arXiv subjects

Ian D. Morris

Publications and source records attributed to Ian D. Morris.

At least 19 recordsLinked to original sources

arXiv2026

A Poincaré-Bendixson theorem for Bebutov shifts and applications to switched systems

We prove a version of the Poincaré-Bendixson theorem for certain classes of curves on the 2-sphere which are not required to be the trajectories of an underlying flow or semiflow on the sphere itself. Using this result we extend the Poincaré-Bendixson theorem to the context of continuous semiflows on compact subsets of the 2-sphere and the projective plane, give new sufficient conditions for the existence of periodic trajectories of certain low-dimensional affine control systems, and give a new criterion for the global uniform exponential stability of switched systems of homogeneous ODEs in dimension three. We prove in particular that periodic asymptotic stability implies global uniform exponential stability for real linear switched systems of dimension three and complex linear switched systems of dimension two. In combination with a recent result of the second author, this resolves a question of R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King and resolves a natural analogue of the Lagarias-Wang finiteness conjecture in continuous time.

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arXiv2025

Stability with respect to periodic switching laws does not imply global stability under arbitrary switching

R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King have asked whether a linear switched system is guaranteed to be globally uniformly stable under arbitrary switching if it is known that every trajectory induced by a periodic switching law converges exponentially to the origin. Positive answers to this question have previously been announced for linear switched systems of order two and three. We answer this question negatively in all higher orders by constructing a fourth-order linear switched system with four switching states which is not uniformly exponentially stable but which has the property that every trajectory defined by a periodic switching law converges exponentially to the origin. We argue informally that positive linear systems with this combination of properties are likely to exist in sufficiently high dimensions.

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arXiv2025

General real measurable Livšic regularity via transfer operators

We prove a general measurable Livšic regularity theorem for real-valued cocycles over non-invertible dynamical systems using only abstract hypotheses on an associated transfer operator. As illustrative applications we derive measurable Livšic regularity results in the analytic regularity class for cocycles over real-analytic expanding maps, in the bounded-variation regularity class for $β$-transformations, and in $C^α$ regularity over the class of virtually expanding maps recently introduced by M. Tsujii.

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arXiv2025

Exceptional projections of self-affine sets: an introduction

We describe some recent results on the dimensions of linear projections of self-affine fractals, focusing in particular on an upper bound for the dimension of the projected image. We give a self-contained treatment of this bound and illustrate it through explicit examples, in the process exhibiting some smooth submanifolds of the Grassmannian which can be contained in the exceptional set in Marstrand's theorem.

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arXiv2025

Thermodynamic formalism of countably generated self-affine sets

In this article, we further develop the thermodynamic formalism of affine iterated function systems with countably many transformations by showing the existence and extending earlier characterisations of the equilibrium states of finite affine iterated function systems to the countably infinite case. As an application, under mild conditions, we prove that the affinity dimension of a countable affine iterated function system is equal to the supremum of the affinity dimensions of its finite subsystems. We deduce corollaries concerning the Hausdorff dimension of countably generated self-affine sets in dimensions $1$, $2$, and $3$ satisfying mild deterministic assumptions and in arbitrary dimension with generic translations.

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arXiv2023

A variational principle relating self-affine measures to self-affine sets

A breakthrough result of Bárány, Hochman and Rapaport published in 2019 established that every self-affine measure on $\mathbb{R}^2$ satisfying certain mild non-degeneracy conditions has Hausdorff dimension equal to its Lyapunov dimension. In combination with a variational principle established earlier by Morris and Shmerkin this result implied as a corollary that the attractor of a planar affine iterated function system satisfying the same conditions necessarily has Hausdorff dimension equal to a value proposed by Falconer in 1988. In this article we extend the variational principle of Morris and Shmerkin from the planar context to the case of affine iterated function systems acting on $\mathbb{R}^d$. This allows a recent theorem of Rapaport on the dimensions of self-affine measures in $\mathbb{R}^3$ to be extended into a characterisation of the dimensions of the corresponding self-affine subsets of $\mathbb{R}^3$. At the core of the present work is an algebraic result concerned with finding large Zariski-dense Schottky semigroups inside a given finitely generated completely reducible semigroup of linear transformations.

