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Anthony Quas

Publications and source records attributed to Anthony Quas.

At least 37 records · Page 2Linked to original sources
arXiv2018

Geometric random graphs and Rado sets in sequence spaces

We consider a random geometric graph model, where pairs of vertices are points in a metric space and edges are formed independently with fixed probability $p$ between pairs within threshold distance $δ$. A countable dense set in a metric space is {\sl Rado} if this random model gives, with probability 1, a graph that is unique up to isomorphism. In earlier work, the first two authors proved that in finite dimensional spaces $\mathbb{R}^n$ equipped with the $\ell_{\infty}$ norm, all countable dense set satisfying a mild non-integrality condition are Rado. In this paper, we extend this result to infinite-dimensional spaces. If the underlying metric space is a separable Banach space, then we show in some cases that we can almost surely recover the Banach space from such a geometric random graph. More precisely, we show that in the sequence spaces $c$ and $c_0$, for measures $μ$ satisfying certain conditions, $μ^\N$-almost all countable sets are Rado. Moreover, with probability 1, in $c$ as in $c_0$, all graphs obtained from the random geometric model with a randomly chosen dense countable vertex set are isomorphic to each other. Finally, we show that representatives of the isomorphism classes obtained in this way from $c$ and $c_0$ are non-isomorphic to each other, and also non-isomorphic to their counterparts obtained from finite dimensional spaces.

math.CO↗
arXiv2017

Hilbert Space Lyapunov Exponent stability

We study cocycles of compact operators acting on a separable Hilbert space, and investigate the stability of the Lyapunov exponents and Oseledets spaces when the operators are subjected to additive Gaussian noise. We show that as the noise is shrunk to 0, the Lyapunov exponents of the perturbed cocycle converge to those of the unperturbed cocycle; and the Oseledets spaces converge in probability to those of the unperturbed cocycle. This is, to our knowledge, the first result of this type with cocycles taking values in operators on infinite-dimensional spaces. The infinite dimensionality gives rise to a number of substantial difficulties that are not present in the finite-dimensional case.

math.DS↗
arXiv2017

Dynamical Analysis of a Repeated Game with Incomplete Information

We study a two player repeated zero-sum game with asymmetric information introduced by Renault in which the underlying state of the game undergoes Markov evolution (parameterized by a transition probability $\frac 12\le p\le 1$). Hörner, Rosenberg, Solan and Vieille identified an optimal strategy, $σ^*$ for the informed player for $p$ in the range $[\frac 12,\frac 23]$. We extend the range on which $σ^*$ is proved to be optimal to about $[\frac 12,0.719]$ and prove that it fails to be optimal at a value around 0.7328. Our techniques make use of tools from dynamical systems, specifically the notion of pressure, introduced by D. Ruelle.

math.OC↗
arXiv2017

On irreducibility of Oseledets subspaces

For a cocycle of invertible real $n$-by-$n$ matrices, the Multiplicative Ergodic Theorem gives an Oseledets subspace decomposition of $\mathbb{R}^n$; that is, above each point in the base space, $\mathbb{R}^n$ is written as a direct sum of equivariant subspaces, one for each Lyapunov exponent of the cocycle. It is natural to ask if these summands may be further decomposed into equivariant subspaces; that is, if the Oseledets subspaces are reducible. We prove a theorem yielding sufficient conditions for irreducibility of the trivial equivariant subspaces $\mathbb{R}^2$ and $\mathbb{C}^2$ for $O_2(\mathbb{R})$-valued cocycles and give explicit examples where the conditions are satisfied.

math.DS↗
arXiv2017

Computing automorphism groups of shifts using atypical equivalence classes

We study the automorphism group of an infinite minimal shift $(X,σ)$ such that the complexity difference function, $p(n+1)-p(n)$, is bounded. We give some new bounds on $\mbox{Aut}(X,σ)/\langle σ\rangle$ and also study the one-sided case. For a class of Toeplitz shifts, including the class of shifts defined by constant length primitive substitutions with a coincidence and with height one, we show that the two-sided automorphism group is a cyclic group. We next focus on shifts generated by primitive constant length substitutions. For these shifts, we give an algorithm that computes their two-sided automorphism group, As a corollary we describe how to compute the set of conjugacies between two such shifts.

math.DS↗
arXiv2016

Bratteli diagrams where random orders are imperfect

For the simple Bratteli diagrams B where there is a single edge connecting any two vertices in consecutive levels, we show that a random order has uncountably many infinite paths if and only if the growth rate of the level-n vertex sets is super-linear. This gives us the dichotomy: a random order on a slowly growing Bratteli diagram admits a homeomorphism, while a random order on a quickly growing Bratteli diagram does not. We also show that for a large family of infinite rank Bratteli diagrams, a random order on B does not admit a continuous Vershik map.

math.DS↗
arXiv2016

A monotone Sinai theorem

Sinai proved that a nonatomic ergodic measure-preserving system has any Bernoulli shift of no greater entropy as a factor. Given a Bernoulli shift, we show that any other Bernoulli shift that is of strictly less entropy and is stochastically dominated by the original measure can be obtained as a monotone factor; that is, the factor map has the property that for each point in the domain, its image under the factor map is coordinatewise smaller than or equal to the original point.

