arXiv subjects

Robin J. Deeley

Publications and source records attributed to Robin J. Deeley.

At least 19 recordsLinked to original sources

arXiv2026

Synchronizing Dynamical Systems: Finitely presented systems and Ruelle algebras

The main goals of the present paper are to determine the structure of the $C^\ast$-algebras associated to a finitely presented system and to develop the basic theory of the Ruelle algebras associated to a general synchronizing system. The later is related to the former in the sense that we show that Ruelle algebras associated to a finitely presented system are explicitly related to the Smale space case. The relevant $C^\ast$-algebras are the synchronizing heteroclinic algebras that were introduced in our previous work on synchronizing systems. They are very much related to previous work of Thomsen, who in turn was building on work of Ruelle, Putnam, and Spielberg.

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arXiv2026

Bordisms and unbounded $KK$-theory

This monograph studies $KK$-theory in its unbounded model. The central object is the $KK$-bordism group obtained by imposing the $KK$-bordism relation on unbounded $KK$-cycles. In the paradigm of noncommutative geometry, an unbounded $KK$-cycle is a noncommutative geometry in its own right and our approach allow for the study of mildly noncommutative geometries (orbifolds, foliations et cetera) as if they were closed manifolds. The techniques we introduce enable us to directly import manifold techniques and arguments into the important yet technical field of unbounded $KK$-theory. Recent decades has seen a tremendous progress in the study of the unbounded model for $KK$ as well as secondary invariants, the first motivated by refining computational tools in Kasparov's $KK$-theory and the second by applications to geometry and topology. The aim of this work is to provide a common framework for these two areas: equipping unbounded $KK$-cycles with a geometrically motivated relation recovering Kasparov's $KK$-theory that is computationally tractable for working with secondary invariants.

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arXiv2024

The rational HK-conjecture: transformation groupoids and a revised version

We prove the rational HK-conjecture for a large class of transformation groupoids in the case when the relevant action has torsion-free stabilizers. A revised version of the rational HK-conjecture in the case of (possibly) torsion stabilizers is introduced and proved for a large class of transformation groupoids. In particular, this revised version holds for Scarparo's counterexamples to the original rational HK-conjecture. The key tools used are the Baum-Connes conjecture and a Chern character defined by Raven.

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arXiv2023

Wieler solenoids: non-Hausdorff expansiveness, Cuntz-Pimsner models, and functorial properties

Building on work of Williams, Wieler proved that every irreducible Smale space with totally disconnected stable sets can be realized via a stationary inverse limit. Using this result, the first and fourth listed authors of the present paper showed that the stable $C^*$-algebra associated to such a Smale space can be obtained from a stationary inductive limit of a Fell algebra. Its spectrum is typically non-Hausdorff and admits a self-map related to the stationary inverse limit. With the goal of understanding the fine structure of the stable algebra and the stable Ruelle algebra, we study said self-map on the spectrum of the Fell algebra as a dynamical system in its own right. Our results can be summarized into the statement that this dynamical system is an expansive, surjective, local homeomorphism of a compact, locally Hausdorff space and from its $K$-theory we can compute $K$-theoretical invariants of the stable and unstable Ruelle algebra of a Smale space with totally disconnected stable sets.

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arXiv2023

Synchronizing Dynamical Systems: their groupoids and $C^*$-algebras

Building on work of Ruelle and Putnam in the Smale space case, Thomsen defined the homoclinic and heteroclinic $C^\ast$-algebras for an expansive dynamical system. In this paper we define a class of expansive dynamical systems, called synchronizing dynamical systems, that exhibit hyperbolic behavior almost everywhere. Synchronizing dynamical systems generalize Smale spaces (and even finitely presented systems). Yet they still have desirable dynamical properties such as having a dense set of periodic points. We study various $C^\ast$-algebras associated with a synchronizing dynamical system. Among other results, we show that the homoclinic algebra of a synchronizing system contains an ideal which behaves like the homoclinic algebra of a Smale space.

