Homotopy minimal periods for nilmanifold maps

Abstract

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

Homotopy minimal periods for maps of three dimensional nilmanifolds

“…Recently the authors extended the main theorem of [12] onto the case of a map f of an arbitrary compact nilmanifold X with the similar qualitative statement ( [10] Thm. A).…”

Averaging Formula for Nielsen Numbers of Maps on Infra-Solvmanifolds of Type (R)

“…Note that Λ is a lattice of S. Then, the induced homomorphism ϕ : π → π restricts to a homomorphism ϕ ′ = ϕ| Λ : Λ → Λ, which extends to an endomorphism of the Lie group S in a unique way. See [8,Theorem 2.2]. The differential of this map is an endomorphism of the Lie algebra, f * : S → S.…”

“…We recall from [6], [8], [12] some definitions about solvable Lie groups and give some basic properties which are necessary for our discussion. A connected solvable Lie group S is called of type (NR) (for "no roots") [12] if the eigenvalues of Ad(x) : S → S are always either equal to 1 or else they are not roots of unity.…”

Nielsen type numbers and homotopy minimal periods for maps on 3–solvmanifolds

“…Since the homotopy minimal period is preserved under a small perturbation of a self-map f on a manifold X , we can say that the set HPer.f / of homotopy minimal periods of f describes the rigid part of the dynamics of f . A complete description of the set of homotopy minimal periods of all self-maps was obtained on 3-nilmanifolds by Jezierski and Marzantowicz [6], and Lee and Zhao [10] and on 3-solvmanifolds by Jezierski, Kȩdra and Marzantowicz [5].…”

“…In [5] (see also Kim, Lee and Yoo [9]), Jezierski, Kȩdra and Marzantowicz carried out a further study of homotopy minimal periods for maps on 3-solvmanifolds. One of motivations of the present work is to correct Proposition 4.3 of [5], in which the entries of the matrices are claimed to be integers. However this is not true; see Example 3.6.…”