arXiv subjects

Tony Samuel

Publications and source records attributed to Tony Samuel.

25 records · Page 2Linked to original sources

arXiv2015

Intermediate β-shifts of finite type

An aim of this article is to highlight dynamical differences between the greedy, and hence the lazy, $β$-shift (transformation) and an intermediate $β$-shift (transformation), for a fixed $β\in (1, 2)$. Specifically, a classification in terms of the kneading invariants of the linear maps $T_{β,α} \colon x \mapsto βx + α\bmod 1$ for which the corresponding intermediate $β$-shift is of finite type is given. This characterisation is then employed to construct a class of pairs $(β,α)$ such that the intermediate $β$-shift associated with $T_{β, α}$ is a subshift of finite type. It is also proved that these maps $T_{β,α}$ are not transitive. This is in contrast to the situation for the corresponding greedy and lazy $β$-shifts and $β$-transformations, for which both of the two properties do not hold.

math.DS↗

arXiv2014

A note on measure-geometric Laplacians

We consider the measure-geometric Laplacians $Δ^μ$ with respect to atomless compactly supported Borel probability measures $μ$ as introduced by Freiberg and Zähle in 2002 and show that the harmonic calculus of $Δ^μ$ can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of $Δ^μ$. Further, it is shown that there exists a measure-geometric Laplacian whose eigenfunctions are the Chebyshev polynomials and illustrate our results through specific examples of fractal measures, namely Salem and inhomogeneous self-similar Cantor measures.

math.FA↗

arXiv2014

On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps

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 classes on which convergence to equilibrium has been previously shown.

math.DS↗

arXiv2014

On the asymptotics of the $α$-Farey transfer operator

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 asymptotic behaviour of iterates of the transfer operator is known on the first element of the partition $α$, then the same asymptotic holds on any compact set bounded away from the indifferent fixed point.

math.DS↗

arXiv2013

Continuous images of Cantor's ternary set

The Hausdorff-Alexandroff Theorem states that any compact metric space is the continuous image of Cantor's ternary set $C$. It is well known that there are compact Hausdorff spaces of cardinality equal to that of $C$ that are not continuous images of Cantor's ternary set. On the other hand, every compact countably infinite Hausdorff space is a continuous image of $C$. Here we present a compact countably infinite non-Hausdorff space which is not the continuous image of Cantor's ternary set.

math.DS↗

arXiv2012

A Simple Proof of Vitali's Theorem for Signed Measures

There are several theorems named after the Italian mathematician Vitali. In this note we provide a simple proof of an extension of Vitali's Theorem on the existence of non-measurable sets. Specifically, we show, without using any decomposition theorems, that there does not exist a non-trivial, atom-less, $σ$-additive and translation invariant set function $\mathcal{L}$ from the power set of the real line to the extended real numbers with $\mathcal{L}([0,1]) = 1$. (Note that $\mathcal{L}$ is not assumed to be non-negative.)

math.DS↗

arXiv2011

Spectral metric spaces for Gibbs measures

We construct spectral metric spaces for Gibbs measures on a one-sided topologically exact subshift of finite type. That is, for a given Gibbs measure we construct a spectral triple and show that Connes' corresponding pseudo-metric is a metric and that its metric topology agrees with the weak-*-topology on the state space over the set of continuous functions defined on the subshift. Moreover, we show that each Gibbs measure can be fully recovered from the noncommutative integration theory and that the noncommutative volume constant of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of the shift invariant Gibbs measure.

math.OA↗