arXiv subjects

Rhiannon Dougall

Publications and source records attributed to Rhiannon Dougall.

7 recordsLinked to original sources

arXiv2026

Relative ($\tau$), Expanders, and Decay of Correlations for certain Expanding Maps

Relative ($\tau$) is equivalent to a statement that the sequence of Cayley graphs associated to group quotients $\Gamma_q=\Gamma/N_q$, $q\in\mathbb{N}$, form an expander family. There is a philosophy that expander graphs give rise to good mixing; for instance, one has exponential mixing for the geodesic flow uniformly the along a family of congruence covers of the modular surface, stemming from the symmetry in the $\mathrm{SL}(2,\mathbb{R})$ action and the uniform spectral gap for the Laplacian. Do we see similar phenomena in less structured settings? We investigate this question for a tower of finite sheeted covers of certain expanding dynamical systems. We use transfer operator machinery that applies in particular in the cases of subshifts of finite type and expanding interval maps. We make fruitful connections with KMS states of Cuntz--Krieger algebras.

math.DS↗

arXiv2023

A non-symmetric Kesten criterion and ratio limit theorem for random walks on amenable groups

We consider random walks on countable groups. A celebrated result of Kesten says that the spectral radius of a symmetric walk (whose support generates the group as a semigroup) is equal to one if and only if the group is amenable. We give an analogue of this result for finitely supported walks which are not symmetric. We also conclude a ratio limit theorem for amenable groups.

math.GR↗

arXiv2021

Drift and Matrix coefficients for discrete group extensions of countable Markov shifts

There has been much interest in generalizing Kesten's criterion for amenability in terms of a random walk to other contexts, such as determining amenability of a deck covering group by the bottom of the spectrum of the Laplacian or entropy of the geodesic flow. One outcome of this work is to generalise the results to so-called discrete group extensions of countable Markov shifts that satisfy a strong positive recurrence hypothesis. The other outcome is to further develop the language of unitary representation theory in this problem, and to bring some of the machinery developed by Coulon-Dougall-Schapira-Tapie [Twisted Patterson-Sullivan measures and applications to amenability and coverings, arXiv:1809.10881, 2018] to the countable Markov shift setting. In particular we recast the problem of determining a drop in Gurevič pressure in terms of eventual almost sure decay for matrix coefficients, and explain that a so-called twisted measure "finds points with the worst decay." We are also able locate the results of Dougall-Sharp [Anosov flows, growth rates on covers and group extensions of subshifts, {\em Inventiones Mathematicae}, 223, 445-483, 2021] within this framework.

math.DS↗

arXiv2019

Anosov flows, growth rates on covers and group extensions of subshifts

The aim of this paper is to study growth properties of group extensions of hyperbolic dynamical systems, where we do not assume that the extension satisfies the symmetry conditions seen, for example, in the work of Stadlbauer on symmetric group extensions and of the authors on geodesic flows. Our main application is to growth rates of periodic orbits for covers of an Anosov flow: we reduce the problem of counting periodic orbits in an amenable cover $X$ to counting in a maximal abelian subcover $X^{\mathrm{ab}}$. In this way, we obtain an equivalence for the Gurevič entropy: $h(X)=h(X^{\mathrm{ab}})$ if and only if the covering group is amenable. In addition, when we project the periodic orbits for amenable covers $X$ to the compact factor $M$, they equidistribute with respect to a natural equilibrium measure -- in the case of the geodesic flow, the measure of maximal entropy.

math.DS↗

arXiv2018

Twisted Patterson-Sullivan measures and applications to amenability and coverings

Let $Γ'<Γ$ be two discrete groups acting properly by isometries on a Gromov-hyperbolic space $X$. We prove that their critical exponents coincide if and only if $Γ'$ is co-amenable in $Γ$, under the assumption that the action of $Γ$ on $X$ is strongly positively recurrent, i.e. has a growth gap at infinity. This generalizes all previously known results on this question, which required either $X$ to be the real hyperbolic space and $Γ$ geometrically finite, or $X$ Gromov hyperbolic and $Γ$ cocompact. This result is optimal: we provide several counterexamples when the action is not strongly positively recurrent.

math.GR↗

arXiv2017

Critical exponents of normal subgroups, the spectrum of group extended transfer operators, and Kazhdan distance

For a pinched Hadamard manifold $X$ and a discrete group of isometries $Γ$ of $X$, the critical exponent $δ_Γ$ is the exponential growth rate of the orbit of a point in $X$ under the action of $Γ$. We show that the critical exponent for any family $\mathcal{N}$ of normal subgroups of $Γ_0$ has the same coarse behaviour as the Kazhdan distances for the right regular representations of the quotients $Γ_0/Γ$. The key tool is to analyse the spectrum of transfer operators associated to subshifts of finite type, for which we obtain a result of independent interest.

math.DS↗

arXiv2015

Amenability, Critical Exponents of Subgroups and Growth of Closed Geodesics

Let $Γ$ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold $X$. We show that a normal subgroup $Γ_0$ has critical exponent equal to the critical exponent of $Γ$ if and only if $Γ/ Γ_0$ is amenable. We prove a similar result for the exponential growth rate of closed geodesics on $X / Γ$. These statements are analogues of classical results of Kesten for random walks on groups and of Brooks for the spectrum of the Laplacian on covers of Riemannian manifolds.

math.DS↗