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arXiv2023

An irreducible linear switching system whose unique Barabanov norm is not strictly convex

We construct a marginally stable linear switching system in continuous time, in four dimensions and with three switching states, which is exponentially stable with respect to constant switching laws and which has a unique Barabanov norm, but such that the Barabanov norm fails to be strictly convex. This resolves a question of Y. Chitour, M. Gaye and P. Mason.

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arXiv2023

A stability dichotomy for discrete-time linear switching systems in dimension two

We prove that for every discrete-time linear switching system in two complex variables and with finitely many switching states, either the system is Lyapunov stable or there exists a trajectory which escapes to infinity with at least linear speed. We also give a checkable algebraic criterion to distinguish these two cases. This dichotomy was previously known to hold for systems in two real variables, but is known to be false in higher dimensions and for systems with infinitely many switching states.

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arXiv2023

A note on the marginal instability rates of two-dimensional linear cocycles

A theorem of Guglielmi and Zennaro implies that if the uniform norm growth of a locally constant GL(2,R)-cocycle on the full shift is not exponential then it must be either bounded or linear, with no other possibilities occurring. We give an alternative proof of this result and demonstrate that its conclusions do not hold for Lipschitz continuous cocycles over the full shift on two symbols.

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arXiv2022

On marginal growth rates of matrix products

In this article we consider the maximum possible growth rate of sequences of long products of $d \times d$ matrices all of which are drawn from some specified compact set which has been normalised so as to have joint spectral radius equal to $1$. We define the marginal instability rate sequence associated to such a set to be the sequence of real numbers whose $n^{th}$ entry is the norm of the largest product of length $n$, and study the general properties of sequences of this form. We describe how new marginal instability rate sequences can be constructed from old ones, extend an earlier example of Protasov and Jungers to obtain marginal instability rate sequences whose limit superior rate of growth matches various non-integer powers of $n$, and investigate the relationship between marginal instability rate sequences arising from finite sets of matrices and those arising from sets of matrices with cardinality $2$. We also give the first example of a finite set whose marginal instability rate sequence is asymptotically similar to a polynomial with non-integer exponent. Previous examples had this property only along a subsequence.

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arXiv2022

Marginally unstable discrete-time linear switched systems with highly irregular trajectory growth

We investigate the uniform stability properties of discrete-time linear switched systems subject to arbitrary switching, focusing on the "marginally unstable" regime in which the system is not Lyapunov stable but in which trajectories cannot escape to infinity at exponential speed. For a discrete linear system of this type without switching the fastest-growing trajectory must grow as an exact polynomial function of time, and a significant body of prior research has focused on investigating how far this intuitive picture can be extended from systems without switching to cases where switching is present. In this note we give an example of a family of discrete linear switched systems in three dimensions, with two switching states, for which this intuition fails badly: for a generic member of this family the maximal rate of uniform growth of escaping trajectories can be made arbitrarily slow along one subsequence of times and yet also faster than any prescribed slower-than-linear function along a complementary subsequence of times. Using this construction we give new counterexamples to a conjecture of Chitour, Mason and Sigalotti and obtain a negative answer to a related question of Jungers, Protasov and Blondel. Our examples have the additional feature that marginal stability and marginal instability are densely intermingled in the same parameter space.

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arXiv2022

On affine iterated function systems which robustly admit an invariant affine subspace

In this note we give a simple sufficient condition for an affine iterated function system to admit an invariant affine subspace persistently with respect to changes in the translation parameters. This yields further examples of tuples of contracting linear maps which do not satisfy the conclusions of Falconer's theorem on the Hausdorff dimension of almost every self-affine set. We also obtain new examples of iterated function systems of similarity transformations which cannot satisfy the open set condition for any choice of translation parameters, and resolve a related question of Peres and Solomyak.

