Essential extension type maps for general classes of maps

This paper discusses admissible (and more general) maps between topological spaces. We show that if F is H-essential and F ≅ G then G has a fixed point.     

Coincidence theory for multivalued maps satisfying compactness conditions on countable sets

ABSTRACT

We present a coincidence theory for set-valued maps which satisfy certain compactness-type conditions on countable sets. Our theory is based on fixed point results for compositions of set-valued self maps.     

Entropy and periodic points for transitive maps

The aim of this paper is to investigate the connection between transitivity, density of the set of periodic points and topological entropy for low dimensional continuous maps. The paper deals with this problem in the case of the -star and the circle among the one-dimensional spaces and in some higher dimensional spaces. Particular attention is paid to triangular maps and to extensions of transitive maps to higher dimensions without increasing topological entropy.

     

Periodic points of nonexpansive maps and nonlinear generalizations of the Perron-Frobenius theory

Let and suppose that f : K ⁿ →K ⁿ is nonexpansive with respect to the l ₁-norm, , and satisfies f (0) = 0. Let P ₃(n) denote the (finite) set of positive integers p such that there exists f as above and a periodic point of f of minimal period p. For each n≥ 1 we use the concept of 'admissible arrays on n symbols' to define a set of positive integers Q(n) which is determined solely by number theoretical and combinatorial constraints and whose computation reduces to a finite problem. In a separate paper the sets Q(n) have been explicitly determined for 1 ≤n≤ 50, and we provide this information in an appendix. In our main theorem (Theorem 3.1) we prove that P ₃(n) = Q(n) for all n≥ 1. We also prove that the set Q(n) and the concept of admissible arrays are intimately connected to the set of periodic points of other classes of nonlinear maps, in particular to periodic points of maps g : D _g→D _g, where is a lattice (or lower semilattice) and g is a lattice (or lower semilattice) homomorphism.     

Fixed point theory for composite maps on almost dominating extension spaces

New fixed point results are presented forU _c ^k (X, X) maps in extension type spaces.     

Some applications of coincidence theory

Summary As a first application we compute the Lefschetz coincidence number of maps between manifolds whose rational singular cohomologyH* has a simple system of generators. For the second let be anH-manifold with multiplicationm. Define for ,m ₂ (x,x)=m(x,x) andm _k (x)=m(x,m _k−1 (x)), for allk>2. All roots of equationm _k (x)=m ₈ (x),k>s such thatm _k (x)=m ₃ (x) butm _i (x)≠m _j (x) for allk>i>j,s≥j≥0 andk−s does not dividei−j, split into a finite number of equivalence classes. We compute precisely the numbers of classes such that their members satisfy the above property. This article was processed by the author using the Springer-Verlag T_EX macro package 1991.     

Common fixed points for weakly compatible maps

The purpose of this paper is to prove a common fixed point theorem, from the class of compatible continuous maps to a larger class of maps having weakly compatible maps without appeal to continuity, which generalizes the results of Jungck [3], Fisher [1], Kang and Kim [8], Jachymski [2], and Rhoades [9].     

Over-rotation numbers for unimodal maps

We introduce twist unimodal maps of the interval and describe their structure. Sufficient conditions for the growth of over-rotation interval in families of maps are given.     

Deterministic and random coincidence point results for f-nonexpansive maps

Some deterministic and random coincidence theorems for f-nonexpansive maps are obtained. As applications, invariant approximation theorems are derived. Our results unify, extend and complement various known results existing in the literature.     

Conjugate maps and duality in multiobjective optimization

This paper considers duality in convex vector optimization. A vector optimization problem requires one to find all the efficient points of the attainable value set for given multiple objective functions. Embedding the primal problem into a family of perturbed problems enables one to define a dual problem in terms of the conjugate map of the perturbed objective function. Every solution of the stable primal problem is associated with a certain solution of the dual problem, which is characterized as a subgradient of the perturbed efficient value map. This pair of solutions also provides a saddle point of the Lagrangian map.     

Topological entropy of standard type monotone twist maps

We study invariant measures of families of monotone twist maps with periodic Morse potential . We prove that there exist a constant such that the topological entropy satisfies . In particular, for . We show also that there exist arbitrary large such that has nonuniformly hyperbolic invariant measures with positive metric entropy. For large , the measures are hyperbolic and, for a class of potentials which includes , the Lyapunov exponent of the map with invariant measure grows monotonically with .

     

A note on the periodic orbits and topological entropy of graph maps

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.

     

Dynamics of elementary maps of dendrites

The notion of elementary map of a dendrite into itself is introduced. Arithmetical relations between the periods of periodic points are given; the structure ofω-limit sets, sets of periodic and nonwandering points is described; the topological entropy of elementary maps is shown to be equal to 0. Examples are given illustrating the differences in the entropic properties of continuous maps of dendrites with a countable set of branch points and continuous maps of their retracts which are finite trees. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 183–195, February, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01755.     

Existence theorems for saddle points of vector-valued maps

In this paper, we prove some new existence theorems for loose saddle points and for saddle points of set-valued maps or vector-valued functions. These theorems generalize the corresponding results of Tanaka and those of Luc and Vargas via different proofs.The authors would like to thank two referees for their careful reading of the paper and helpful comments.     

Invariant measures for interval maps with critical points and singularities

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.     

On planar zero-diagonal and zero-trace iterated maps

We call an iterated map zero-diagonal, if it has a zero-diagonal Jacobi matrix for all x,y. Similarly, zero-trace iterated maps are the maps with zero-trace Jacobi matrix. In this paper, we present some of the geometric and algebraic properties of zero-diagonal planar maps. However, the main focus of this paper is the analysis of the zero-trace planar maps by linear transforming them to a zero-diagonal ones. Some sufficient conditions for the transformation are obtained. Stability for non-hyperbolic fixed points, two types of codim-2 bifurcations, and the local/global invariant manifolds for zero-diagonal and zero-trace maps are investigated.     

On the dynamics of composition of commuting interval maps

Let be two commuting continuous maps. We establish some results on the topological dynamic shared by both maps and state some conditions to get that the topological entropy of the composition f○g will be positive.     

Analytic smoothing of geometric maps with applications to KAM theory

We show that finitely differentiable diffeomorphisms which are either symplectic, volume-preserving, or contact can be approximated with analytic diffeomorphisms that are, respectively, symplectic, volume-preserving or contact. We prove that the approximating functions are uniformly bounded on some complex domains and that the rate of convergence, in Cr-norms, of the approximation can be estimated in terms of the size of such complex domains and the order of differentiability of the approximated function. As an application to this result, we give a proof of the existence, the local uniqueness and the bootstrap of regularity of KAM tori for finitely differentiable symplectic maps. The symplectic maps considered here are not assumed either to be written in action-angle variables or to be perturbations of integrable systems. Our main assumption is the existence of a finitely differentiable parameterization of a maximal dimensional torus that satisfies a non-degeneracy condition and that is approximately invariant. The symplectic, volume-preserving and contact forms are assumed to be analytic.