Cantor Space Research Articles

BOREL LINE GRAPHS

Abstract We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the nine finite graphs from the classical result of Beineke together with a 10th infinite graph associated with the equivalence relation $\mathbb {E}_0$ on the Cantor space. As a corollary, we prove a partial converse to the Feldman–Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers.

SUFFICIENT CONDITIONS FOR A GROUP OF HOMEOMORPHISMS OF THE CANTOR SET TO BE TWO-GENERATED

AbstractWe introduce and study two conditions on groups of homeomorphisms of Cantor space, namely the conditions of being vigorous and of being flawless. These concepts are dynamical in nature, and allow us to study a certain interplay between the dynamics of an action and the algebraic properties of the acting group. A group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is vigorous if for any clopen set A and proper clopen subsets B and C of A, there is $\gamma \in G$ in the pointwise stabiliser of $\mathfrak {C}\backslash A$ with $B\gamma \subseteq C$ . A nontrivial group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is flawless if for all k and w a nontrivial freely reduced product expression on k variables (including inverse symbols), a particular subgroup $w(G)_\circ $ of the verbal subgroup $w(G)$ is the whole group. We show: 1) simple vigorous groups are either two-generated by torsion elements, or not finitely generated, 2) flawless groups are both perfect and lawless, 3) vigorous groups are simple if and only if they are flawless, and, 4) the class of vigorous simple subgroups of $\operatorname {Homeo}(\mathfrak {C})$ is fairly broad (the class is closed under various natural constructions and contains many well known groups, such as the commutator subgroups of the Higman–Thompson groups $G_{n,r}$ , the Brin-Thompson groups $nV$ , Röver’s group $V(\Gamma )$ , and others of Nekrashevych’s ‘simple groups of dynamical origin’).

Computably and punctually universal spaces

We prove that the standard computable presentation of the space C[0,1] of continuous real-valued functions on the unit interval is computably and punctually (primitively recursively) universal. From the perspective of modern computability theory, this settles a problem raised by Sierpiński in the 1940s.We prove that the original Urysohn's construction of the universal separable Polish space U is punctually universal. We also show that effectively compact, punctual Stone spaces are punctually homeomorphically embeddable into Cantor space 2ω; note that we do not require effective compactness be primitive recursive. We also prove that effective compactness cannot be dropped from the premises by constructing a counterexample.

Searching problems above arithmetical transfinite recursion

We investigate some Weihrauch problems between ATR2 and Cωω. We show that the fixed point theorem for monotone operators on the Cantor space (a weaker version of the Knaster-Tarski theorem) is not Weihrauch reducible to ATR2. Furthermore, we introduce the ω-model reflection ATR2rfn of ATR2 and show that it is an upper bound for problems provable from the axiomatic system ATR0 which are of the form ∀X(θ(X)→∃Yη(X,Y)) with arithmetical formulas θ,η. We also show that Weihrauch degrees of relativized least fixed point theorems for monotone operators on the Cantor space form a linear hierarchy between ATR2rfn and Cωω.

Dynamic Tangled Derivative Logic of Metric Spaces

Dynamical systems are abstract models of interaction between space and time. They are often used in fields such as physics and engineering to understand complex processes, but due to their general nature, they have found applications for studying computational processes, interaction in multi-agent systems, machine learning algorithms and other computer science related phenomena. In the vast majority of applications, a dynamical system consists of the action of a continuous `transition function' on a metric space. In this work, we consider decidable formal systems for reasoning about such structures. Spatial logics can be traced back to the 1940's, but our work follows a more dynamic turn that these logics have taken due to two recent developments: the study of the topological mu-calculus, and the the integration of linear temporal logic with logics based on the Cantor derivative. In this paper, we combine dynamic topological logics based on the Cantor derivative and the `next point in time' operators with an expressively complete fixed point operator to produce a combination of the topological mu-calculus with linear temporal logic. We show that the resulting logics are decidable and have a natural axiomatisation. Moreover, we prove that these logics are complete for interpretations on the Cantor space, the rational numbers, and subspaces thereof.

Computability and non-monotone induction

Non-monotone inductive definitions were studied in the late 1960’s and early 1970’s with the aim of understanding connections between the complexity of the formulas defining the induction steps and the ordinals measuring the rank of the inductive process. In general, any type 2 functional will generate an inductive process, and in this paper we will view non-monotone induction as a functional of type 3. We investigate the associated computation theory of this functional inherited from the Kleene schemes and we investigate the associated companion of sets with codes computable in the functional of non-monotone induction. The interest in this functional is motivated from observing that constructions via non-monotone induction appear to be natural in classical analysis. There are two groups of results: We establish strong closure properties of the least ordinal without a code computable in the functional of non-monotone induction, and we provide a characterisation of the class of functionals of type 3 computable from this functional, a characterisation in terms of sequential operators working in transfinite time. We will also see that the full computational power of non-monotone induction is required when this principle is used to construct functionals witnessing the compactness of the Cantor space and of closed, bounded intervals.

