arXiv subjects

Scott Schmieding

Publications and source records attributed to Scott Schmieding.

At least 19 recordsLinked to original sources

arXiv2026

Isogenies of minimal Cantor systems: from Sturmian to Denjoy and interval exchanges

This work is motivated by the study of continued fraction expansions of real numbers: we describe in dynamical terms their orbits under the action of $\mathrm{PGL}_2(\mathbb{Q})$. A real number gives rise to a Sturmian system encoding a rotation of the circle. It is well known that $\mathrm{PGL}_2(\mathbb{Z})$-equivalence of real numbers, characterized by the tails of their continued fraction expansions, amounts to flow equivalence of Sturmian systems. We show that the multiplicative action of $m\in \mathbb{Z}$ on a real number corresponds to taking the $m$th-power followed by what we call an infinitesimal 2-asymptotic factor of its Sturmian system. This leads us to introduce the notion of isogeny between zero-dimensional systems: it combines virtual flow equivalences and infinitesimal asymptotic equivalences. We develop tools for classifying systems up to isogeny involving cohomological invariants and states. We then use this to give a complete description of $\mathrm{PSL}_2(\mathbb{Q})$-equivalence of real numbers in terms of Sturmian systems. We classify Denjoy systems up to isogenies within this class via the action of $\mathrm{PGL}_{2}(\mathbb{Q})$ on their invariants. We also investigate eventual flow equivalence of Sturmian systems: we show that for non-quadratic parameters it amounts to topological conjugacy and for quadratic parameters it implies total flow equivalence and other arithmetic constraints. In another direction, we consider interval exchanges satisfying Keane's condition. We characterize flow equivalence in terms of interval-induced subsystems (or the tails of their paths in the bilateral Rauzy induction diagram). Finally we find rational invariants for isogeny involving the length modules and SAF invariants of the associated ergodic measures. This leads to a conjecture for their classification up to isogeny, which we prove in the totally ergodic case.

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arXiv2026

Invariant random compacts

For a compact metric space $X$ with a group $G$ acting on it continuously, an invariant random compact is a Borel probability measure on the space of nonempty compact subsets of $X$ that is invariant under the action of $G$. The action is IC-rigid if, with respect to every invariant random compact, every compact set is almost surely either finite or $X$. We give sufficient conditions for an action to be IC-rigid, and show there are natural examples of such actions. We further consider a notion of weak IC-rigidity, and prove that the Chacon system is weakly IC-rigid but not IC-rigid. As an application, we prove results concerning multiplicative largeness of dilations of sets on the circle.

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arXiv2025

Finitary Ryan's and local $\mathcal{Q}$ entropy for $\mathbb{Z}^{d}$ subshifts

For the action of a group $G$ by homeomorphisms on a space $X$, the automorphism group $\mathrm{Aut}(X,G)$ consists of all self-homeomorphisms of $X$ which commute with $x \mapsto g \cdot x$ for every $g \in G$. A theorem of Ryan shows that for an irreducible $\mathbb{Z}$-shift of finite type $(X,σ_{X})$, the center of $\mathrm{Aut}(X,σ_{X})$ is generated by the shift $σ_{X}$. A finitary version of this for $\mathbb{Z}$-shifts of finite type was proved by the second author for certain full shifts, and later generalized by Kopra to irreducible $\mathbb{Z}$-shifts of finite type. We generalize these finitary Ryan's theorems to shifts of finite type over more general groups. We prove that for contractible $\mathbb{Z}^{d}$-shifts of finite type with a fixed point, there is a finitely generated subgroup of the automorphism group whose centralizer in the group of homeomorphisms is the subgroup of shifts. We also prove versions of this for full shifts over any infinite, finitely generated group on sufficiently nice alphabet sizes. The stabilized automorphism group $\mathrm{Aut}^{(\infty)}(X,G)$ is the union of $\mathrm{Aut}(X,H)$ over all finite index subgroups $H \subset G$. Aimed at studying stabilized automorphism groups for shifts of finite type, we introduce an entropy-like quantity for pointed groups which we call local $\mathcal{Q}$ entropy, a generalization of a notion called local $\mathcal{P}$ entropy previously introduced by the first author. Using the finitary Ryan's theorems, we prove that the local $\mathcal{Q}$ entropy of the stabilized automorphism group of a contractible $\mathbb{Z}^{d}$-shift of finite type recovers the topological entropy of the underlying shift system up to a rational multiple. We then use this to give a complete classification up to isomorphism of the stabilized automorphism groups of full shifts over $\mathbb{Z}^{d}$.

