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

Crystallographic groups of Sol

“…This implies that A is a hyperbolic matrix; it has different real eigenvalues: one is greater than 1 and the other is less than 1. Furthermore, neither ℓ 12 nor ℓ 21 vanishes, see for example .…”

“…Recall for example from [, Lemma 2.1] that the set of all lattices of Sol is the same as the set of all hyperbolic 2 × 2‐integer matrices with trace >2. We can refine this relation further.…”

“…In particular, we observe that the affine group of Sol imbeds into the affine group . In the following two sections we review the lattices of Sol mainly from and then we review structure theorems for the so‐called solvable space forms problem of type (R). We study the structure of SC‐ groups modeled on Sol (see the next section for its definition) and find out in particular intrinsic actions induced from SC‐groups.…”

Minimal number of periodic points of smooth boundary-preserving self-maps of simply-connected manifolds

“…Finding minimal number of r -periodic points in the homotopy class (for a fixed r ) is an important challenge in modern homotopy periodic point theory, with an increasing number of valuable results obtained in the last decade in many particular cases [1][2][3][4][5][6][7].…”

Combinatorial scheme of finding minimal number of periodic points for smooth self-maps of simply connected manifolds

“…Graff and J. Jezierski JFPTA introduced by Jiang [18] in 1983 as a lower bound for the number of r-periodic points in the homotopy class, and it was proved in 2006 that NF r (f ) is the best such lower bound, i.e., it is equal to the minimum in (1.1); see [16]. During the last decade, NF r (f ) was computed in many particular cases; see [13,14,15,19,20,21]. Recent investigations of the authors showed that the smooth and continuous theories do not coincide.…”