STRONG MEASURE ZERO SETS ON FOR INACCESSIBLE

Abstract We investigate the notion of strong measure zero sets in the context of the higher Cantor space $2^\kappa $ for $\kappa $ at least inaccessible. Using an iteration of perfect tree forcings, we give two proofs of the relative consistency of $$\begin{align*}|2^\kappa| = \kappa^{++} + \forall X \subseteq 2^\kappa:\ X \textrm{ is strong measure zero if and only if } |X| \leq \kappa^+. \end{align*}$$ Furthermore, we also investigate the stronger notion of stationary strong measure zero and show that the equivalence of the two notions is undecidable in ZFC.

Topologies for Finite Words: Compatibility with the Cantor Topology

Infinite words are often considered as limits of finite words. As topological methods have been proved to be useful in the theory of ω-languages it seems to be providing to include finite and infinite words into one (topological) space. In most cases this results in a poor topological structure induced on the subspace of finite words. In the present paper we investigate the possibility to link topologies in the space of finite words with a topology in the space of infinite words via a natural mapping. A requirement in this linking of topologies consists in the compatibility of the topological properties (openness, closedness etc.) of images with preimages and vice versa. Here, we show that choosing for infinite words the natural topology of the CANTOR space and the δ-limit as linking mapping there are several natural topologies on the space of finite words compatible with the topology of the CANTOR space. It is interesting to observe that besides the well-known prefix topology there are at least two more whose origin is from language theory – centers and supercenters of languages. We show that several of these topologies on the space of finite words fit into a class of ℒ-topologies and exhibit their special properties w.r.t. to the compatibility with the CANTOR topology.

Tomiyama’s K-commutative diagrams of minimal dynamical systems

Let [Formula: see text] be the Cantor space and [Formula: see text] be an even-dimensional sphere. By applying a result of the existence of minimal skew products, we show that, associated with any Cantor minimal system [Formula: see text], there is a class [Formula: see text] of minimal skew products on [Formula: see text], such that for any two rigid homeomorphisms [Formula: see text] and [Formula: see text], the notions of approximate [Formula: see text]-conjugacy and [Formula: see text]-strongly approximate conjugacy coincide, which are also equivalent to a [Formula: see text]-version of Tomiyama’s commutative diagram. In fact, this is also the case if [Formula: see text] is replaced by any (infinite) connected finite CW-complex with torsion free [Formula: see text]-group, vanished [Formula: see text]-group and the so-called Lipschitz-minimal-property.

Computability Theory on Polish Metric Spaces

AbstractComputability theoretic aspects of Polish metric spaces are studied by adapting notions and methods of computable structure theory. In this dissertation, we mainly investigate index sets and classification problems for computably presentable Polish metric spaces. We find the complexity of a number of index sets, isomorphism problems, and embedding problems for computably presentable metric spaces. We also provide several computable structure theory results related to some classical Polish metric spaces such as the Urysohn space $\mathbb {U}$ , the Cantor space $2^{\mathbb {N}}$ , the Baire space $\mathbb {N}^{\mathbb {N}}$ , and spaces of continuous functions.Abstract prepared by Teerawat Thewmorakot.E-mail: teerawat.thew@hotmail.com

Complexity profiles and generic Muchnik reducibility

We study the relative computational complexity of expansions of Cantor and Baire space in terms of generic Muchnik reducibility. We show that no expansion of Cantor space by countably many unary or closed relations can give a generic Muchnik degree strictly between the degree of Cantor space and the degree of Baire space. Similarly, assuming Δ21-Wadge determinacy we show that no expansion of Baire space by countably many unary or closed relations can give a generic Muchnik degree strictly between the degree of Baire space and the Borel complete degree. On the other hand, we provide a construction of a degree strictly between the degree of Cantor space and the degree of Baire space and also between the degree of Baire space and the Borel complete degree.

Amenable wreath products with non almost finite actions of mean dimension zero

Almost finiteness was introduced in the seminal work of Kerr [J. Eur. Math. Soc. (JEMS) 22 (2020), pp. 3697–3745] as an dynamical analogue of Z \mathcal {Z} -stability in the Toms-Winter conjecture. In this article, we provide the first examples of minimal, topologically free actions of amenable groups that have mean dimension zero but are not almost finite. More precisely, we prove that there exists an infinite family of amenable wreath products that admit topologically free, minimal profinite actions on the Cantor space which fail to be almost finite. Furthermore, these actions have dynamical comparison. This intriguing new phenomenon shows that Kerr’s dynamical analogue of Toms-Winter conjecture fails for minimal, topologically free actions of amenable groups. The notion of allosteric group holds a significant position in our study. A group is allosteric if it admits a minimal action on a compact space with an invariant ergodic probability measure that is topologically free but not essentially free. We study allostery of wreath products and provide the first examples of allosteric amenable groups.

