Fixed points of endomorphisms over special confluent rewriting systems

Cayley graphs of monoids defined through special confluent rewriting systems are known to be hyperbolic metric spaces which admit a compact completion given by irreducible finite and infinite words. In this paper, we prove that the fixed point submonoids for endomorphisms of these monoids which are boundary injective (or have bounded length decrease) are rational, with similar results holding for infinite fixed points. Decidability of these properties is proved, and constructibility is proved for the case of bounded length decrease. These results are applied to free products of cyclic groups, providing a new generalization for the case of infinite fixed points.     

Fixed points of endomorphisms of trace monoids

It is proved that the fixed point submonoid and the periodic point submonoid of a trace monoid endomorphism are always finitely generated. If the dependence alphabet is a transitive forest, it is proved that the set of regular fixed points of the (Scott) continuous extension of an endomorphism to real traces is Ω-rational for every endomorphism if and only if the monoid is a free product of free commutative monoids.     

不动点与平衡点

介绍Brouwer不动点定理、Kakutani不动点定理与数理经济学中平衡点和博弈论中Nash平衡点存在性定理的等价性结果.     

Periodic points of endomorphisms on solenoids and related groups

This paper investigates the problem of finding the possiblesequences of periodic point counts for endomorphisms of solenoids.For an ergodic epimorphism of a solenoid, a closed formula isgiven that expresses the number of points of any given periodin terms of sets of places of finitely many algebraic numberfields and distinguished elements of those fields. The resultextends to more general epimorphisms of compact abelian groups.     

Existence of periodic points of endomorphisms of a circle

A theorem of Block and Franke is improved on the existence of periodic points for a map of a circle to itself and a proof which seems more understandable is given. Project supported by the National Natural Science Foundation of China (Grant No. 19531070).     

Homological invariants of modules over contracting endomorphisms

It is proved that when R is a local ring of positive characteristic, \({\phi\colon R{\to} R}\) is its Frobenius endomorphism, and some non-zero finite R-module has finite flat dimension or finite injective dimension for the R-module structure induced through \({\phi}\) , then R is regular. This broad generalization of Kunz’s characterization of regularity in positive characteristic is deduced from a theorem concerning a local ring R with residue field of k of arbitrary characteristic: If \({\phi}\) is a contracting endomorphism of R, then the Betti numbers and the Bass numbers over \({\phi}\) of any non-zero finitely generated R-module grow at the same rate, on an exponential scale, as the Betti numbers of k over R.     

Algorithms for confluent Vandermonde systems

Two compact algorithms are developed for solving systems of linear equationsV x=b andV ^T a=f, whereV=V( ₀, ₁, ..., _n ) is a confluent Vandermonde matrix of Hermite type. The solution is obtained by one forward and one backward vector recursion, starting with the right hand side. The total amount of storage is only 2n. The number of arithmetic operations needed isO(n ²) and compares favourably with other proposed methods.     

Fixed points vs. coupled fixed points

In this paper, we will show some connections between fixed point and coupled fixed point problems in Banach and metric spaces.     

Fixed points and matching points in partitions

The Ramanujan Journal - We study fixed points in integer partitions viewed, respectively, as weakly increasing or weakly decreasing structures. A fixed point is a point with value i in position i....     

Fixed points and products

We prove that if the finite ordered setsP andX have the fixed point property then so too doesP×X.Supported by NSERC Operating Grant 41702.     

Free operated monoids and rewriting systems

The construction of bases for quotients is an important problem. In this paper, applying the method of rewriting systems, we give a unified approach to construct sections—an alternative name for bases in semigroup theory—for quotients of free operated monoids. As applications, we capture sections of free \(*\)-monoids and free groups, respectively.     

Fixed points of non-Hamiltonian symplectomorphisms

In this article we study a version of the Arnold conjecture for symplectic maps that are not Hamiltonian. That is, we give a lower bound for the number of fixed points such a map must have. We achieve the result for symplectic maps with sufficiently small Calabi invariant.     

Fixed points of asymptotic contractions

Let be a contractive gauge function in the sense that φ is continuous, φ(s)<s for s>0, and if f:M→M satisfies d(f(x),f(y))?φ(d(x,y)) for all x,y in a complete metric space (M,d), then f always has a unique fixed point. It is proved that if T:M→M satisfies

     

Fixed points of approximable maps

We present a simple proof of the Leray-Schauder type theorem for approximable multimaps given recently by Ben-El-Mechaiekh and Idzik. We apply this theorem to obtain a Schaefer type theorem, the Birkhoff-Kellogg type theorems, a Penot type theorem for non-self-maps, and quasi-variational inequalities, all related to compact closed approximable maps.