Some remarks on topological horseshoes and applications

An investigation on topological methods proving chaotic dynamics is presented: the relationships between Kennedy, Koçak and Yorke’s “chaos lemma” and Medio, Pireddu and Zanolin’s “stretching along the paths” put in evidence a double way to prove that a discrete dynamical system is chaotic according to Block and Coppel’s definition of chaos.Particular relevance is given to non-injective discrete systems, such as Lotka–Volterra with Holling Type II, since they are strongly involved in semi-conjugacy.