Searcharxiv⌕ Search

arXiv subjects

Darren Creutz

Publications and source records attributed to Darren Creutz.

15 recordsLinked to original sources
arXiv2023

On minimal subshifts of linear word complexity with slope less than 3/2

We prove that every infinite minimal subshift with word complexity $p(q)$ satisfying $\limsup p(q)/q < 3/2$ is measure-theoretically isomorphic to its maximal equicontinuous factor; in particular, it has measurably discrete spectrum. Among other applications, this provides a proof of Sarnak's conjecture for all subshifts with $\limsup p(q)/q < 3/2$ (which can be thought of as a much stronger version of zero entropy). As in \cite{creutzpavlov}, our main technique is proving that all low-complexity minimal subshifts have a specific type of representation via a sequence $\{τ_k\}$ of substitutions, usually called an S-adic decomposition. The maximal equicontinuous factor is the product of an odometer with a rotation on a compact abelian connected one-dimensional group, for which we can give an explicit description in terms of the substitutions $τ_k$. We also prove that all such odometers and groups may appear for minimal subshifts with $\limsup p(q)/q = 1$, demonstrating that lower complexity thresholds do not further restrict the possible structure.

math.DS↗
arXiv2023

Low Complexity Subshifts have Discrete Spectrum

We prove results about subshifts with linear (word) complexity, meaning that $\limsup \frac{p(n)}{n} < \infty$, where for every $n$, $p(n)$ is the number of $n$-letter words appearing in sequences in the subshift. Denoting this limsup by $C$, we show that when $C < \frac{4}{3}$, the subshift has discrete spectrum, i.e. is measurably isomorphic to a rotation of a compact abelian group with Haar measure. We also give an example with $C = \frac{3}{2}$ which has a weak mixing measure. This partially answers an open question of Ferenczi, who asked whether $C = \frac{5}{3}$ was the minimum possible among such subshifts; our results show that the infimum in fact lies in $[\frac{4}{3}, \frac{3}{2}]$. All results are consequences of a general S-adic/substitutive structure proved when $C < \frac{4}{3}$.

math.DS↗
arXiv2023

Word Complexity of (Measure-Theoretically) Weakly Mixing Rank-One Subshifts

We exhibit subshifts admitting weakly mixing (probability) measures, for arbitrary $ε> 0$, with word complexity $p$ satisfying $\limsup \frac{p(q)}{q} < 1.5 + ε$. For arbitrary $f(q) \to \infty$, said subshifts can be made to satisfy $p(q) < q + f(q)$ infinitely often. We establish that every subshift associated to a rank-one transformation (on a probability space) which is not an odometer satisfies $\limsup p(q) - 1.5q = \infty$ and that this is optimal for rank-ones.

math.DS↗
arXiv2022

Measure-Theoretically Mixing Subshifts of Minimal Word Complexity

We resolve a long-standing open question on the relationship between measure-theoretic dynamical complexity and symbolic complexity by establishing the exact word complexity at which measure-theoretic strong mixing manifests: For every superlinear $f : \mathbb{N} \to \mathbb{N}$, i.e. $f(q)/q \to \infty$, there exists a subshift admitting a (strongly) mixing of all orders probability measure with word complexity $p$ such that $p(q)/f(q) \to 0$. For a subshift with word complexity $p$ which is non-superlinear, i.e. $\liminf p(q)/q < \infty$, every ergodic probability measure is partially rigid.

math.DS↗
arXiv2022

Measure-Theoretically Mixing Subshifts with Low Complexity

We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any $f : \mathbb{N} \to \mathbb{N}$ with $f(n)/n$ increasing and $\sum 1/f(n) < \infty$, that there exists an extremely elevated staircase with word complexity $p(n) = o(f(n))$. This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.

math.DS↗
arXiv2022

Character rigidity for lattices and commensurators

We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, in certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.

