Invariant Equivalence Relations Research Articles

On mean sensitive tuples

In this paper we introduce and study several mean forms of sensitive tuples. It is shown that the topological or measure-theoretical entropy tuples are correspondingly mean sensitive tuples under certain conditions (minimal in the topological setting or ergodic in the measure-theoretical setting). Characterizations of the question when every non-diagonal tuple is mean sensitive are presented. Among other results we show that under minimality assumption a topological dynamical system is weakly mixing if and only if every non-diagonal tuple is mean sensitive and so as a consequence every minimal weakly mixing topological dynamical system is mean n -sensitive for any integer n ≥ 2 . Moreover, the notion of weakly sensitive in the mean tuple is introduced and it turns out that this property has some special lift property. As an application we obtain that the maximal mean equicontinuous factor for any topological dynamical system can be induced by the smallest closed invariant equivalence relation containing all weakly sensitive in the mean pairs.

The action of Γ K (N) and its suborbital graphs

For any natural number N, Γ K (N) where K is a subgroup of is a congruence subgroup of the modular group Γ acting on . We determine orbits of the action of Γ K (N) on and present a Γ K (N) invariant equivalence relation by imprimitive action. Then, we investigate the suborbital graph arising from the action of Γ K (N) on the orbit of ∞ and give conditions for two adjacent vertices in graph. In addition, we obtain connectedness properties of the subgraph .

Left is Better Than Right for Reducing Nondeterminism of NFAs

We study the NFA reductions by invariant equivalences and preorders. It is well-known that the NFA minimization problem is PSPACE-complete. Therefore, there have been many approaches to reduce the size of NFAs in low polynomial time by computing invariant equivalence or preorder relation and merging the states within same equivalence class. Here we consider the nondeterminism reduction of NFAs by invariant equivalences and preorders. We, in particular, show that computing equivalence and preorder relation from the left is more useful than the right for reducing the degree of nondeterminism in NFAs. We also present experimental evidence for showing that NFA reduction from the left achieves the better reduction of nondeterminism than reduction from the right.

On quadrilaterals in the suborbital graphs of the normalizer

n this paper‎, ‎we investigate suborbital graphs formed by $Nbig(Gamma_0(N)big)$-invariant equivalence relation induced on $hat{mathbb{Q}}$‎. ‎Conditions for being an edge are obtained as a main tool‎, ‎then necessary and sufficient conditions for the suborbital graphs to contain a circuit are investigated‎.

A Note on Mean Equicontinuity

In this note, it is shown that several results concerning mean equicontinuity proved before for minimal systems are actually held for general topological dynamical systems. Particularly, it turns out that a dynamical system is mean equicontinuous if and only if it is equicontinuous in the mean if and only if it is Banach (or Weyl) mean equicontinuous if and only if its regionally proximal relation is equal to the Banach proximal relation. Meanwhile, a relation is introduced such that the smallest closed invariant equivalence relation containing this relation induces the maximal mean equicontinuous factor for any system.

Topological dynamics and the complexity of strong types

We develop topological dynamics for the group of automorphisms of a monster model of any given theory. In particular, we find strong relationships between objects from topological dynamics (such as the generalized Bohr compactification introduced by Glasner) and various Galois groups of the theory in question, obtaining essentially new information about them, e.g., we present the closure of the identity in the Lascar Galois group of the theory as the quotient of a compact, Hausdorff group by a dense subgroup. We apply this to describe the complexity of bounded, invariant equivalence relations, obtaining comprehensive results, subsuming and extending the existing results and answering some open questions from earlier papers. We show that, in a countable theory, any such relation restricted to the set of realizations of a complete type over O is type-definable if and only if it is smooth. Then we show a counterpart of this result for theories in an arbitrary (not necessarily countable) language, obtaining also new information involving relative definability of the relation in question. As a final conclusion we get the following trichotomy. Let $$\mathfrak{C}$$ be a monster model of a countable theory, p ∈ S(O), and E be a bounded, (invariant) Borel (or, more generally, analytic) equivalence relation on p( $$\mathfrak{C}$$ ). Then, exactly one of the following holds: (1) E is relatively definable (on p( $$\mathfrak{C}$$ )), smooth, and has finitely many classes, (2) E is not relatively definable, but it is type-definable, smooth, and has 2ℵ0 classes, (3) E is not type definable and not smooth, and has 2ℵ0 classes. All the results which we obtain for bounded, invariant equivalence relations carry over to the case of bounded index, invariant subgroups of definable groups.

EQUIVALENCE RELATIONS INVARIANT UNDER GROUP ACTIONS

AbstractWe extend some recent results about bounded invariant equivalence relations and invariant subgroups of definable groups: we show that type-definability and smoothness are equivalent conditions in a wider class of relations than heretofore considered, which includes all the cases for which the equivalence was proved before.As a by-product, we show some analogous results in purely topological context (without direct use of model theory).

