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A natural number m is called the homotopy minimal period of a map f : X → X if it is a minimal period for every map g homotopic to f. In this paper we show that the complete description of the sets of homotopy minimal periods of a torus map given by Jiang and Llibre extends to the case of a map of compact nilmanifold. The proof follows the approach of Jiang and Llibre and uses the Nielsen theory. The main geometric ingredient is a theorem on cancelling m-periodic points of a local homeomorphism. For a map of nilmanifold the general case reduces to it by a homotopy argument. (2000): 55M20, 57N05, 54H25 Mathematics Subject Classification

Abstract. We present a new approach to an equivariant version of Farber's topological complexity called invariant topological complexity. It seems that the presented approach is more adequate for the analysis of impact of a symmetry on a motion planning algorithm than the one introduced and studied by Colman and Grant. We show many bounds for the invariant topological complexity comparing it with already known invariants and prove that in the case of a free action it is equal to the topological complexity of the orbit space. We define the Whitehead version of it.

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We give a complete description of the behaviour of the sequence of displacements η n (z) = Φ n (x)−Φ n−1 (x) mod 1, z = exp(2πix), along a trajectory {ϕ n (z)}, where ϕ is an orientation preserving circle homeomorphism and Φ : R → R its lift. If the rotation number ̺(ϕ) = p q is rational then η n (z) is asymptotically periodic with semi-period q. This convergence to a periodic sequence is uniform in z if we admit that some points are iterated backward instead of taking only forward iterations for all z. If ̺(ϕ) / ∈ Q then the values of η n (z) are dense in a set which depends on the map γ (semi-)conjugating ϕ with the rotation by ̺(ϕ) and which is the support of the displacements distribution. We provide an effective formula for the displacement distribution if ϕ is C 1 -diffeomorphism and show approximation of the displacement distribution by sample displacements measured along a trajectory of any other circle homeomorphism which is sufficiently close to the initial homeomorphism ϕ. Finally, we prove that even for the irrational rotation number ̺ the displacement sequence exhibits some regularity properties.

In 1974 Michael Shub asked the following question [29] : When is the topological entropy of a continuous mapping of a compact manifold into itself is estimated from below by the logarithm of the spectral radius of the linear mapping induced in the cohomologies with real coefficients? This estimate has been called the Entropy Conjecture (EC). In 1977 the second author and Micha l Misiurewicz proved [23] that EC holds for all continuous mappings of tori. Here we prove EC for all continuous mappings of compact nilmanifolds. Also generalizations for maps of some solvmanifolds and another proof via Lefschetz and Nielsen numbers, under the assumption the map is not homotopic to a fixed points free map, are provided.

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