Conjugacy Relation Research Articles

The conjugacy problem in ergodic theory

All common probability preserving transformations can be represented as elements of MPT, the group of measure preserving transformations of the unit interval with Lebesgue measure. This group has a natural Polish topology and the induced topology on the set of ergodic transformations is also Polish. Our main result is that the set of ergodic elements T in MPT that are isomorphic to their inverse is a complete analytic set. This has as a consequence the fact that the isomorphism relation is also a complete analytic set and in particular is not Borel. This is in stark contrast to the situation of unitary operators where the spectral theorem can be used to show that conjugacy relation in the unitary group is Borel. This result explains, perhaps, why the problem of determining whether ergodic transformations are isomorphic or not has proven to be so intractable. The construction that we use is general enough to show that the set of ergodic T ’s with nontrivial centralizer is also complete analytic. On the positive side we show that the isomorphism relation is Borel when restricted to the rank one transformations, which form a generic subset of MPT. It remains an open problem to nd a good explicit method of checking when two rank one transformations are isomorphic. In Memoriam: Prior to the

Flows of Continuous-Time Dynamical Systems with No Periodic Orbit as an Equivalence Class under Topological Conjugacy Relation

Problem statement: Flows of continuous-time dynamical systems with the same number of equilibrium points and trajectories, and which has no periodic orbit form an equivalence class under the topological conjugacy relation. Approach: Arbitrarily, two trajectories resulting from two distinct flows of this type of dynamical systems were written as a set of points (orbit). A homeomorphism which maps between these two sets is then built. Using the notion of topological conjugacy, they were shown to conjugate topologically. By the arbitrariness in selection of flows and their respective initial states, the results were extended to all the flows of dynamical system of that type. Results: Any two flows of such dynamical systems were shown to share the same dynamics temporally along with other properties such as order isomorphic and homeomorphic. Conclusion: Topological conjugacy serves as an equivalence relation in the set of flows of continuous-time dynamical systems which have same number of equilibrium points and trajectories, and has no periodic orbit.

Remnant inequalities and doubly-twisted conjugacy in free groups

We give two results for computing doubly-twisted conjugacy relations in free groups with respect to homomorphisms φ and ψ such that certain remnant words from φ are longer than the images of generators under ψ . Our first result is a remnant inequality condition which implies that two words u and v are not doubly-twisted conjugate. Further we show that if ψ is given and φ , u , and v are chosen at random, then the asymptotic probability that u and v are not doubly-twisted conjugate is 1. In the particular case of singly-twisted conjugacy, this means that if φ , u , and v are chosen at random, then u and v are not in the same singly-twisted conjugacy class with asymptotic probability 1. Our second result generalizes Kim’s “bounded solution length”. We give an algorithm for deciding doubly-twisted conjugacy relations in the case where φ and ψ satisfy a similar remnant inequality. In the particular case of singly-twisted conjugacy, our algorithm suffices to decide any twisted conjugacy relation if φ has remnant words of length at least 2. As a consequence of our generic properties we give an elementary proof of a recent result of Martino, Turner, and Ventura, that computes the densities of the sets of injective and surjective homomorphisms from one free group to another. We further compute the expected value of the density of the image of a homomorphism.

INVERSE SEMIGROUPS OF PARTIAL AUTOMATON PERMUTATIONS

The inverse semigroup of partial automaton permutations over a finite alphabet is characterized in terms of wreath products. The permutation conjugacy relation in this semigroup and the Green's relations are described. Criteria of primary conjugacy and conjugacy are given for certain naturally defined families of partial automaton permutations. Sufficient conditions under which an inverse semigroup admits a level transitive action are presented. We give explicit examples (monogenic inverse semigroups and some commutative Clifford semigroups) of inverse semigroups generated by finite automata.

A characterization of (regular) circular languages generated by monotone complete splicing systems

Circular splicing systems are a formal model of a generative mechanism of circular words, inspired by a recombinant behaviour of circular DNA. Some unanswered questions are related to the computational power of such systems, and finding a characterization of the class of circular languages generated by circular splicing systems is still an open problem. In this paper we solve this problem for monotone complete systems, which are finite circular splicing systems with rules of a simpler form. We show that a circular language L is generated by a monotone complete system if and only if the set L i n ( L ) of all words corresponding to L is a pure unitary language generated by a set closed under the conjugacy relation. The class of pure unitary languages was introduced by A. Ehrenfeucht, D. Haussler, G. Rozenberg in 1983, as a subclass of the class of context-free languages, together with a characterization of regular pure unitary languages by means of a decidable property. As a direct consequence, we characterize (regular) circular languages generated by monotone complete systems. We can also decide whether the language generated by a monotone complete system is regular. Finally, we point out that monotone complete systems have the same computational power as finite simple systems, an easy type of circular splicing system defined in the literature from the very beginning, when only one rule of a specific type is allowed. From our results on monotone complete systems, it follows that finite simple systems generate a class of languages containing non-regular languages, showing the incorrectness of a longstanding result on simple systems.