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arXiv2024

Hyperspatiality for isomorphisms of stabilized automorphism groups of shifts of finite type

Given a homeomorphism $T \colon X \to X$ of a compact metric space $X$, the stabilized automorphism group $\textrm{Aut}^{\infty}(T)$ of the system $(X,T)$ is the group of self-homeomorphisms of $X$ which commute with some power of $T$. We study the question of spatiality for stabilized automorphism groups of shifts of finite type. We prove that any isomorphism $Ψ\colon \textrm{Aut}^{\infty}(σ_{m}) \to \textrm{Aut}^{\infty}(σ_{n})$ between stabilized automorphism groups of full shifts is spatially induced by a homeomorphism $\hatΨ$ between respective stabilized spaces of chain recurrent subshifts. This spatialization in particular gives a bijection between the sets of periodic points which intertwines some powers of the shifts, and this bijection recovers the isomorphism at the level of the faithful actions on the sets of periodic points. We also prove that the outer automorphism group of $\textrm{Aut}^{\infty}(σ_{n})$ is uncountable, and deduce several other properties of $\textrm{Aut}^{\infty}(σ_{n})$ using the spatiality results.

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arXiv2024

Chaotic almost minimal actions

Motivated by Furstenberg's Theorem on sets in the circle invariant under multiplication by a non-lacunary semigroup, we define a general class of dynamical systems possessing similar topological dynamical properties. We call such systems chaotic almost minimal, reflecting that these systems are chaotic, but in some sense are close to minimal. We study properties of the acting group needed to admit such an action, and show the existence of a chaotic almost minimal $\mathbb{Z}$-action. We show there exists chaotic almost minimal $\mathbb{Z}^{d}$-actions which support multiple distinct nonatomic ergodic probability measures.

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arXiv2024

Geometry and Transcendence of the Hexponential

The modular group $\operatorname{PSL}_2(\mathbb{Z})$ acts on the upper-half plane $\mathbb{HP}$ with quotient the modular orbifold, uniformized by the function $\mathfrak{j} \colon \mathbb{HP}\to \mathbb{C}$. We first show that second derived subgroup $\operatorname{PSL}_2(\mathbb{Z})''$ corresponds to a $\mathbb{Z}^2\rtimes \mathbb{Z}/6$ Galois cover of the modular orbifold by a hexpunctured plane, uniformized by the hexponential map $\operatorname{hexp} \colon \mathbb{HP} \to \mathbb{C} \setminus (ω_0\mathbb{Z}[j])$, which is a primitive of $Cη^4$ where $ω_0\in i\mathbb{R}$ and $C\in \mathbb{R}$ are explicit constants and $η$ is Dedekind eta function. We describe the values of the cusp-compactification $\partial \operatorname{hexp}\colon \mathbb{QP}^1\to ω_0 \mathbb{Z}[j]$. After defining the radial-compactification $\operatorname{Shexp} \colon \mathscr{R} \to \mathbb{R}/(2π\mathbb{Z})$, we construct a simple section $\operatorname{InSh} \colon \mathbb{R}/(2π\mathbb{Z}) \to \mathscr{S} \bmod{\operatorname{PSL}_2(\mathbb{Z})'}$ where $\mathscr{S} \subset \mathbb{RP}^1$ is a set of numbers whose continued fraction expansions arise from Sturmian sequences, which contains the set $\mathscr{M}$ of Markov quadratic irrationals as those numbers arising from periodic Sturmian sequences. We will show that the values of $\operatorname{InSh}$ are either Markov quadratic irrationals or transcendental. Finally we provide a continued fraction expansion for $\operatorname{hexp}$, and discuss its monodromy.

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arXiv2023

Symbolic dynamics and the stable algebra of matrices

We give an introduction to the "stable algebra of matrices" as related to certain problems in symbolic dynamics. We consider this stable algebra (especially, shift equivalence and strong shift equivalence) for matrices over general rings as well as various specific rings. This algebra is of independent interest and can be followed with little attention to the symbolic dynamics. We include strong connectionsto algebraic K-theory and the inverse spectral problem for nonnegative matrices. We also review key features of the automorphism group of a shift of finite type, and the work of Kim, Roush and Wagoner giving counterexamples to Williams' Shift Equivalence Conjecture.

