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Oliver Jenkinson

Publications and source records attributed to Oliver Jenkinson.

15 recordsLinked to original sources
arXiv2026

Explicit exposure of Haar measure

We solve the problem of explicitly constructing a continuous function whose unique maximizing measure for the doubling map is Lebesgue measure. More generally, given a nontrivial compact metrizable abelian group and a continuous surjective endomorphism for which normalised Haar measure is ergodic, we explicitly construct a continuous function on the group for which Haar measure is the unique invariant maximizing measure. The function is the uniform limit of a recursively defined sequence of trigonometric polynomials with rational coefficients; every parameter of the recursion is given by a closed formula, every step is exact, and the rate of convergence is explicit. In specific cases, we further obtain a uniformly convergent Fourier expansion in the classical frequency order, each of whose coefficients is rational and computable exactly, by a finite procedure.

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arXiv2026

Joint typical periodic optimization: systems with stable hyperbolicity

The framework of joint typical periodic optimization, in which both the dynamical system and the potential function are allowed to vary simultaneously, was introduced in [HHJL25], in a direction motivated by the work of Yang, Hunt & Ott [YHO00]. For certain classes of hyperbolic systems, it was shown there that optimizing periodic orbits persist under simultaneous perturbation, yielding joint locking sets that contain open dense subsets of the relevant product spaces. In the present article we broaden the scope of this theory, by developing an axiomatic joint perturbation framework that accommodates a wider class of stably hyperbolic systems, and by establishing new joint typical periodic optimization results for several natural and important families: Axiom A diffeomorphisms with the no-cycle property, hyperbolic rational maps on the Riemann sphere, real quadratic polynomials, and $C^r$ maps in one dimension.

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arXiv2026

Typical periodic optimization for dynamical systems: symbolic dynamics

We develop a new theory of maximizing sets in dynamical systems, for the study of ergodic optimization in systems with weak hyperbolicity but where the Ma\~n\'e cohomology lemma does not hold. This leads to new solutions of the Typical Periodic Optimization problem in the Lipschitz category: existence of an open dense set of Lipschitz functions such that each member has a unique maximizing measure and this measure is periodic (an equi-distribution on a single periodic orbit). The theory yields a structural theorem, that isolates the part of the system responsible for any robust non-periodic optimization. The structural theorem is developed further in the setting of symbolic dynamics: given any shift space, for typical Lipschitz functions the maximizing measure is shown to be either periodic or supported on the Markov boundary of the shift space. It follows that Contreras' Typical Periodic Optimization theorem for shifts of finite type can be extended to a wide class of shift spaces, including every sofic shift. The structural theorem is used to provide the first known example of a shift space where Typical Periodic Optimization fails despite periodic measures being dense in the set of all invariant measures.

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arXiv2025

Ergodic optimization for Gauss's continued fraction map

The theory of ergodic optimization for distance-expanding maps is extended to Gauss's continued fraction map. Since the set of invariant probability measures is not weak$^*$ closed, we establish a characterisation of the closure of this set, and investigate limit-maximizing measures for H\"older continuous functions. Although a Ma\~n\'e cohomology lemma is shown to hold, the typical periodic optimization conjecture is shown to fail, as a consequence of the typical finite optimization property established for a certain class of (rationally maximized) functions. The typical periodic optimization (TPO) property is shown to hold, however, for the class of $\alpha$-H\"older essentially compact functions.

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arXiv2025

Typical Uniqueness in Ergodic Optimization

For ergodic optimization on any topological dynamical system, with real-valued potential function $f$ belonging to any separable Banach space $B$ of continuous functions, we show that the $f$-maximizing measure is typically unique, in the strong sense that a countable collection of hypersurfaces contains the exceptional set of those $f\in B$ with non-unique maximizing measure. This strengthens previous results asserting that the uniqueness set is both residual and prevalent.

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arXiv2025

Ergodic Optimization for Open Expanding Multi-valued Topological Dynamical Systems

We study the optimization of ergodic averages for multi-valued dynamical systems, i.e. where points may have multiple different forward orbits. Under upper semi-continuity assumptions, we show that the maximum space average with respect to invariant probability measures for such systems can be characterised in terms of maximum time averages on an auxiliary shift space. For all multi-valued expanding systems that are open mappings, we show that every H\"older continuous real-valued function can be modified by a coboundary, of the same H\"older exponent, such that the resulting function is dominated by its maximum ergodic average.