SMOOTHNESS OF BOUNDED INVARIANT EQUIVALENCE RELATIONS

AbstractWe generalise the main theorems from the paper “The Borel cardinality of Lascar strong types” by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (concerning connected components in definable group extensions) from that paper is not only sufficient but also necessary to obtain the conclusion.

Suborbital graphs for a special subgroup of the SL(3,Z)

In this paper we examine some properties of suborbital graphs for the group SL*(3,Z). We first introduce an invariant equivalence relation by using the congruence subgroup SL*(3,Z) instead of ?0(n) and obtain some results for the newly constructed subgraphs Fu,n whose vertices form the block [?]. We obtain edge and circuit conditions and some relations between lengths of circuits in Fu,n and elliptic elements of ?0(n).

Plus ultra

We define a reasonably well-behaved class of ultraimaginaries, i.e. classes modulo [Formula: see text]-invariant equivalence relations, called tame, and establish some basic simplicity-theoretic facts. We also show feeble elimination of supersimple ultraimaginaries: If [Formula: see text] is an ultraimaginary definable over a tuple [Formula: see text] with [Formula: see text], then [Formula: see text] is eliminable up to rank [Formula: see text]. Finally, we prove some uniform versions of the weak canonical base property.

LR Rotation Rule for Creating Minimal NFA

The problem of creating a minimal NFA is a primal (fundamental) problem. Reducing the size of NFA by using LR rotation rule has been shown to reduce importantly the search time. In [1] Ilie and Yu describe a construction of a right invariant equivalence relation on the states of a nondeterministic finite state automaton. We give a more efficient LR Rotation rule for constructing the minimal NFA. In this paper we represent new LR Rotation rule for the state minimization of NFA. The description of the proposed methods is given and we also shown the results of the numerical experiments. We conceive the problem of reducing the number of state and transition of Non Deterministic Finite Automata. Numerical experiments show that NFA reduction algorithm produces a minimal automation in all most cases. NFA reduction algorithm also reduces the complexity of Kameda-Weiner algorithm. We have shown empirically that these algorithms are effective in largely reducing the memory requirement of NFA minimization algorithm and algorithm minimization of the number of rules for NFA grows each year.

Laminations in the language of leaves

Thurston defined invariant laminations, i.e. collections of chords of the unit circle S (called leaves) that are pairwise disjoint inside the open unit disk and satisfy a few dynamical properties. To be directly associated to a polynomial, a lamination has to be generated by an equivalence relation with specific properties on S; then it is called a q-lamination. Since not all laminations are q-laminations, then from the point of view of studying polynomials the most interesting are those of them which are limits of q-laminations. In this paper we introduce an alternative definition of an invariant lamination, which involves only conditions on the leaves (and avoids gap invariance). The new class of laminations is slightly smaller than that defined by Thurston and is closed. We use this notion to elucidate the connection between invariant laminations and invariant equivalence relations on S.

Simultaneity as an Invariant Equivalence Relation

This paper deals with the concept of simultaneity in classical and relativistic physics as construed in terms of group-invariant equivalence relations. A full examination of Newton, Galilei and Poincaré invariant equivalence relations in ℝ4 is presented, which provides alternative proofs, additions and occasionally corrections of results in the literature, including Malament’s theorem and some of its variants. It is argued that the interpretation of simultaneity as an invariant equivalence relation, although interesting for its own sake, does not cut in the debate concerning the conventionality of simultaneity in special relativity.

Subdegree growth rates of infinite primitive permutation groups

A transitive group $G$ of permutations of a set $\Omega$ is primitive if the only $G$-invariant equivalence relations on $\Omega$ are the trivial and universal relations. If $\alpha \in \Omega$, then the orbits of the stabiliser $G_\alpha$ on $\Omega$ are called the $\alpha$-suborbits of $G$; when $G$ acts transitively the cardinalities of these $\alpha$-suborbits are the subdegrees of $G$. If $G$ acts primitively on an infinite set $\Omega$, and all the suborbits of $G$ are finite, Adeleke and Neumann asked if, after enumerating the subdegrees of $G$ as a non-decreasing sequence $1 = m_0 \leq m_1 \leq ...$, the subdegree growth rates of infinite primitive groups that act distance-transitively on locally finite distance-transitive graphs are extremal, and conjecture there might exist a number $c$ which perhaps depends upon $G$, perhaps only on $m$, such that $m_r \leq c(m-2)^{r-1}$. In this paper it is shown that such an enumeration is not desirable, as there exist infinite primitive permutation groups possessing no infinite subdegree, in which two distinct subdegrees are each equal to the cardinality of infinitely many suborbits. The examples used to show this provide several novel methods for constructing infinite primitive graphs. A revised enumeration method is then proposed, and it is shown that, under this, Adeleke and Neumann's question may be answered, at least for groups exhibiting suitable rates of growth.