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arXiv2022

On dense intermingling of exact overlaps and the open set condition

We prove that certain families of homogenous affine iterated function systems in $\mathbb{R}^d$ have the property that the open set condition and the existence of exact overlaps both occur densely in the space of translation parameters. These examples demonstrate that in the theorems of Falconer and Jordan-Pollicott-Simon on the almost sure dimensions of self-affine sets and measures, the set of exceptional translation parameters can be a dense set. The proof combines results from the literature on self-affine tilings of $\mathbb{R}^d$ with an adaptation of a classic argument of Erdős on the singularity of certain Bernoulli convolutions. Our result encompasses a one-dimensional example due to Kenyon which arises as a special case.

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arXiv2021

Fast approximation of the affinity dimension for dominated affine iterated function systems

In 1988 K. Falconer introduced a formula which predicts the value of the Hausdorff dimension of the attractor of an affine iterated function system. The value given by this formula -- sometimes referred to as the affinity dimension -- is known to agree with the Hausdorff dimension both generically and in an increasing range of explicit cases. It is however a nontrivial problem to estimate the numerical value of the affinity dimension for specific iterated function systems. In this article we substantially extend an earlier result of M. Pollicott and P. Vytnova on the computation of the affinity dimension. Pollicott and Vytnova's work applies to planar invertible affine contractions with positive linear parts under several additional conditions which among other things constrain the affinity dimension to be between 0 and 1. We extend this result by passing from planar self-affine sets to self-affine sets in arbitrary dimensions, relaxing the positivity hypothesis to a domination condition, and removing all other constraints including that on the range of values of the affinity dimension. We provide some explicit examples of two- and three-dimensional affine iterated function systems for which the affinity dimension can be calculated to more than 30 decimal places.

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arXiv2021

A converse statement to Hutchinson's theorem and a dimension gap for self-affine measures

A well-known theorem of J.E. Hutchinson states that if an iterated function system consists of similarity transformations and satisfies the open set condition then its attractor supports a self-similar measure with Hausdorff dimension equal to the similarity dimension. In this article we prove the following result which may be regarded as a form of partial converse: if an iterated function system consists of invertible affine transformations whose linear parts do not preserve a common invariant subspace, and its attractor supports a self-affine measure with Hausdorff dimension equal to the affinity dimension, then the system necessarily consists of similarity transformations. We obtain this result by showing that the equilibrium measures of an affine iterated function system are never Bernoulli measures unless the system either is reducible or consists of similarity transformations. The proof builds on earlier results in the thermodynamic formalism of affine iterated function systems due to Bochi, Feng, Käenmäki, Shmerkin and the first named author and also relies on the work of Benoist on the spectral properties of Zariski-dense subsemigroups of reductive linear groups.

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arXiv2020

Totally ergodic generalised matrix equilibrium states have the Bernoulli property

We show that every totally ergodic generalised matrix equilibrium state is psi-mixing with respect to the natural partition into cylinders and hence is measurably isomorphic to a Bernoulli shift in its natural extension. This implies that the natural extensions of ergodic generalised matrix equilibrium states are measurably isomorphic to Bernoulli processes extended by finite rotations. This resolves a question of Gatzouras and Peres in the special case of self-affine repelling sets with generic translations.

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arXiv2020

How long is the Chaos Game?

In the 1988 textbook "Fractals Everywhere" M. Barnsley introduced an algorithm for generating fractals through a random procedure which he called the "chaos game". Using ideas from the classical theory of covering times of Markov chains we prove an asymptotic formula for the expected time taken by this procedure to generate a $δ$-dense subset of a given self-similar fractal satisfying the open set condition.

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arXiv2020

Prevalent uniqueness in ergodic optimisation

One of the fundamental results of ergodic optimisation asserts that for any dynamical system on a compact metric space $X$ and for any Banach space of continuous real-valued functions on $X$ which embeds densely in $C(X)$ there exists a residual set of functions in that Banach space for which the maximising measure is unique. We extend this result by showing that this residual set is additionally prevalent, answering a question of J. Bochi and Y. Zhang.

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