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arXiv2025

Joint typical periodic optimization

We prove a generalised Yuan--Hunt--Ma\~n\'e Conjecture: if $\mathcal{F}$ is the Banach space of $\alpha$-H\"older functions, and $\mathcal{T}$ is either a space of Lipschitz expanding maps, or of Anosov diffeomorphisms, or the family of beta-transformations on the interval, there is an open dense subset of $\mathcal{T}\times\mathcal{F}$ consisting of map-function pairs whose maximizing invariant measure is unique and supported on a periodic orbit.

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arXiv2022

A dynamical approach to validated numerics

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. This leads to validated numerical estimates on their values

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arXiv2020

How many inflections are there in the Lyapunov spectrum?

Iommi & Kiwi showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture of Iommi & Kiwi by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question of Iommi & Kiwi by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

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arXiv2017

Ergodic optimization in dynamical systems

Ergodic optimization is the study of problems relating to maximizing orbits, maximizing invariant measures and maximum ergodic averages. An orbit of a dynamical system is called f-maximizing if the time average of the real-valued function f along the orbit is larger than along all other orbits, and an invariant probability measure is called f-maximizing if it gives f a larger space average than does any other invariant probability measure. In this survey we consider the main strands of ergodic optimization, beginning with an influential model problem, and the interpretation of ergodic optimization as the zero temperature limit of thermodynamic formalism. We describe typical properties of maximizing measures for various spaces of functions, the key tool of adding a coboundary so as to reveal properties of these measures, as well as certain classes of functions where the maximizing measure is known to be Sturmian.

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arXiv2016

Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: a hundred decimal digits for the dimension of $E_2$

We prove that the algorithm of [13] for approximating the Hausdorff dimension of dynamically defined Cantor sets, using periodic points of the underlying dynamical system, can be used to establish completely rigorous high accuracy bounds on the dimension. The effectiveness of these rigorous estimates is illustrated for Cantor sets consisting of continued fraction expansions with restricted digits. For example the Hausdorff dimension of the set $E_2$ (of those reals whose continued fraction expansion only contains digits 1 and 2) can be rigorously approximated, with an accuracy of over 100 decimal places, using points of period up to 25. The method for establishing rigorous dimension bounds involves the holomorphic extension of mappings associated to the allowed continued fraction digits, an appropriate disc which is contracted by these mappings, and an associated transfer operator acting on the Hilbert Hardy space of analytic functions on this disc. We introduce methods for rigorously bounding the approximation numbers for the transfer operators, showing that this leads to effective estimates on the Taylor coefficients of the associated determinant, and hence to explicit bounds on the Hausdorff dimension.

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arXiv2015

Joint spectral radius, Sturmian measures, and the finiteness conjecture

The joint spectral radius of a pair of 2x2 real matrices $(A_0,A_1)\in M_2(\mathbb{R})^2$ is defined to be $r(A_0,A_1)= \limsup_{n\to\infty} \max \{\|A_{i_1}...A_{i_n}\|^{1/n}: i_j\in\{0,1\}\}$, the optimal growth rate of the norm of products of these matrices. The Lagarias-Wang finiteness conjecture, asserting that $r(A_0,A_1)$ is always the nth root of the spectral radius of some length-n product $A_{i_1}...A_{i_n}$, has been refuted by Bousch & Mairesse, with subsequent counterexamples presented by Blondel, Theys & Vladimirov; Kozyakin; Hare, Morris, Sidorov & Theys. In this article we introduce a new approach to generating finiteness counterexamples, and use this to exhibit an open subset of $M_2(\mathbb{R})^2$ with the property that each member $(A_0,A_1)$ of the subset generates uncountably many counterexamples of the form $(A_0, tA_1)$. Our methods employ ergodic theory, in particular the analysis of Sturmian invariant measures; this approach allows a short proof that the relation between the parameter $t$ and the Sturmian parameter $P(t)$ is a devil's staircase.

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arXiv2008

Explicit eigenvalue estimates for transfer operators

We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if D is any open set in C^d, and L is a suitable transfer operator acting on Bergman space A^2(D), its eigenvalue sequence lambda_n(L) is bounded by |lambda_n(L)| \leq A\exp(-a n^{1/d}), where a, A are explicitly given.

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arXiv2008

On the Ruelle eigenvalue sequence

For certain real analytic data, we show that the eigenvalue sequence of the associated transfer operator L is insensitive to the holomorphic function space on which L acts. Explicit bounds on this eigenvalue sequence are established.

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