Small retractions of smooth dendroids onto trees

On genericity of shadowing in one dimension

“…Indeed, this question seems intricately related to the open problem concerning the characterization of dendroids mentioned in Section 2. As such, answers to this question are naturally correlated with analogous questions for dendroids [11,13,14,21,22].…”

“…for all x ∈ T , π −1 (x) has diameter less than ). While some progress has been made in demonstrating the equivalence of these definitions [13,14], it remains an open problem [30]. It is from this latter notion of dendroid that we draw our analogy.…”

On genericity of shadowing in one dimension

“…Indeed, this question seems intricately related to the open problem concerning the characterization of dendroids mentioned in Section 2. As such, answers to this question are naturally correlated with analogous questions for dendroids [10,12,13,19,20].…”

“…for all x ∈ T , π −1 (x) has diameter less than ǫ). While some progress has been made in demonstrating the equivalence of these definitions [12,13], it remains an open problem [28]. It is from this latter notion of dendroid that we draw our analogy.…”

Cylinders over λ-dendroids have the fixed point property

“…Therefore, by a standard argument of fixed point theory, the segment I can be replaced by any λ-dendroid which can be approximated from within by dendrites, for instance, it can be replaced by any smooth dendroid (cf. Fugate [4]). …”