explicit construction of the bifurcation locus E_KU for this family, showing it is parametrized by Farey words and it has Hausdorff dimension zero. As a consequence, we prove that the metric entropy of K_alpha is analytic outside the bifurcation set but not differentiable at points of E_KU, and that the entropy is monotone as a function of the parameter. Finally, we prove that the bifurcation set is combinatorially isomorphic to the main cardioid in the Mandelbrot set, providing one more entry to the dictionary developed by the authors between continued fractions and complex dynamics.

math.DS↗

arXiv2012

Dynamics of continued fractions and kneading sequences of unimodal maps

In this paper we construct a correspondence between the parameter spaces of two families of one-dimensional dynamical systems, the alpha-continued fraction transformations T_alpha and unimodal maps. This correspondence identifies bifurcation parameters in the two families, and allows one to transfer topological and metric properties from one setting to the other. As an application, we recover results about the real slice of the Mandelbrot set, and the set of univoque numbers.

math.DS↗

arXiv2011

A thermodynamic approach to two-variable Ruelle and Selberg zeta functions via the Farey map

In this paper we consider the transfer operator approach to the Ruelle and Selberg zeta functions associated to continued fractions transformations and the geodesic flow on the full modular surface. We extend the results by Ruelle and Mayer to two-variable zeta functions, $ζ(q,z)$ and $Z(q,z)$. The $q$ variable plays the role of the inverse temperature and the introduction of the "geometric variable" $z$ is essential in the tentative to provide a general approach, based on the Farey map, to the correspondence between the analytic properties of the zeta functions themselves, the spectral properties of a class of generalised transfer operators and the theory of a generalisation of the three-term functional equations studied by Lewis and Zagier. The first step in this direction is a detailed study of the spectral properties of a family of signed transfer operators $\PP^{\pm}_{q}$ associated to the Farey map.

math.DS↗

arXiv2008

On ergodic and mixing properties of the triangle map

In this paper we study in detail, both analytically and numerically, the dynamical properties of the triangle map, a piecewise parabolic automorphism of the two-dimensional torus, for different values of the two independent parameters defining the map. The dynamics is studied numerically by means of two different symbolic encoding schemes, both relying on the fact that it maps polygons to polygons: in the first scheme we consider dynamically generated partitions made out of suitable sets of disjoint polygons, in the second we consider the standard binary partition of the torus induced by the discontinuity set. These encoding schemes are studied in detail and shown to be compatible, although not equivalent. The ergodic properties of the triangle map are then investigated in terms of the Markov transition matrices associated to the above schemes and furthermore compared to the spectral properties of the Koopman operator in L2(T2). Finally, a stochastic version of the triangle map is introduced and studied. A simple heuristic analysis of the latter yields the correct statistical and scaling behaviours of the correlation functions of the original map.

nlin.CD↗

arXiv2008

Orderings of the rationals and dynamical systems

This paper is devoted to a systematic study of a class of binary trees encoding the structure of rational numbers both from arithmetic and dynamical point of view. The paper is divided into two parts. The first one is a critical review of rather standard topics such as Stern-Brocot and Farey trees and their connections with continued fraction expansion and the question mark function. In the second part we introduce a class of one-dimensional maps which can be used to generate the binary trees in different ways and study their ergodic properties. This also leads us to study some random processes (Markov chains and martingales) arising in a natural way in this context.

math.DS↗

arXiv2007

Spectral analysis of transfer operators associated to Farey fractions

The spectrum of a one-parameter family of signed transfer operators associated to the Farey map is studied in detail. We show that when acting on a suitable Hilbert space of analytic functions they are self-adjoint and exhibit absolutely continuous spectrum and no non-zero point spectrum. Polynomial eigenfunctions when the parameter is a negative half-integer are also discussed.

math-ph↗

arXiv2007

A renormalization approach to irrational rotations

We introduce a renormalization procedure which allows us to study in a unified and concise way different properties of the irrational rotations on the unit circle $β\mapsto \set{α+β}$, $α\in \R\setminus \Q$. In particular we obtain sharp results for the diffusion of the walk on $\Z$ generated by the location of points of the sequence $\{nα+β\}$ on a binary partition of the unit interval. Finally we give some applications of our method.

math.DS↗

arXiv2006

Generalized Farey trees, transfer Operators and phase transitions

We consider a family of Markov maps on the unit interval, interpolating between the tent map and the Farey map. The latter map is not uniformly expanding. Each map being composed of two fractional linear transformations, the family generalizes many particular properties which for the case of the Farey map have been successfully exploited in number theory. We analyze the dynamics through the spectral analysis of generalized transfer operators. Application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function.

math-ph↗

arXiv2003

On the rate of convergence to equilibrium for countable ergodic Markov chains

Using elementary methods, we prove that for a countable Markov chain $P$ of ergodic degree $d > 0$ the rate of convergence towards the stationary distribution is subgeometric of order $n^{-d}$, provided the initial distribution satisfies certain conditions of asymptotic decay. An example, modelling a renewal process and providing a markovian approximation scheme in dynamical system theory, is worked out in detail, illustrating the relationships between convergence behaviour, analytic properties of the generating functions associated to transition probabilities and spectral properties of the Markov operator $P$ on the Banach space $\ell_1$. Explicit conditions allowing to obtain the actual asymptotics for the rate of convergence are also discussed.

math.PR↗

arXiv2003

On systems with finite ergodic degree

In this paper we study the ergodic theory of a class of symbolic dynamical systems $(Ø, T, μ)$ where $T:Ø\to Ø$ the left shift transformation on $Ø=\prod_0^\infty\{0,1\}$ and $μ$ is a $\s$-finite $T$-invariant measure having the property that one can find a real number $d$ so that $μ(τ^d)=\infty$ but $μ(τ^{d-ε})<\infty$ for all $ε>0$, where $τ$ is the first passage time function in the reference state 1. In particular we shall consider invariant measures $μ$ arising from a potential $V$ which is uniformly continuous but not of summable variation. If $d>0$ then $μ$ can be normalized to give the unique non-atomic equilibrium probability measure of $V$ for which we compute the (asymptotically) exact mixing rate, of order $n^{-d}$. We also establish the weak-Bernoulli property and a polynomial cluster property (decay of correlations) for observables of polynomial variation. If instead $d\leq 0$ then $μ$ is an infinite measure with scaling rate of order $n^d$. Moreover, the analytic properties of the weighted dynamical zeta function and those of the Fourier transform of correlation functions are shown to be related to one another via the spectral properties of an operator-valued power series which naturally arises from a standard inducing procedure. A detailed control of the singular behaviour of these functions in the vicinity of their non-polar singularity at $z=1$ is achieved through an approximation scheme which uses generating functions of a suitable renewal process. In the perspective of differentiable dynamics, these are statements about the unique absolutely continuous invariant measure of a class of piecewise smooth interval maps with an indifferent fixed point.

math.DS↗

arXiv2003

A one-parameter family of analytic Markov maps with an intermittency transition

In this paper we introduce and study a one-parameter family of piecewise analytic interval maps having the tent map and the Farey map as extrema. Among other things, we construct a Hilbert space of analytic functions left invariant by the Perron-Frobenius operator of all these maps and study the transition between discrete and continuous spectrum when approaching the intermittent situation.

math.DS↗