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arXiv2022

Invariants for the Smale space associated to an expanding endomorphism of a flat manifold

We study invariants associated to Smale spaces obtained from an expanding endomorphism on a (closed connected Riemannian) flat manifold. Specifically, the relevant invariants are the $K$-theory of the associated $C^*$-algebras and Putnam's homology theory for Smale spaces. The latter is isomorphic to the groupoid homology of the groupoids used to construct the $C^*$-algebras.

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arXiv2022

A counterexample to the HK-conjecture that is principal

Scarparo has constructed counterexamples to Matui's HK-conjecture. These counterexample and other known counterexamples are essentially principal but not principal. In the present paper, a counterexample to the HK-conjecture that is principal is given. Like Scarparo's original counterexample, our counterexample is the transformation groupoid associated to a particular odometer. However, the relevant group is the fundamental group of a flat manifold (and hence is torsion-free) and the associated odometer action is free. The examples discussed here do satisfy the rational version of the HK-conjecture.

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arXiv2022

Fell algebras, groupoids, and projections

Examples of Fell algebras with compact spectrum and trivial Dixmier-Douady invariant are constructed to illustrate differences with the case of continuous trace $C^*$-algebras. At the level of the spectrum, this translates to only assuming the spectrum is locally Hausdorff (rather than Hausdorff). The existence of (full) projections is the fundamental question considered. The class of Fell algebras studied here arise naturally in the study of Wieler solenoids and applications to dynamical systems will be discussed in a separate paper.

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arXiv2022

Minimal homeomorphisms and topological $K$-theory

The Lefschetz fixed point theorem provides a powerful obstruction to the existence of minimal homeomorphisms on well-behaved spaces such as finite CW-complexes. We show that these obstructions do not hold for more general spaces. More precisely, minimal homeomorphisms are constructed on space with prescribed $K$-theory or cohomology. We also allow for some control of the map on $K$-theory and cohomology induced from these minimal homeomorphisms. This allows for the construction of many minimal homeomorphisms that are not homotopic to the identity. Applications to $C^*$-algebras will be discussed in another paper.

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arXiv2020

Classifiable $\mathrm{C}^*$-algebras from minimal $\mathbb{Z}$-actions and their orbit-breaking subalgebras

In this paper we consider the question of what abelian groups can arise as the $K$-theory of $\mathrm{C}^*$-algebras arising from minimal dynamical systems. We completely characterize the $K$-theory of the crossed product of a space $X$ with finitely generated $K$-theory by an action of the integers and show that crossed products by a minimal homeomorphisms exhaust the range of these possible $K$-theories. Moreover, we may arrange that the minimal systems involved are uniquely ergodic, so that their $\mathrm{C}^*$-algebras are classified by their Elliott invariants. We also investigate the $K$-theory and the Elliott invariants of orbit-breaking algebras. We show that given arbitrary countable abelian groups $G_0$ and $G_1$ and any Choquet simplex $Δ$ with finitely many extreme points, we can find a minimal orbit-breaking relation such that the associated $\mathrm{C}^*$-algebra has $K$-theory given by this pair of groups and tracial state space affinely homeomorphic to $Δ$. We also improve on the second author's previous results by using our orbit-breaking construction to $\mathrm{C}^*$-algebras of minimal amenable equivalence relations with real rank zero that allow torsion in both $K_0$ and $K_1$. These results have important applications to the Elliott classification program for $\mathrm{C}^*$-algebras. In particular, we make a step towards determining the range of the Elliott invariant of the $\mathrm{C}^*$-algebras associated to étale equivalence relations.

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arXiv2020

Non-homogeneous extensions of Cantor minimal systems

Floyd gave an example of a minimal dynamical system which was an extension of an odometer and the fibres of the associated factor map were either singletons or intervals. Gjerde and Johansen showed that the odometer could be replaced by any Cantor minimal system. Here, we show further that the intervals can be generalized to cubes of arbitrary dimension and to attractors of certain iterated function systems. We discuss applications.

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arXiv2019

Smale space C*-algebras have nonzero projections

The main result of the present paper is that the stable and unstable C*-algebras associated to a mixing Smale space always contain nonzero projections. This gives a positive answer to a question of the first listed author and Karen Strung and has implications for the structure of these algebras in light of the Elliott program for simple C*-algebras. Using our main result, we also show that the homoclinic, stable, and unstable algebras each have real rank zero.

