Periodic points of symmetric product mappings
Periodic problems for ODEs via multivalued Poincaré operators
We shall consider periodic problems for ordinary differential equations of the form where satisfies suitable assumptions. To study the above problem we shall follow an approach based on the topological degree theory. Roughly speaking, if on some ball of , the topological degree of, associated to (), multivalued Poincaré operator turns out to be different from zero, then problem () has solutions. Next by using the multivalued version of the classical Liapunov-Krasnoselskǐ guiding potential...
Periodic segments and Nielsen numbers
We prove that the Poincaré map has at least fixed points (whose trajectories are contained inside the segment W) where the homeomorphism is given by the segment W.
Poincaré and domain invariance theorem
Properties of fixed point set of a multivalued map.
Ranges of -homogeneous operators and their perturbations
Realization of fixed point sets of relative maps
Given a relative map f: (X,A) → (X,A) on a pair (X,A) of compact polyhedra and a closed subset Y of X, we shall give some criteria for Y to be the fixed point set of some map relatively homotopic to f.
Reducing the number of fixed points of some homeomorphisms on nonprime 3-manifolds.
Reducing the number of periodic points in the smooth homotopy class of a self-map of a simply-connected manifold with periodic sequence of Lefschetz numbers
Let f be a smooth self-map of an m-dimensional (m ≥ 4) closed connected and simply-connected manifold such that the sequence of the Lefschetz numbers of its iterations is periodic. For a fixed natural r we wish to minimize, in the smooth homotopy class, the number of periodic points with periods less than or equal to r. The resulting number is given by a topological invariant J[f] which is defined in combinatorial terms and is constant for all sufficiently large r. We compute J[f] for self-maps...
Reidemeister conjugacy for finitely generated free fundamental groups
Let X be a space with the homotopy type of a bouquet of k circles, and let f: X → X be a map. In certain cases, algebraic techniques can be used to calculate N(f), the Nielsen number of f, which is a homotopy invariant lower bound on the number of fixed points for maps homotopic to f. Given two fixed points of f, x and y, and their corresponding group elements, and , the fixed points are Nielsen equivalent if and only if there is a solution z ∈ π₁(X) to the equation . The Nielsen number is the...
Reidemeister orbit sets
The Reidemeister orbit set plays a crucial role in the Nielsen type theory of periodic orbits, much as the Reidemeister set does in Nielsen fixed point theory. Extending Ferrario's work on Reidemeister sets, we obtain algebraic results such as addition formulae for Reidemeister orbit sets. Similar formulae for Nielsen type essential orbit numbers are also proved for fibre preserving maps.
Relative Borsuk-Ulam Theorems for Spaces with a Free ℤ₂-action
Let (X,A) be a pair of topological spaces, T : X → X a free involution and A a T-invariant subset of X. In this context, a question that naturally arises is whether or not all continuous maps have a T-coincidence point, that is, a point x ∈ X with f(x) = f(T(x)). In this paper, we obtain results of this nature under cohomological conditions on the spaces A and X.
Repulsive Fixed Points of Multivalued Transformations and the Fixed Point Index.
Rétractes Absolus de Voisinage Algébriques
2000 Mathematics Subject Classification: 54C55, 54H25, 55M20.We introduce the class of algebraic ANRs. It is defined by replacing continuous maps by chain mappings in Lefschetz’s characterization of ANRs. To a large extent, the theory of algebraic ANRs parallels the classical theory of ANRs. Every ANR is an algebraic ANR, but the class of algebraic ANRs is much larger; the most striking difference between these classes is that every locally equiconnected metrisable space is an algebraic ANR, whereas...
Revisiting Cauty's proof of the Schauder conjecture.
Rigidity Properties of Compact Lie Groups Modulo Maximal Tori.
Roots of mappings from manifolds.
Sandwich type theorems
Some applications of Coincidence theory.
Some elementary applications of topological degree.