Minimal set of periods for continuous self-maps of the eight space

Abstract

AbstractLet $G_{k}$ G k be a bouquet of circles, i.e., the quotient space of the interval $[0,k]$ [ 0 , k ] obtained by identifying all points of integer coordinates to a single point, called the branching point of $G_{k}$ G k . Thus, $G_{1}$ G 1 is the circle, $G_{2}$ G 2 is the eight space, and $G_{3}$ G 3 is the trefoil. Let $f: G_{k} \to G_{k}$ f : G k → G k be a continuous map such that, for $k>1$ k > 1 , the branching point is fixed.If $\operatorname{Per}(f)$ Per ( f ) denotes the set of periods of f, the minimal set of periods of f, denoted by $\operatorname{MPer}(f)$ MPer ( f ) , is defined as $\bigcap_{g\simeq f} \operatorname{Per}(g)$ ⋂ g ≃ f Per ( g ) where $g:G_{k}\to G_{k}$ g : G k → G k is homological to f.The sets $\operatorname{MPer}(f)$ MPer ( f ) are well known for circle maps. Here, we classify all the sets $\operatorname{MPer}(f)$ MPer ( f ) for self-maps of the eight space.