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arXiv2019

Constructions in minimal amenable dynamics and applications to the classification of $\mathrm{C}^*$-algebras

We study the existence of minimal dynamical systems, their orbit and minimal orbit-breaking equivalence relations, and their applications to C*-algebras and K-theory. We show that given any finite CW-complex there exists a space with the same K-theory and cohomology that admits a minimal homeomorphism. The proof relies on the existence of homeomorphisms on point-like spaces constructed by the authors in previous work, together with existence results for skew product systems due to Glasner and Weiss. To any minimal dynamical system one can associate minimal equivalence relations by breaking orbits at small subsets. Using Renault's groupoid C*-algebra construction we can associate K-theory groups to minimal dynamical systems and orbit-breaking equivalence relations. We show that given arbitrary countable abelian groups $G_0$ and $G_1$ we can find a minimal orbit-breaking relation such that the K-theory of the associated C*-algebra is exactly this pair. These results have important applications to the Elliott classification program for C*-algebras. In particular, we make a step towards determining the range of the Elliott invariant of the C*-algebras associated to minimal dynamical systems with mean dimension zero and their minimal orbit-breaking relations

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arXiv2019

The stable algebra of a Wieler solenoid: inductive limits and K-theory

Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Using her construction, we show that the associated stable C*-algebra is the stationary inductive limit of a C*-stable Fell algebra that has compact spectrum and trivial Dixmier-Douady invariant. This result applies in particular to Williams solenoids along with other examples. Beyond the structural implications of this inductive limit, one can use this result to in principle compute the K-theory of the stable C*-algebra. A specific one-dimensional Smale space (the aab/ab-solenoid) is considered as an illustrative running example throughout.

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arXiv2019

Group actions on Smale space C*-algebras

Group actions on a Smale space and the actions induced on the C*-algebras associated to such a dynamical system are studied. We show that an effective action of a discrete group on a mixing Smale space produces a strongly outer action on the homoclinic algebra. We then show that for irreducible Smale spaces, the property of finite Rokhlin dimension passes from the induced action on the homoclinic algbera to the induced actions on the stable and unstable C*-algebras. In each of these cases, we discuss the preservation of properties---such as finite nuclear dimension, Z-stability, and classification by Elliott invariants---in the resulting crossed products.

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arXiv2018

Relative geometric assembly and mapping cones, Part II: Chern characters and the Novikov property

We study Chern characters and the assembly mapping for free actions using the framework of geometric $K$-homology. The focus is on the relative groups associated with a group homomorphism $ϕ:Γ_1\to Γ_2$ along with applications to Novikov type properties. In particular, we prove a relative strong Novikov property for homomorphisms of hyperbolic groups and a relative strong $\ell^1$-Novikov property for polynomially bounded homomorphisms of groups with polynomially bounded cohomology in $\C$. As a corollary, relative higher signatures on a manifold with boundary $W$, with $π_1(\partial W)\to π_1(W)$ belonging to the class above, are homotopy invariant.

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arXiv2018

Relative geometric assembly and mapping cones, Part I: The geometric model and applications

Inspired by an analytic construction of Chang, Weinberger and Yu, we define an assembly map in relative geometric $K$-homology. The properties of the geometric assembly map are studied using a variety of index theoretic tools (e.g., the localized index and higher Atiyah-Patodi-Singer index theory). As an application we obtain a vanishing result in the context of manifolds with boundary and positive scalar curvature; this result is also inspired and connected to work of Chang, Weinberger and Yu. Furthermore, we use results of Wahl to show that rational injectivity of the relative assembly map implies homotopy invariance of the relative higher signatures of a manifold with boundary.

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arXiv2017

Nuclear dimension and classification of C*-algebras associated to Smale spaces

We show that the homoclinic C*-algebras of mixing Smale spaces are classifiable by the Elliott invariant. To obtain this result, we prove that the stable, unstable, and homoclinic C*-algebras associated to such Smale spaces have finite nuclear dimension. Our proof of finite nuclear dimension relies on Guentner, Willett, and Yu's notion of dynamic asymptotic dimension.

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