Infinite primitive directed graphs

A group $G$ of permutations of a set $\Omega$ is {\em primitive} if it acts transitively on $\Omega$, and the only $G$-invariant equivalence relations on $\Omega$ are the trivial and universal relations. A graph $\Gamma$ is {\em primitive} if its automorphism group acts primitively on its vertex set. A graph $\Gamma$ has {\em connectivity one} if it is connected and there exists a vertex $\alpha$ of $\Gamma$, such that the induced graph $\Gamma \setminus \{\alpha\}$ is not connected. If $\Gamma$ has connectivity one, a {\em block} of $\Gamma$ is a connected subgraph that is maximal subject to the condition that it does not have connectivity one. The primitive undirected graphs with connectivity one have been fully classified by Jung and Watkins: the blocks of such graphs are primitive, pairwise-isomorphic and have at least three vertices. When one considers the general case of a directed primitive graph with connectivity one, however, this result no longer holds. In this paper we investigate the structure of these directed graphs, and obtain a complete characterisation.

N-ARY OPERATIONS WITH LONG PRE-PERIODS

Iterating a unary operation f defined on the finite set A with |A| = k one obtains the descending chain [Formula: see text] The least integer λ(f) with Imfλ(f) = Imfλ(f)+1 is called the pre-period of f. The pre-period of f is an integer between 0 and k - 1. If λ(f) = k - 1 and k ≥ 1, then f is called a long-tailed (LT)-operation and if λ(f) = k - 2 for k ≥ 2, f is said to be an (LT1)-operation. Unary (LT)- and (LT1)-operations and their invariant equivalence relations are characterized in [5]. In [6] these results are extended to partial operations. In this paper we consider the iteration of n-ary operations for n > 1, define and characterize (LT)- and (LT1)- operations and their invariant equivalence relations. The results can be applied in all fields where iteration and recursion plays a role.

Relative entropy tuples, relative u.p.e. and c.p.e. extensions

Relative entropy tuples both in topological and measure-theoretical settings, relative uniformly positive entropy (rel.-u.p.e.) and relative completely positive entropy (rel.-c.p.e.) are studied. It is shown that a relative topological Pinsker factor can be deduced by the smallest closed invariant equivalence relation containing the set of relative entropy pairs. A relative disjointness theorem involving relative topological entropy is proved. Moreover, it is shown that the product of finite rel.-c.p.e. extensions is also rel.-c.p.e..

Towards the Definition of Metric Hyperbolicity

We introduce measure-theoretic definitions of {\it hyperbolic structure for measure-preserving automorphisms}. A wide class of $K$-automorphisms possesses a hyperbolic structure; we prove that all $K$-automorphisms have a slightly weaker structure of {\it semi-hyperbolicity}. Instead of the notions of stable and unstable foliations and other notions from smooth theory, we use the tools of the theory of polymorphisms. The central role is played by {\it polymorphisms} associated with a special invariant equivalence relation, more exactly, with a homoclinic equivalence relation. We call an automorphism with given hyperbolic structure a hyperbolic automorphism and prove that it is canonically quasi-similar to a so-called prime nonmixing polymorphism. We present a short but necessary vocabulary of polymorphisms and Markov operators from \cite{V1,V2}.

Null systems and sequence entropy pairs

A measure-preserving transformation (respectively a topological system) is null if the metric (respectively topological) sequence entropy is zero for any sequence. Kushnirenko has shown that an ergodic measure-preserving transformation has a discrete spectrum if and only if it is null. We prove that for a minimal system this statement remains true modulo an almost one-to-one extension. It allows us to show that a scattering system is disjoint from any null minimal system. Moreover, some necessary conditions for a transitive non-minimal system to be null are obtained.Localizing the notion of sequence entropy, we define sequence entropy pairs and show that there is a maximal null factor for any system. Meanwhile, we define a weaker notion, namely weak mixing pairs. It turns out that a system is weakly mixing if and only if any pair not in the diagonal is a sequence entropy pair if and only if the same holds for a weak mixing pair, answering a question in Blanchard et al (F. Blanchard, B. Host and A. Maass, Topological complexity. Ergod. Th. & Dynam. Sys., 20 (2000), 641–662). For a group action we give a direct proof of the fact that the factor induced by the smallest invariant equivalence relation containing the regionally proximal relation is equicontinuous. Furthermore, we show that a non-equicontinuous minimal distal system is not null.

$F$-minimal sets

An $F$-minimal set is the simplest proximal extension of an equicontinuous minimal set. It has one interesting proximal cell and all the points in this proximal cell are uniformly asymptotic. The Sturmian minimal sets are the best known examples of $F$-minimal sets. Our analysis of them is in terms of their maximal equicontinuous factors. Algebraically speaking $F$-minimal sets are obtained by taking an invariant *-closed algebra of almost periodic functions and adjoining some suitable functions to it. Our point of view is to obtain these functions from the maximal equicontinuous factor. In §3 we consider a subclass of $F$-minimal sets which generalize the classical Sturmian minimal sets, and in §4 we examine the class of minimal sets obtained by taking the minimal right ideal of an $F$-minimal set and factoring by a closed invariant equivalence relation which is smaller than the proximal relation.