Elliptic complex numbers with dual multiplication

Investigated is a number system in which the square of a basis number: ( w) 2, and the square of its additive inverse: (− w) 2, are not equal. Termed W space, a vector space over the reals, this number system will be introduced by restating defining relations for complex space C , then changing a defining conjugacy relation from conj( z) + z = 0 in the complexes to conj( z) + z = 1 for W space. This change produces a dual-represented vector space consisting of two dual, isomorphic fields, which are unified under one “context-sensitive” multiplication. Fundamental algebraic and geometric properties will be investigated. W space can be interpreted as a generalization of the complexes but is characterized by an interacting duality which seems to produce two of everything: two representations, two multiplications, two norm values, and two solutions to a linear equation. W space will be compared to a previous suggestion of a similar algebra, and then possible applications will be offered, including a W space fractal.

Circular Languages Generated by Complete Splicing Systems and Pure Unitary Languages

Circular splicing systems are a formal model of a generative mechanism of circular words, inspired by a recombinant behaviour of circular DNA. Some unanswered questions are related to the computational power of such systems, and finding a characterization of the class of circular languages generated by circular splicing systems is still an open problem. In this paper we solve this problem for complete systems, which are special finite circular splicing systems. We show that a circular language L is generated by a complete system if and only if the set Lin(L) of all words corresponding to L is a pure unitary language generated by a set closed under the conjugacy relation. The class of pure unitary languages was introduced by A. Ehrenfeucht, D. Haussler, G. Rozenberg in 1983, as a subclass of the class of context-free languages, together with a characterization of regular pure unitary languages by means of a decidable property. As a direct consequence, we characterize (regular) circular languages generated by complete systems. We can also decide whether the language generated by a complete system is regular. Finally, we point out that complete systems have the same computational power as finite simple systems, an easy type of circular splicing system defined in the literature from the very beginning, when only one rule is allowed. From our results on complete systems, it follows that finite simple systems generate a class of context-free languages containing non-regular languages, showing the incorrectness of a longstanding result on simple systems.

Unfolding of resonant saddles and the Dulac time

In this work we study unfoldings of planar vector fields in a neighbourhood of a resonant saddle. We give a $\mathcal C^k$ normal form for the unfolding with respect to the conjugacy relation. Using our normal form we determine an asymptotic development, uniform with respect to the parameters, of the Dulac time of a resonant saddle deformation. Conjugacy relation instead of weaker equivalence relation is necessary when studying the time function. The Dulac time of a resonant saddle can be seen as the basic building block of the total period function of an unfolding of a hyperbolic polycycle.

On the conjugacy relation in ergodic theory

The set of pairs of transformations on the interval [ 0 , 1 ] can be equipped with a standard Borel structure. We prove that the relation of conjugacy is not a Borel subset of this space, in fact it is complete analytic. Moreover, our construction proves that the two sets, { T : T is conjugate of T −1 } , and { T : the centralizer of T is non-trivial } are complete analytic sets. To cite this article: M.D. Foreman et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Circular splicing and regularity

Circular splicing has been very recently introduced to model a specific recombinant behaviour of circular DNA, continuing the investigation initiated with linear splicing. In this paper we restrict our study to the relationship between regular circular languages and languages generated by finite circular splicing systems and provide some results towards a characterization of the intersection between these two classes. We consider the class of languages X*, called here star languages, which are closed under conjugacy relation and with X being a regular language. Using automata theory and combinatorial techniques on words, we show that for a subclass of star languages the corresponding circular languages are (Paun) circular splicing languages. For example, star languages belong to this subclass when X* is a free monoid or X is a finite set. We also prove that each (Paun) circular splicing language L over a one-letter alphabet has the form L = X+ ∪ Y, with X,Y finite sets satisfying particular hypotheses. Cyclic and weak cyclic languages, which will be introduced in this paper, show that this result does not hold when we increase the size of alphabets, even if we restrict ourselves to regular languages.

Conjugate Scaling Algorithm for Fenchel-Type Duality in Discrete Convex Optimization

This paper presents a polynomial time algorithm for solving submodular flow problems with a class of discrete convex cost functions. This class of problems is a common generalization of the submodular flow and valuated matroid intersection problems. The algorithm adopts a new scaling technique that scales the discrete convex cost functions via the conjugacy relation. The algorithm can be used to find a pair of optima in the form of the Fenchel-type duality theorem in discrete convex analysis.