math.DS↗
arXiv2014

Ergodic universality of some topological dynamical systems

The Krieger generator theorem says that every invertible ergodic measure-preserving system with finite measure-theoretic entropy can be embedded into a full shift with strictly greater topological entropy. We extend Krieger's theorem to include toral automorphisms, and more generally, any topological dynamical system on a compact metric space that satisfies almost weak specification, asymptotic entropy expansiveness, and the small boundary property. As a corollary, one obtains a complete solution to a natural generalization of an open problem in Halmos's 1956 book regarding an isomorphism invariant that he proposed.

math.DS↗
arXiv2014

Rudolph's Two-Step Coding Theorem and Alpern's Lemma for R^d Actions

Rudolph showed that the orbits of any measurable, measure preserving $\mathbb R^d$ action can be measurably tiled by $2^d$ rectangles and asked if this number of tiles is optimal for $d>1$. In this paper, using a tiling of $\mathbb R^d$ by notched cubes, we show that $d+1$ tiles suffice. Furthermore, using a detailed analysis of the set of invariant measures on tilings of $\mathbb R^2$ by two rectangles, we show that while for $\mathbb R^2$ actions with completely positive entropy this bound is optimal there exist mixing $\mathbb R^2$ actions whose orbits can be tiled by 2 tiles.

math.DS↗
arXiv2012

Stability and approximation of random invariant densities for Lasota-Yorke map cocycles

We establish stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota-Yorke maps under a variety of perturbations. Our family of random maps need not be close to a fixed map; thus, our results can handle very general driving mechanisms. We consider (i) perturbations via convolutions, (ii) perturbations arising from finite-rank transfer operator approximation schemes and (iii) static perturbations, perturbing to a nearby cocycle of Lasota-Yorke maps. The former two results provide a rigorous framework for the numerical approximation of random acims using a Fourier-based approach and Ulam's method, respectively; we also demonstrate the efficacy of these schemes.

math.DS↗
arXiv2012

A semi-invertible operator Oseledets theorem

Semi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we establish a semi-invertible multiplicative ergodic theorem that for the first time can be applied to the study of transfer operators associated to the composition of piecewise expanding interval maps randomly chosen from a set of cardinality of the continuum.

math.DS↗
arXiv2012

Weak mixing suspension flows over shifts of finite type are universal

Let S be an ergodic measure-preserving automorphism on a non-atomic probability space, and let T be the time-one map of a topologically weak mixing suspension flow over an irreducible subshift of finite type under a Holder ceiling function. We show that if the measure-theoretic entropy of S is strictly less than the topological entropy of T, then there exists an embedding from the measure-preserving automorphism into the suspension flow. As a corollary of this result and the symbolic dynamics for geodesic flows on compact surfaces of negative curvature developed by Bowen and Ratner, we also obtain an embedding from the measure-preserving automorphism into a geodesic flow, whenever the measure-theoretic entropy of S is strictly less than the topological entropy of the time-one map of the geodesic flow.

math.DS↗
arXiv2011

Ergodic Optimization of Super-continuous Functions in the Shift

Ergodic Optimization is the process of finding invariant probability measures that maximize the integral of a given function. It has been conjectured that "most" functions are optimized by measures supported on a periodic orbit, and it has been proved in several separable spaces that an open and dense subset of functions is optimized by measures supported on a periodic orbit. We add to these positive results by presenting a non-separable space, the class of super-continuous functions, where the set of functions optimized by periodic orbit measures contains an open subset dense in super-continuous functions.

math.DS↗
arXiv2011

Class Degree and Relative Maximal Entropy

Given a factor code $π$ from a one-dimensional shift of finite type $X$ onto an irreducible sofic shift $Y$, if $π$ is finite-to-one there is an invariant called the degree of $π$ which is defined the number of preimages of a typical point in $Y$. We generalize the notion of the degree to the class degree which is defined for any factor code on a one-dimensional shift of finite type. Given an ergodic measure $ν$ on $Y$, we find an invariant upper bound on the number of ergodic measures on $X$ which project to $ν$ and have maximal entropy among all measures in the fibre $π^{-1}\{ν\}$. We show that this bound and the class degree of the code agree when $ν$ is ergodic and fully supported. One of the main ingredients of the proof is a uniform distribution property for ergodic measures of relative maximal entropy.

math.DS↗
arXiv2010

Entropy Rate for Hidden Markov Chains with rare transitions

We consider Hidden Markov Chains obtained by passing a Markov Chain with rare transitions through a noisy memoryless channel. We obtain asymptotic estimates for the entropy of the resulting Hidden Markov Chain as the transition rate is reduced to zero.

cs.IT↗
arXiv2010

A semi-invertible Oseledets Theorem with applications to transfer operator cocycles

Oseledets' celebrated Multiplicative Ergodic Theorem (MET) is concerned with the exponential growth rates of vectors under the action of a linear cocycle on R^d. When the linear actions are invertible, the MET guarantees an almost-everywhere pointwise splitting of R^d into subspaces of distinct exponential growth rates (called Lyapunov exponents). When the linear actions are non-invertible, Oseledets' MET only yields the existence of a filtration of subspaces, the elements of which contain all vectors that grow no faster than exponential rates given by the Lyapunov exponents. The authors recently demonstrated that a splitting over R^d is guaranteed even without the invertibility assumption on the linear actions. Motivated by applications of the MET to cocycles of (non-invertible) transfer operators arising from random dynamical systems, we demonstrate the existence of an Oseledets splitting for cocycles of quasi-compact non-invertible linear operators on Banach spaces.

math.DS↗