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arXiv2022

On the structure of generic subshifts

We investigate generic properties (i.e. properties corresponding to residual sets) in the space of subshifts with the Hausdorff metric. Our results deal with four spaces: the space $\mathbf{S}$ of all subshifts, the space $\mathbf{S}^{\prime}$ of non-isolated subshifts, the closure $\overline{\mathbf{T}^{\prime}}$ of the infinite transitive subshifts, and the closure $\overline{\mathbf{T}\mathbf{T}^{\prime}}$ of the infinite totally transitive subshifts. In the first two settings, we prove that generic subshifts are fairly degenerate; for instance, all points in a generic subshift are biasymptotic to periodic orbits. In contrast, generic subshifts in the latter two spaces possess more interesting dynamical behavior. Notably, generic subshifts in both $\overline{\mathbf{T}^{\prime}}$ and $\overline{\mathbf{T}\mathbf{T}^{\prime}}$ are zero entropy, minimal, uniquely ergodic, and have word complexity which realizes any possible subexponential growth rate along a subsequence. In addition, a generic subshift in $\overline{\mathbf{T}^{\prime}}$ is a regular Toeplitz subshift which is strongly orbit equivalent to the universal odometer.

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arXiv2021

Local finiteness and automorphism groups of low complexity subshifts

We prove that for any transitive subshift $X$ with word complexity function $c_n(X)$, if $\liminf \frac{\log (c_n(X)/n)}{\log \log \log n} = 0$, then the quotient group $\textrm{Aut}(X,σ) / \langle σ\rangle$ of the automorphism group of $X$ by the subgroup generated by the shift $σ$ is locally finite. We prove that significantly weaker upper bounds on $c_n(X)$ imply the same conclusion if the Gap Conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of $X$ of range $n$ in terms of word complexity, which may be of independent interest. As an application, we are also able to prove that for any subshift $X$, if $\frac{c_n(X)}{n^2 (\log n)^{-1}} \rightarrow 0$, then $\textrm{Aut}(X,σ)$ is amenable, improving a result of Cyr and Kra. In the opposite direction, we show that for any countable infinite locally finite group $G$ and any unbounded increasing $f: \mathbb{N} \rightarrow \mathbb{N}$, there exists a minimal subshift $X$ with $\textrm{Aut}(X,σ) / \langle σ\rangle$ isomorphic to $G$ and $\frac{c_n(X)}{nf(n)} \rightarrow 0$.

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arXiv2020

Local $\mathcal{P}$ entropy and stabilized automorphism groups of subshifts

For a homeomorphism $T \colon X \to X$ of a compact metric space $X$, the stabilized automorphism group $\text{Aut}^{(\infty)}(T)$ consists of all self-homeomorphisms of $X$ which commute with some power of $T$. Motivated by the study of these groups in the context of shifts of finite type, we introduce a certain entropy for groups called local $\mathcal{P}$ entropy. We show that when $(X,T)$ is a non-trivial mixing shift of finite type, the local $\mathcal{P}$ entropy of the group $\text{Aut}^{(\infty)}(T)$ is determined by the topological entropy of $(X,T)$. We use this to give a complete classification of the isomorphism type of the stabilized automorphism groups of full shifts.

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arXiv2020

The stabilized automorphism group of a subshift

For a mixing shift of finite type, the associated automorphism group has a rich algebraic structure, and yet we have few criteria to distinguish when two such groups are isomorphic. We introduce a stabilization of the automorphism group, study its algebraic properties, and use them to distinguish many of the stabilized automorphism groups. We also show that for a full shift, the subgroup of the stabilized automorphism group generated by elements of finite order is simple, and that the stabilized automorphism group is an extension of a free abelian group of finite rank by this simple group.

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arXiv2019

Random Substitution Tilings and Deviation Phenomena

Suppose a set of prototiles allows $N$ different substitution rules. In this paper we study tilings of $\mathbb{R}^d$ constructed from random application of the substitution rules. The space of all possible tilings obtained from all possible combinations of these substitutions is the union of all possible tilings spaces coming from these substitutions and has the structure of a Cantor set. The renormalization cocycle on the cohomology bundle over this space determines the statistical properties of the tilings through its Lyapunov spectrum by controlling the deviation of ergodic averages of the $\mathbb{R}^d$ action on the tiling spaces.