math.OA↗
arXiv2022

Harmonic Functions on Compactly Generated Groups

A compactly generated group is noncompact if and only if it admits a nonconstant harmonic function (for some, equivalently for every, reasonable measure). This generalizes the known fact that a finitely generated group is infinite if and only if it admits a nonconstant harmonic function (for some, equivalently every, reasonable measure).

math.GR↗
arXiv2021

Mixing on Stochastic Staircase Transformations

We prove mixing on a general class of rank-one transformations containing all known examples of rank-one mixing, including staircase transformations and Ornstein's constructions, and a variety of new constructions.

math.DS↗
arXiv2016

Contractive Spaces and Relatively Contractive Maps

We present an exposition of contractive spaces and of relatively contractive maps. Contractive spaces are the natural opposite of measure-preserving actions and relatively contractive maps the natural opposite of relatively measure-preserving maps. These concepts play a central role in the work of the author and J.~Peterson on the rigidity of actions of semisimple groups and their lattices and have also appeared in recent work of various other authors. We present detailed definitions and explore the relationship of these phenomena with other aspects of the ergodic theory of group actions, proving along the way several new results, with an eye towards explaining how contractiveness is intimately connected with rigidity phenomena.

math.DS↗
arXiv2015

Stabilizers of Actions of Lattices in Products of Groups

We prove that any ergodic nonatomic probability-preserving action of an irreducible lattice in a semisimple group, at least one factor being connected and higher-rank, is essentially free. This generalizes the result of Stuck and Zimmer that the same statement holds when the ambient group is a semisimple real Lie group and every simple factor is higher-rank. We also prove a generalization of a result of Bader and Shalom by showing that any probability-preserving action of a product of simple groups, at least one with property (T), which is ergodic for each simple subgroup is either essentially free or essentially transitive. Our method involves the study of relatively contractive maps and the Howe-Moore property, rather than the relaying on algebraic properties of semisimple groups and Poisson boundaries, and introduces a generalization of the ergodic decomposition to invariant random subgroups of independent interest.

math.DS↗
arXiv2015

Stabilizers of Ergodic Actions of Lattices and Commensurators

We prove that any ergodic measure-preserving action of an irreducible lattice in a semisimple group, with finite center and each simple factor having rank at least two, either has finite orbits or has finite stabilizers. The same dichotomy holds for many commensurators of such lattices. The above are derived from more general results on groups with the Howe-Moore property and property $(T)$. We prove similar results for commensurators in such groups and for irreducible lattices (and commensurators) in products of at least two such groups, at least one of which is totally disconnected.

math.DS↗
arXiv2014

A Normal Subgroup Theorem for Commensurators of Lattices

We establish a general normal subgroup theorem for commensurators of lattices in locally compact groups. While the statement is completely elementary, its proof, which rests on the original strategy of Margulis in the case of higher rank lattices, relies heavily on analytic tools pertaining to amenability and Kazhdan's property (T). It is a counterpart to the normal subgroup theorem for irreducible lattices of Bader and the second named author, and may also be used to sharpen that result when one of the ambient factors is totally disconnected.

math.GR↗
arXiv2006

Mixing on Rank-One Transformations

We prove that mixing on rank-one transformations is equivalent to the spacer sequence being slice-ergodic. Slice-ergodicity, introduced in this paper, generalizes the notion of ergodic sequence to the uniform convergence of ergodic averages (as in the mean ergodic theorem) over subsequences of partial sums. We show that polynomial staircase transformations satisfy this condition and therefore are mixing.

math.DS↗
arXiv2001

Mixing on a class of rank one transformations

We prove that a rank one transformation satisfying a condition called restricted growth is a mixing transformation if and only if the spacer sequence for the transformation is uniformly ergodic. Uniform ergodicity is a generalization of the notion of ergodicity for sequences in the sense that the mean ergodic theorem holds for what we call dynamical sequences. In particular, Adams' class of staircase transformations and Ornstein's class constructed using ``random spacers'' have both restricted growth and uniformly ergodic spacer sequences.

math.DS↗