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Suppose $f\colon M\to M$ on a compact manifold. Let $m$ be a natural number. One of the most important questions in the topological theory of periodic points is whether the Nielsen-Jiang periodic number $NF_m(f)$ is a sharp lower bound on $\# \text{\rm Fix}(g^m)$ over all $g\sim f$. This question has a positive answer if $\text{\rm dim} M\geq 3$ but in general a negative answer for self maps of compact surfaces. However, we show the answer to be positive when $M={\mathbb K}$ is the Klein bottle. As a consequence, we reconfirm a result of Llibre and compute the set $\text{\rm HPer} (f)$ of homotopy minimal periods on the Klein bottle.

We give a formula for the Nielsen coincidence number of a pair of maps from a surface into the 2-torus. We also show that a similar formula gives the lower bound of the number of connected components of the coincidence set of a pair of maps between two tori (possibly of different dimensions). Mathematics Subject Classification (1991): 55M20 Keywords: coincidence index, coincidence Nielsen number, fixed points and coincidences tori, Nielsen Quaestiones Mathematicae 24(2) 2001, 217-223

The relative Lefschetz and Nielsen fixed-point theorems are generalized for compact absorbing contractions on ANR-spaces and nilmanifolds. The nontrivial Lefschetz number implies the existence of a fixed-point in the closure of the complementary domain. The relative Nielsen numbers improve the lower estimate of the number of coincidences on the total space or indicate the location of fixed-points on the complement. Nontrivial applications of these topological invariants (under homotopy) are given to admissible semi-flows and differential inclusions.

Shub and Sullivan noticed that there is a large gap between the least number of periodic points, taken over the family of continuous maps homotopic to a given self-map f and the analogous number taken over the family of smooth maps homotopic to f . In this paper we show that in the case of self-maps of solvmanifolds the minimal numbers of n-periodic points, in both categories, are equal.

Let M be a closed smooth connected and simply connected manifold of dimension m at least 3, and let r be a fixed natural number. The topological invariant \({{D^m_r} [f]}\) , defined by the authors in [Forum Math. 21 (2009), 491–509], is equal to the minimal number of r-periodic points in the smooth homotopy class of f, a given self-map of M. In this paper, we present a general combinatorial scheme of computing \({{D^m_r} [f]}\) for arbitrary dimension m ≥ 4. Using this approach we calculate the invariant in case r is a product of different odd primes. We also obtain an estimate for \({{D^m_r} [f]}\) from below and above for some other natural numbers r.

A natural number m is called the homotopyminimal period of a map f : X → X if it is a minimal period for everymap g homotopic to f. The set HPer (f ) of all minimal homotopy periods is an invariant of the dynamics of f which is the same for a small perturbation of f. In this paper we give a complete description of the sets of homotopyminimal periods of selfmaps of nonabelian three dimensional nilmanifold which is a counterpart of the corresponding characterization for three dimensional torus proved byJiang and Llibre. As a corollary we show that if 2 ∈ HPer (f ) then HPer (f )= N for such a map.

The Lefschetz and the Nielsen periodic point theorems are developed for compact absorbing contractions on ANRs. These results are reformulated in terms of discrete multivalued semi-dynamical systems. They are also applied to Caratheodory differential inclusions on tori for obtaining the existence and multiplicity results for boundary value problems.