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arXiv2018

The Mapping Class Group of a Minimal Subshift

For a homeomorphism $T \colon X \to X$ of a Cantor set $X$, the mapping class group $\mathcal{M}(T)$ is the group of isotopy classes of orientation-preserving self-homeomorphisms of the suspension $Σ_{T}X$. The group $\mathcal{M}(T)$ can be interpreted as the symmetry group of the system $(X,T)$ with respect to the flow equivalence relation. We study $\mathcal{M}(T)$, focusing on the case when $(X,T)$ is a minimal subshift. We show that when $(X,T)$ is a subshift associated to a substitution, the group $\mathcal{M}(T)$ is an extension of $\mathbb{Z}$ by a finite group; for a large class of substitutions including Pisot type, this finite group is a quotient of the automorphism group of $(X,T)$. When $(X,T)$ is a minimal subshift of linear complexity satisfying a no-infinitesimals condition, we show that $\mathcal{M}(T)$ is virtually abelian. We also show that when $(X,T)$ is minimal, $\mathcal{M}(T)$ embeds into the Picard group of the crossed product algebra $C(X) \rtimes_{T} \mathbb{Z}$.

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arXiv2018

Traces of random operators associated with self-affine Delone sets and Shubin's formula

We study operators defined on a Hilbert space defined by a self-affine Delone set $Λ$ and show that the usual trace of a restriction of the operator to finite-dimensional subspaces satisfies a certain $\limsup$ law controlled by traces on a certain subalgebra. The asymptotic traces are defined through asymptotic cycles, or $\mathbb{R}^d$-invariant distributions of a dynamical system defined by $Λ$. We use this to refine Shubin's trace formula for self-adjoint operators and show that the errors of convergence in Shubin's formula are given by these traces.

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arXiv2018

Automorphisms of the shift: Lyapunov exponents, entropy, and the dimension representation

Let $(X_{A},σ_{A})$ be a shift of finite type and $\text{Aut}(σ_{A})$ its corresponding automorphism group. Associated to $ϕ\in \text{Aut}(σ_{A})$ are certain Lyapunov exponents $α^{-}(ϕ), α^{+}(ϕ)$ which describe asymptotic behavior of the sequence of coding ranges of $ϕ^{n}$. We give lower bounds on $α^{-}(ϕ), α^{+}(ϕ)$ in terms of the spectral radius of the corresponding action of $ϕ$ on the dimension group associated to $(X_{A},σ_{A})$. We also give lower bounds on the topological entropy $h_{top}(ϕ)$ in terms of a distinguished part of the spectrum of the action of $ϕ$ on the dimension group, but show that in general $h_{top}(ϕ)$ is not bounded below by the logarithm of the spectral radius of the action of $ϕ$ on the dimension group.

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arXiv2017

Self affine Delone sets and deviation phenomena

We study the growth of norms of ergodic integrals for the translation action on spaces coming from expansive, self-affine Delone sets. The linear map giving the self-affinity induces a renormalization map on the pattern space and we show that the rate of growth of ergodic integrals is controlled by the induced action of the renormalizing map on the cohomology of the pattern space up to boundary errors. We explore the consequences for the diffraction of such Delone sets, and explore in detail what the picture is for substitution tilings as well as for cut and project sets which are self-affine. We also explicitly compute some examples.

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arXiv2016

Strong shift equivalence and the generalized spectral conjecture for nonnegative matrices

We show that the weak and strong forms of the Generalized Spectral Conjecture (GSC) of Boyle and Handelman are equivalent. The GSC asserts that well understood necessary spectral conditions on a square matrix A over a subring S of the reals are sufficient for that matrix to be shift equivalent over S (in the weak form) or strong shift equivalent over S (in the strong form) to a primitive matrix over S. The foundation of this work is the recent result that the group NK_1(S) of algebraic K-theory exactly captures the refinement of shift equivalence over S by strong shift equivalence over S. The GSC remains open in general even in the case that S equals the real numbers.

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arXiv2016

Finite group extensions of shifts of finite type: K-theory, Parry and Livšic

This paper extends and applies algebraic invariants and constructions for mixing finite group extensions of shifts of finite type. For a finite abelian group G, Parry showed how to define a G-extension S_A from a square matrix A over Z_+G, and classified the extensions up to topological conjugacy by the strong shift equivalence class of A over Z_+G. Parry asked in this case if the det(I-tA) (which captures the "periodic data" of the extension) would classify up to finitely many topological conjugacy classes the extensions by G of a fixed mixing shift of finite type. When the algebraic K-theory group NK_1(ZG) is nontrivial (e.g., for G=Z/4), we show the dynamical zeta function for any such extension is consistent with infinitely many topological conjugacy classes. Independent of NK_1(ZG): for every nontrivial abelian G we show there exists a shift of finite type with an infinite family of mixing nonconjugate G extensions with the same dynamical zeta function. We define computable complete invariants for the periodic data of the extension for G not necessarily abelian, and extend all the above results to the nonabelian case. There is other work on basic invariants. The constructions require the "positive K-theory" setting for positive equivalence of matrices over ZG[t].

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