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Sara Munday

Publications and source records attributed to Sara Munday.

14 recordsLinked to original sources
arXiv2020

A slow triangle map with a segment of indifferent fixed points and a complete tree of rational pairs

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 sequence, the expansion associated to $T$. Finally, we confirm the role of the map $S$ as a two-dimensional version of the Farey map by introducing a complete tree of rational pairs, constructed using the inverse branches of $S$, in the same way as the Farey tree is generated by the Farey map, and then, equivalently, generated by a generalised mediant operation.

math.DS↗
arXiv2019

Matching for a family of infinite measure continued fraction transformations

As a natural counterpart to Nakada's $α$-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension for a large part of the parameter space. We use this to obtain an explicit expression for the density of the unique infinite $σ$-finite absolutely continuous invariant measure and to compute the Krengel entropy, return sequence and wandering rate of the corresponding maps.

math.DS↗
arXiv2018

Pointwise convergence of Birkhoff averages for global observables

It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in infinite ergodic theory is trivial; it states that for any infinite-measure-preserving ergodic system the Birkhoff average of every integrable function is almost everywhere zero. Nor does a different rescaling of the Birkhoff sum that leads to a non-degenerate pointwise limit exist. In this paper we give a version of Birkhoff's theorem for conservative, ergodic, infinite-measure-preserving dynamical systems where instead of integrable functions we use certain elements of $L^\infty$, which we generically call global observables. Our main theorem applies to general systems but requires an hypothesis of "approximate partial averaging" on the observables. The idea behind the result, however, applies to more general situations, as we show with an example. Finally, by means of counterexamples and numerical simulations, we discuss the question of finding the optimal class of observables for which a Birkhoff theorem holds for infinite-measure-preserving systems.

math.DS↗
arXiv2018

A multifractal analysis for cuspidal windings on hyperbolic surfaces

In this paper we investigate the multifractal decomposition of the limit set of a finitely generated, free Fuchsian group with respect to the mean cusp winding number. We will completely determine its multifractal spectrum by means of a certain free energy function and show that the Hausdorff dimension of sets consisting of limit points with the same scaling exponent coincides with the Legendre transform of this free energy function. As a by-product we generalise previously obtained results on the multifractal formalism for infinite iterated function systems to the setting of infinite graph directed Markov systems.

math.DS↗
arXiv2016

Escape rate scaling in infinite measure preserving systems

We investigate the scaling of the escape rate from piecewise-linear dynamical systems displaying intermittency due to the presence of an indifferent fixed-point. Strong intermittent behaviour in the dynamics can result in the system preserving an infinite measure. We define a neighbourhood of the indifferent fixed point to be a hole through which points escape and investigate the scaling of the rate of this escape as the length of the hole decreases, both in the finite measure preserving case and infinite measure preserving case. In the infinite measure preserving systems we observe logarithmic corrections to and polynomial scaling of the escape rate with hole length. Finally we conjecture a relationship between the wandering rate and the observed scaling of the escape rate.

nlin.CD↗
arXiv2016

Pointwise perturbations of countable Markov maps

We study the pointwise perturbations of countable Markov maps with infinitely many inverse branches and establish the following continuity theorem: Let $T_k$ and $T$ be expanding countable Markov maps such that the inverse branches of $T_k$ converge pointwise to the inverse branches of $T$ as $k \to \infty$. Then under suitable regularity assumptions on the maps $T_k$ and $T$ the following limit exists: $$\lim_{k \to \infty} \dim_\mathrm{H} \{x : θ_k'(x) \neq 0\} = 1,$$ where $θ_k$ is the topological conjugacy between $T_k$ and $T$ and $\dim_\mathrm{H}$ stands for the Hausdorff dimension. This is in contrast with the fact that other natural quantities measuring the singularity of $θ_k$ fail to be continuous in this manner under pointwise convergence such as the Hölder exponent of $θ_k$ or the Hausdorff dimension $\dim_\mathrm{H} (μ\circ θ_k)$ for the preimage of the absolutely continuous invariant measure $μ$ for $T$. As an application we obtain a perturbation theorem in non-uniformly hyperbolic dynamics for conjugacies between intermittent Manneville-Pomeau maps $x \mapsto x + x^{1+α} \mod 1$ when varying the parameter $α$.

math.DS↗
arXiv2014

On two conjectures for M&m sequences

In this paper, the recently introduced M&m sequences and associated mean-median map are studied. These sequences are built by adding new points to a set of real numbers by balancing the mean of the new set with the median of the original. This process, although seemingly simple, gives rise to complicated dynamics. The main result is that two conjectures put forward by Chamberland and Martelli are shown to be true for a subset of possible starting conditions.

math.CO↗
arXiv2014

Diophantine approximation and coloring

We demonstrate how connections between graph theory and Diophantine approximation can be used in conjunction to give simple and accessible proofs of seemingly difficult results in both subjects.

math.NT↗
arXiv2013

Density of orbits of semigroups of endomorphisms acting on the Adeles

We investigate the question of whether or not the orbit of a point in A/Q, under the natural action of a subset S of Q, is dense in A/Q. We prove that if the set S is a multiplicative semigroup which contains at least two multiplicatively independent elements, one of which is an integer, then the orbit under S of any point with irrational real coordinate is dense.

math.NT↗
arXiv2012

On the derivative of the α-Farey-Minkowski function

In this paper we study the family of $α$-Farey-Minkowski functions $θ_α$, for an arbitrary countable partition $α$ of the unit interval with atoms which accumulate only at the origin, which are the conjugating homeomorphisms between each of the $α$-Farey systems and the tent map. We first show that each function $θ_α$ is singular with respect to the Lebesgue measure and then demonstrate that the unit interval can be written as the disjoint union of the following three sets: $Θ_0:={x\in\U:θ_α'(x)=0}, Θ_\infty:={x\in\U:θ_α'(x)=\infty} and Θ_\sim:=\U\setminus(Θ_0\cupΘ_\infty)$. The main result is that [\dim_{\mathrm{H}}(Θ_\infty)=\dim_{\mathrm{H}}(Θ_\sim)=σ_α(\log2)<\dim_{\mathrm{H}}(Θ_0)=1,] where $σ_α(\log2)$ is the Hausdorff dimension of the level set ${x\in \U:Λ(F_α, x)=s}$, where $Λ(F_α, x)$ is the Lyapunov exponent of the map $F_α$ at the point $x$. The proof of the theorem employs the multifractal formalism for $α$-Farey systems.

math.DS↗
arXiv2011

On Hausdorff dimension and cusp excursions for Fuchsian groups

Certain subsets of limit sets of geometrically finite Fuchsian groups with parabolic elements are considered. It is known that Jarn\'ık limit sets determine a "weak multifractal spectrum" of the Patterson measure in this situation. This paper will describe a natural generalisation of these sets, called strict Jarn\'ık limit sets, and show how these give rise to another weak multifractal spectrum. Number-theoretical interpretations of these results in terms of continued fractions will also be given.

math.DS↗
arXiv2011

Strong renewal theorems and Lyapunov spectra for $α$-Farey and $α$-Lüroth systems

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 $α$-sum-level sets for the $α$-Lüroth map. Similar results have previously been obtained for the Farey map and the Gauss map, by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of the $α$-Farey map and the $α$-Lüroth map in terms of the thermodynamical formalism. We show how to derive these spectra, and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen partition $α$.

math.DS↗