Apartness, sharp elements, and the Scott topology of domains

AbstractWorking constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of being unequal, similar to how inhabitedness is a positive formulation of nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and only if it is located. Our first main result is that for a large class of continuous dcpos, the Bridges–Vîţǎ apartness topology and the Scott topology coincide. Although we cannot expect a tight or cotransitive apartness on nontrivial dcpos, we prove that the intrinsic apartness is both tight and cotransitive when restricted to the sharp elements of a continuous dcpo. These include the strongly maximal elements, as studied by Smyth and Heckmann. We develop the theory of strongly maximal elements highlighting its connection to sharpness and the Lawson topology. Finally, we illustrate the intrinsic apartness, sharpness, and strong maximality by considering several natural examples of continuous dcpos: the Cantor and Baire domains, the partial Dedekind reals, the lower reals and, finally, an embedding of Cantor space into an exponential of lifted sets.

Towards logical foundations for probabilistic computation

The overall purpose of the present work is to lay the foundations for a new approach to bridge logic and probabilistic computation. To this aim we introduce extensions of classical and intuitionistic propositional logic with counting quantifiers, that is, quantifiers that measure to which extent a formula is true. The resulting systems, called cCPL and iCPL, respectively, admit a natural semantics, based on the Borel σ-algebra of the Cantor space, together with a sound and complete proof system. Our main results consist in relating cCPL and iCPL with some central concepts in the study of probabilistic computation. On the one hand, the validity of cCPL-formulae in prenex form characterizes the corresponding level of Wagner's hierarchy of counting complexity classes, closely related to probabilistic complexity. On the other hand, proofs in iCPL correspond, in the sense of Curry and Howard, to typing derivations for a randomized extension of the λ-calculus, so that counting quantifiers reveal the probability of termination of the underlying probabilistic programs.

An n-adic generalization of the Lodha–Moore group

We generalized the Lodha–Moore group , similar to the case of the Thompson group F, into an n-adic version . This generalization is given by regarding the Lodha–Moore group as the group consisting of homeomorphisms on the Cantor space. Just as the case of F and , we also showed that and share many properties.

Lipschitz and Wadge binary games in second order arithmetic

We present a detailed formalization of Lipschitz and Wadge games in the context of second order arithmetic and we investigate the logical strength of Lipschitz and Wadge determinacy, and the tightly related Semi-Linear Ordering principle, for the first levels of the Hausdorff difference hierarchy in the Cantor space. As a result, we obtain characterizations of WKL0 and ACA0 in terms of these determinacy principles.

On the Cardinality of Future Worldlines in Discrete Spacetime Structures

We give an analysis over a variation of causal sets where the light cone of an event is represented by finitely branching trees with respect to any given arbitrary dynamics. We argue through basic topological properties of Cantor space that under certain assumptions about the universe, spacetime structure and causation, given any event x, the number of all possible future worldlines of x within the many-worlds interpretation is uncountable. However, if all worldlines extending the event x are ‘eventually deterministic’, then the cardinality of the set of future worldlines with respect to x is exactly $$\aleph _0$$ , i.e., countably infinite. We also observe that if there are countably many future worldlines with respect to x, then at least one of them must be necessarily ‘decidable’ in the sense that there is an algorithm which determines whether or not any given event belongs to that worldline. We then show that if there are only finitely many worldlines in the future of an event x, then they are all decidable. We finally point out the fact that there can be only countably many terminating worldlines.

Ideals with Smital properties

A sigma -ideal mathcal {I} on a Polish group (X,+) has the Smital Property if for every dense set D and a Borel mathcal {I}-positive set B the algebraic sum D+B is a complement of a set from mathcal {I}. We consider several variants of this property and study their connections with the countable chain condition, maximality and how well they are preserved via Fubini products. In particular we show that there are mathfrak {c} many maximal invariant sigma -ideals with Borel bases on the Cantor space 2^omega .

Complete topological descriptions of certain Morse boundaries

We study direct limits of embedded Cantor sets and embedded Sierpiński curves. We show that under appropriate conditions on the embeddings, all limits of Cantor spaces give rise to homeomorphic spaces, called \omega -Cantor spaces, and, similarly, all limits of Sierpiński curves give homeomorphic spaces, called \omega -Sierpiński curves. We then show that the former occur naturally as Morse boundaries of right-angled Artin groups and fundamental groups of non-geometric graph manifolds, while the latter occur as Morse boundaries of fundamental groups of finite-volume, cusped hyperbolic 3-manifolds.

Common fixed point, Baire's and Cantor's theorems in neutrosophic 2-metric spaces

<abstract> <p>These fundamental Theorems of classical analysis, namely Baire's Theorem and Cantor's Intersection Theorem in the context of Neutrosophic 2-metric spaces, are demonstrated in this article. Naschie discussed high energy physics in relation to the Baire's Theorem and the Cantor space in descriptive set theory. We describe, how to demonstrate the validity and uniqueness of the common fixed-point theorem for four mappings in Neutrosophic 2-metric spaces.</p> </abstract>