arXiv subjects
Publications and source records attributed to Mark Pollicott.
In this note we propose an effective method to estimate the dimension of the Feigenbaum attractor for the period doubling phenomenon. In particular, we will describe a way to convert the highly accurate estimates for $g$ into better estimates on $\dim(X)$.
We describe the anlogue of the Einsidler, Kaydev, Polo and Sarig on Intrinsic Ergodicity in the simple setting of expanding maps and, in particular, $\beta$-transformations.
In this article we consider a restricted orbital counting problem for the action of certain discrete groups on suitable spaces. In particular, we present asymptotics for counting those points in an orbit restricted to a single conjugacy class. A classical example would be cocompact actions of a discrete group acting isometrically on a simply connected manifold with pinched negative curvature. More generally, we obtain results for convex cocompact actions on $CAT(-1)$ spaces.
We show that the abelian Liv\v{s}ic theorem recently obtained by A. Gogolev and F. Rodriguez Hertz for null-homologous periodic orbits of homologically full Anosov flows continues to hold when restricted to periodic orbits which are trivial with respect to any regular cover for which the lifted flow is transitive.
We statistically compare the relationships between frequencies of digits in continued fraction expansions of typical rational points in the unit interval and higher dimensional generalisations. This takes the form of a Large Deviation and Central Limit Theorem, including multidimensional results for random vectors. These results apply to classical multidimensional continued fraction transformations including Brun's algorithm and the Jacobi--Perron algorithm, and more generally for maps satisfying mild contraction hypothesis on the inverse branches. We prove in particular that the finite trajectories capture the generic ergodic behaviour of infinite trajectories.
We study counting limit laws that compare length functions on infinite graphs. We then apply these results to flat surfaces to obtain a statistical comparison between the geometric length and the number of singularities visited by geodesic paths.
We prove extreme value laws for cusp excursions of the horocycle flow in the case of surfaces of constant negative curvature. The key idea of our approach is to study the hitting time distribution for shrinking Poincar\'e sections that have a particularly simple scaling property under the action of the geodesic flow. This extends the extreme value law of Kirsebom and Mallahi-Karai [arXiv:2209.07283] for cusp excursions for the modular surface. Here we show that the limit law can be expressed in terms of Hall's formula for the gap distribution of the Farey sequence.
In this article, we give explicit conditions for compact group extensions of hyperbolic flows (including geodesic flows on negatively curved manifolds) to exhibit quantifiable rates of mixing (or decay of correlations) with respect to the natural probability measures, which are locally the product of a Gibbs measure for a H\"older potential and the Haar measure. More precisely, we show that the mixing rate with respect to H\"older functions will be faster than any given polynomial (i.e., rapid mixing). We also give error estimates on the equidistribution of the holonomy around closed orbits. In particular, these results apply to some frame flows for manifolds with negative sectional curvatures.
Suppose we have two finitely supported, admissible, probability measures on a hyperbolic group $\Gamma$. In this article we prove that the corresponding two Green metrics satisfy a counting central limit theorem when we order the elements of $\Gamma$ according to one of the metrics. Our results also apply to various other metrics including length functions associated to Anosov representations and to group actions on hyperbolic metric spaces.
There are several known constructions of equilibrium states for H\"older continuous potentials in the context of both subshifts of finite type and uniformly hyperbolic systems. In this article we present another method of building such measures, formulated in the unified and more general setting of Smale spaces. This simultaneously extends the authors' previous work for hyperbolic attractors (modelled after Sinai's classical approach for SRB-measures) and gives a new and original construction of equilibrium states for subshifts of finite type.
For the each of the five Euclidean rings of complex quadratic integers, we consider a complex continued fraction algorithm with digits in the ring. We show for each algorithm that the maximal digit obeys a Fr\'echet distribution. We use this to find a limiting distribution for cusp excursions on Bianchi orbifolds associated with the aforementioned rings of quadratic integers.
In this paper we prove asymptotic estimates for closed geodesic loops on compact surfaces with no conjugate points. These generalize the classical counting results of Huber and Margulis and sector theorems for surfaces of strictly negative curvature. We will also prove more general sector theorems, generalizing results of Nicholls and Sharp for special case of surfaces of strictly negative curvature.
We present a practical and effective method for rigorously estimating quantities associated to top eigenvalues of transfer operators to very high precision. The method combines explicit error bounds of the Lagrange-Chebyshev approximation with an established min-max method. We illustrate its applicability by significantly improving rigorous estimates on various ergodic quantities associated to the Bolyai-R\'enyi map.
We estabish rigorous estimates for the Hausdorff dimension of the spectra of Laplacians associated to Sierpi\'nski lattices and infinite Sierpi\'nski gaskets and other post-critically finite self-similar sets.
In this article we address a well known problem at the intersection of ergodic theory and statistical mechanics. We prove that there exists a continuous eigenfunction for the transfer operator corresponding to pair potentials that satisfy a square summability condition on the variations, when the inverse temperature is subcritical. As a corollary we obtain a continuous eigenfunction for the classical Dyson model, with interactions $\J(k)=\beta \, k^{-\alpha}$, $k\ge1$, in the whole subcritical regime $\beta<\beta_c$ for which the parameter $\alpha$ is greater than $3/2$.
The first part of this work is devoted to the study of higher differentials of pressure functions of H\"older potentials on shift spaces of finite type. By describing the differentials of pressure functions via the Central Limit Theorem for the associated random processes, we discover some rigid relationships between differentials of various orders. The rigidity imposes obstructions on fitting candidate convex analytic functions by pressure functions of H\"older potentials globally, which answers a question of Kucherenko-Quas. In the second part of the work we consider fitting candidate analytic germs by pressure functions of locally constant potentials. We prove that all 1-level candidate germs can be realised by pressures of some locally constant potentials, as long as number of the symbolic set is large enough. There are also some results on fitting 2-level germs by pressures of locally constant potentials obtained in the work.
We shall describe a new construction of equilibrium states for a class of partially hyperbolic systems. This generalises our construction for Gibbs measures in the uniformly hyperbolic setting. This more general setting introduces new issues that we need to address carefully, in particular requiring additional assumptions on the transformation. We treat two cases: either the centre-stable manifold satisfies a bounded expansion condition; or the centre-unstable manifold satisfies a subexponential contraction condition which appears new in the context of equilibrium state constructions. The problem of constructing equilibrium states was previously raised by Pesin-Sinai and Dolgopyat for the particular case of u-Gibbs measures, and by Climenhaga, Pesin and Zelerowicz for other equilibrium states.
We present a purely elementary proof that the Rauzy gasket has Hausdorff dimension strictly smaller than two.