Mark Pollicott
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View article: Multidimensional statistics for finite orbits of generalised continued fractions
Multidimensional statistics for finite orbits of generalised continued fractions Open
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 …
View article: Continuous eigenfunctions of the transfer operator for Dyson models
Continuous eigenfunctions of the transfer operator for Dyson models Open
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 …
View article: Extreme events for horocycle flows
Extreme events for horocycle flows Open
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 Poincaré sections that have …
View article: Effective estimates of ergodic quantities illustrated on the Bolyai-Rényi map
Effective estimates of ergodic quantities illustrated on the Bolyai-Rényi map Open
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 …
View article: Rapid mixing for compact group extensions of hyperbolic flows
Rapid mixing for compact group extensions of hyperbolic flows Open
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 …
View article: Central limit theorems for Green metrics on hyperbolic groups
Central limit theorems for Green metrics on hyperbolic groups Open
Suppose we have two finitely supported, admissible, probability measures on a hyperbolic group $Γ$. In this article we prove that the corresponding two Green metrics satisfy a counting central limit theorem when we order the elements of $Γ…
View article: Rigidity of pressures of Hölder potentials and the fitting of analytic functions through them
Rigidity of pressures of Hölder potentials and the fitting of analytic functions through them Open
The first part of this work is devoted to the study of higher derivatives of pressure functions of Hölder potentials on shift spaces with finitely many symbols. By describing the derivatives of pressure functions via the central limit theo…
View article: Constructing equilibrium states for Smale spaces
Constructing equilibrium states for Smale spaces Open
There are several known constructions of equilibrium states for Hölder 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 measu…
View article: Complex continued fractions, Kleinian and extremal theory for cusp excursions
Complex continued fractions, Kleinian and extremal theory for cusp excursions Open
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 Fréchet distribution. We use this…
View article: An infinite interval version of the α-Kakutani equidistribution problem
An infinite interval version of the α-Kakutani equidistribution problem Open
In this article we extend results of Kakutani, Adler–Flatto, Smilansky and others on the classical α -Kakutani equidistribution result for sequences arising from finite partitions of the interval. In particular, we describe a generalizatio…
View article: Counting geodesic loops on surfaces of genus at least 2 without conjugate points
Counting geodesic loops on surfaces of genus at least 2 without conjugate points Open
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…
View article: Effective estimates of ergodic quantities illustrated on the Bolyai-Rényi map
Effective estimates of ergodic quantities illustrated on the Bolyai-Rényi map Open
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 …
View article: Sierpiński Fractals and the Dimension of Their Laplacian Spectrum
Sierpiński Fractals and the Dimension of Their Laplacian Spectrum Open
We establish rigorous estimates for the Hausdorff dimension of the spectra of Laplacians associated with Sierpiński lattices and infinite Sierpiński gaskets and other post-critically finite self-similar sets.
View article: Continuous eigenfunctions of the transfer operator for Dyson models
Continuous eigenfunctions of the transfer operator for Dyson models Open
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 …
View article: Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups
Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups Open
In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable attent…
View article: Explicit examples of resonances for Anosov maps of the torus
Explicit examples of resonances for Anosov maps of the torus Open
In (2017 Nonlinearity 30 2667–86) Slipantschuk, Bandtlow and Just gave concrete examples of Anosov diffeomorphisms of for which their resonances could be completely described. Their approach was based on composition operators acting …
View article: Accurate Bounds on Lyapunov Exponents for Expanding Maps of the Interval
Accurate Bounds on Lyapunov Exponents for Expanding Maps of the Interval Open
In this short note we describe a simple but remarkably effective method for rigorously estimating Lyapunov exponents for expanding maps of the interval. We illustrate the applicability of this method with some standard examples.
View article: Rigidity of pressures of Hölder potentials and the fitting of analytic functions via them
Rigidity of pressures of Hölder potentials and the fitting of analytic functions via them Open
The first part of this work is devoted to the study of higher differentials of pressure functions of Hölder potentials on shift spaces of finite type. By describing the differentials of pressure functions via the Central Limit Theorem for …
View article: Constructing equilibrium states for some partially hyperbolic attractors via densities
Constructing equilibrium states for some partially hyperbolic attractors via densities Open
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 issu…
View article: Minimizing entropy for translation surfaces
Minimizing entropy for translation surfaces Open
In this note we consider the entropy by Dankwart [On the large-scale geometry of flat surfaces, 2014, PhD thesis. https://bib.math.uni-bonn.de/downloads/bms/BMS-401.pdf] of unit area translation surfaces in theorbits of square tiled surfac…
View article: Comparison theorems for closed geodesics on negatively curved surfaces
Comparison theorems for closed geodesics on negatively curved surfaces Open
In this note, we present new asymptotic estimates comparing the word length and geodesic length of closed geodesics on surfaces with (variable) negative sectional curvatures. In particular, we provide an averaged comparison of these two im…
View article: An elementary proof that the Rauzy gasket is fractal
An elementary proof that the Rauzy gasket is fractal Open
We present a purely elementary proof that the Rauzy gasket has Hausdorff dimension strictly smaller than two.
View article: The Bowen$\unicode{x2013}$Series coding and zeros of zeta functions
The Bowen$\unicode{x2013}$Series coding and zeros of zeta functions Open
We give a discussion of the classical Bowen$\unicode{x2013}$Series coding and, in particular, its application to the study of zeta functions associated to geodesic flows and their zeros. In the case of compact surfaces of constant negative…
View article: Groups, drift and harmonic measures
Groups, drift and harmonic measures Open
In this short note we will describe an old problem and a new approach which casts light upon it. The old problem is to understand the nature of harmonic measures for cocompact Fuchsian groups. The new approach is to compute numerically the…
View article: Maximizing dimension for Bernoulli measures and the Gauss map
Maximizing dimension for Bernoulli measures and the Gauss map Open
We give a short proof that there exists a countable state Bernoulli measure maximizing the dimension of their images under the continued fraction expansion.
View article: Explicit examples of resonances for Anosov maps of the torus
Explicit examples of resonances for Anosov maps of the torus Open
In [23], Slipantschuk, Bandtlow and Just gave concrete examples of Anosov diffeomorphisms of the two-torus for which their resonances could be completely described. Their approach was based on composition operators acting on analytic aniso…
View article: Zeta functions in higher Teichmuller theory
Zeta functions in higher Teichmuller theory Open
In this note we introduce zeta functions and L-functions for discrete and faithful representations of surface groups in PSL(d, R), for d >= 3. These are natural generalizations of the wellknown classical Selberg zeta function and L-functio…
View article: A dynamical approach to validated numerics
A dynamical approach to validated numerics Open
We describe a method, using periodic points and determinants, for giving alternative expressions for dynamical quantities (including Lyapunov exponents and Hausdorff dimension of invariant sets) associated to analytic hyperbolic systems. T…
View article: Anosov Flows and Dynamical Zeta Functions (Errata)
Anosov Flows and Dynamical Zeta Functions (Errata) Open
This errata fixes a mistake in the part of Giulietti, P.; Liverani, C.; Pollicott, M. Anosov flows and dynamical zeta functions. Ann. of Math. (2) {\bf 178} (2013), no. 2, 687--773, which proves a spectral gap for contact Anosov flows with…