Conjugacy Relation on Coxeter Elements

Let (W, S, Γ) be an irreducible finitely presented Coxeter system. The present paper is mainly concerned with a conjugacy relation on Coxeter elements in the case where Γ contains just one circle, in particular when Γ is itself a circle. In the cases where Γ is either a three multiple circle or a circle with three nodes, we show that the ss-equivalence relation on Coxeter elements of W is the same as the W-conjugacy relation. An explicit formula is given for the characteristic polynomial of a Coxeter element in the natural reflection representation of W when Γ is a circle. We also give the answers to some questions raised by Coleman and extend some results of Geck and Pfeiffer concerning conjugacy relation in Coxeter groups.

Coding Subset Shift by Subgroup Conjugacy

We present a Borel reduction from a subset shift equivalence relation of a countable group to a subgroup conjugacy relation of a free product. The technique gives a much shorter proof of an earlier result of Thomas and Velickovic. 1991 Mathematics Subject Classification 04A15, 20A10.

On the Complexity of the Isomorphism Relation for Finitely Generated Groups

Confirming a conjecture of G. Hjorth and A. Kechris (1996, Ann. Pure Appl. Logic82, 221–272) we prove that the isomorphism relation for finitely generated groups is a universal essentially countable Borel equivalence relation. We also prove the corresponding result for the conjugacy relation for subgroups of the free group F2 on two generators.

Bounds on the overall properties of composites with debonded frictionless interfaces

In the present work the homogenization problem of periodic composites with nonlinear hyperelastic constituents and debonded frictionless interfaces is formulated as a minimum problem of the free- or complementary-energy of the unit cell of the composite over the convex set of the admissible strain or stress fields, respectively. Because of the unilateral contact condition at the debonded interfaces, the overall constitutive behavior of the composite turns out to be nonlinear even when the constituents are linearly hyperelastic. In this particular case, the homogenized free- and complementary-energy density functionals of the composite are positively homogeneous of degree two, i.e. from a mechanical point of view, the overall constitutive behavior of the composite is nonlinear conewise multimodular. Approximations of the proposed variational principles, suitable for numerical computations, are built up by adopting the finite-element method. In this way, rigorous upper bounds on the homogenized free- and complementary-energy density functionals are obtained. Then, lower bounds are derived by using the conjugacy relation between the homogenized energy density functionals.

Plethysm for the non-compact group and newS-function identities

Methods of computing plethysms of the fundamental unitary irreducible representations of the non-compact symplectic group Sp(2n,R) are considered. Complete results are given for the symmetrized second powers. A number of new S-function identities are reported. The stability properties of the Sp(2n,R) plethysms are noted as well as a remarkable conjugacy relation. The application of the plethysms to N-particles in an isotropic harmonic oscillator is briefly outlined.

A novel interpretation of the two-dimensional discrete Hartley transform

In this paper, a novel view of the two-dimensional discrete Hartley transform (2-D DHT) is proposed. We show that the 2-D DHT can be obtained by projecting the two-dimensional discrete Fourier transform (2-D DFT) from the extension field to the basefield. The conjugacy relation and the trace function are applied to perform the projection operator. It is quite different from the traditional treatment of the 2-D DHT, which is trigonometric decomposition based.

CONJUGACY AND TRANSPOSITION FOR INVERSE MONOID PRESENTATIONS

The transposition problem is shown to be decidable for inverse monoid presentations of the form Inv <X; e=1>, where e is a Dyck word on the alphabet X∪X−1. If e is a, restricted Dyck word, the left conjugacy problem is solved as well, and the (left) conjugacy relation is shown to be the union of the transitive closure of the transposition relation with the universal relation on the semilattice of idempotents, a result which does not hold for a general Dyck word e.

Cross-Ratios Over Local Alternative Rings

It is shown that the conjugacy relation is an equivalence relation on the alternative field A and on the local alternative ring R = A + Aε of dual numbers over A, but on no other alternative ring of a certain class. On the projective lines over A and R the cross-ratio of four points is defined as a conjugacy class. Its elementary properties and applications to chain geometries are investigated.

A topological classification of the periodic orbits of the Hénon family

Since the Henon map,\(H_{a,b} \left[ \begin{array}{l} x \\ y \\ \end{array} \right] = \left[ \begin{array}{l} y + 1 - ax^2 \\ bx \\ \end{array} \right]\), is a diffeomorphism onR2 whenb∈0, we can regard the periodic orbit ofHa, b as a “braid”. It is shown that two homeomorphisms on a disk are isotopic, preserving their periodic orbits, if and only if the corresponding braids are conjugate with each other (”r-conjugate”, when they are orientation reversing). Being motivated by the global bifurcation structure on the 2-parameter space of the Henon family, we consider a question: what kind of relation exists, when we classify the periodic orbits of the Henon family using such isotopy? By the above fact, we investigate the conjugacy relation between corresponding braids. A special relation pattern in a certain class of periodic orbits is obtained. This pattern has a self-similar like structure related to a hyperbolic set of period 2.