In this paper we introduce Hilbert spaces of holomorphic functions given by generalized Borel and Laplace transforms which are left invariant by the transfer operators of the Farey map and its induced version, the Gauss map, respectively. By means of a suitable operator-valued power series we are able to study simultaneously the spectrum of both these operators along with the analytic properties of the associated dynamical zeta functions.
We define the Farey web --- a collection of loci in the parameter plane of families of simple non-invertible maps of the circle. We prove some results about the arrangement of these loci and their relationships with other dynamically significant features of the parameter plane. The results enable us to provide short proofs for a number of theorems about the organization of frequency-locking.
Based on Broise-Alamichel and Paulin's work on the Gauss map corresponding to the principal convergents, we continue the study of the Gauss map via Farey maps to contain all the intermediate convergents. We define the geometric Farey map, which is given by time-1 map of the geodesic flow. We also define algebraic Farey maps, better suited for arithmetic properties, which produce all the intermediate convergents. Then we obtain the ergodic invariant measures for the Farey maps and the convergent speed.
Based on symbolic dynamics of Lorenz maps, we prove that, pro- vided one conjecture due to Morton is true, then Lorenz knots asso- ciated to orbits of points in the renormalization intervals of Lorenz maps with reducible kneading invariant of type (X,Y) * S, where the sequences X and Y are Farey neighbors verifying some conditions, are hyperbolic.
The classical Farey sequence of height $Q$ is the set of rational numbers in reduced form with denominator less than $Q$. In this paper we introduce the concept of a generalized Farey sequence. While these sequences arise naturally in the study of discrete (and in particular thin) subgroups, they can be used to study interesting number theoretic sequences - for example rationals whose continued fraction partial quotients are subject to congruence conditions. We show that these sequences equidistribute, that the gap distribution converges, and we answer an associated problem in Diophantine appr…
We study skew-products of the form (x,ω)\mapsto (Tx, ω+ϕ(x)) where T is a nonuniformly expanding map on a space X, preserving a (possibly singular) probability measure \tildeμ, and ϕ:X\to S^1 is a C^1 function. Under mild assumptions on \tildeμand ϕ, we prove that such a map is exponentially mixing, and satisfies the central and local limit theorems. These results apply to a random walk related to the Farey sequence, thereby answering a question of Guivarc'h and Raugi.
We consider the problem of showing that 1 is an eigenvalue for a family of generalised transfer operators of the Farey map. This problem is related to the spectral theory of the modular surface via the Selberg Zeta function and the theory of dynamical zeta functions of maps. After briefly recalling these connections, we show that the problem can be formulated for operators on an appropriate Hilbert space and translated into a linear algebra problem for infinite matrices.
In this paper we introduce and study the $α$-Farey map and its associated jump transformation, the $α$-Lüroth map, for an arbitrary countable partition $α$ of the unit interval with atoms which accumulate only at the origin. These maps represent linearised generalisations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic-theoretic properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have called $α$…
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.
We expand the cross section of the geodesic flow in the tangent bundle of the modular surface given by Series to produce another section whose return map under the geodesic flow is a double cover of the natural extension of the Farey map. We use this cross section to extend the correspondence between the closed geodesics on the modular surface and the periodic points of the Gauss map to include the periodic points of the Farey map. Then, analogous to the work of Pollicott, we prove an equidistribution result for the periodic points of the Farey map when they are ordered according to the length…
We study the asymptotics of iterates of the transfer operator for non-uniformly hyperbolic $α$-Farey maps. We provide a family of observables which are Riemann integrable, locally constant and of bounded variation, and for which the iterates of the transfer operator, when applied to one of these observables, is not asymptotic to a constant times the wandering rate on the first element of the partition $α$. Subsequently, sufficient conditions on observables are given under which this expected asymptotic holds. In particular, we obtain an extension theorem which establishes that, if the asymptot…
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 info…
We study the two-dimensional continued fraction algorithm introduced in \cite{garr} and the associated \emph{triangle map} $T$, defined on a triangle $\triangle\subset \R^2$. We introduce a slow version of the triangle map, the map $S$, which is ergodic with respect to the Lebesgue measure and preserves an infinite Lebesgue-absolutely continuous invariant measure. We discuss the properties that the two maps $T$ and $S$ share with the classical Gauss and Farey maps on the interval, including an analogue of the weak law of large numbers and of Khinchin's weak law for the digits of the triangle s…
We consider a family $\{ T_{r} \colon [0, 1] \circlearrowleft \}_{r \in [0, 1]}$ of Markov interval maps interpolating between the Tent map $T_{0}$ and the Farey map $T_{1}$. Letting $\mathcal{P}_{r}$ denote the Perron-Frobenius operator of $T_{r}$, we show, for $β\in [0, 1]$ and $α\in (0, 1)$, that the asymptotic behaviour of the iterates of $\mathcal{P}_{r}$ applied to observables with a singularity at $β$ of order $α$ is dependent on the structure of the $ω$-limit set of $β$ with respect to $T_{r}$. Having a singularity it seems that such observables do not fall into any of the function cla…
Given a Farey-type map F with full branches in the extended Hecke group Gamma_m, its dual F_# results from constructing the natural extension of F, letting time go backwards, and projecting. Although numerical simulations may suggest otherwise, we show that the domain of F_# is always tame, that is, it always contains intervals. As a main technical tool we construct, for every m=3,4,5,..., a homeomorphism M_m that simultaneously linearizes all maps with branches in Gamma_m, and show that the resulting dual linearized iterated function system satisfies the strong open set condition. We explicit…
We will establish several arithmetic and geometric properties regarding the bi-sequences of approximation coefficients (BAC) associated with the two one-parameter families of piecewise-continuous Mobius transformations introduced by Haas and Molnar. The Gauss and Renyi maps, which lead to the expansions of irrational numbers on the interval as regular and backwards continued fractions, are realized as special cases. The results are natural generalizations of theorems from Diophantine approximation.
We propose an extension of ergodic theory which focuses on the identification of ergodicity in terms of the uniqueness of the invariant measure. We first explain the concept for the doubling maps, which can be analyzed using Fourier methods. We then proceed to the Gauss-type maps of interest, of the form $x\mapsto -β/x$ mod $2\mathbb Z$ on the symmetric interval $[-1,1]$, for $0<β\le1$. We study an extended state space on the interval, formed as the restriction to the interval $[-1,1]$ of functions of the form $f+\mathbf{H}g$, where $f$ and $g$ are $L^1$-functions. We then look for invariant s…
