The order of coexistence of homoclinic trajectories for the maps of an interval

Abstract

A nonperiodic trajectory of a discrete dynamical system is called n-homoclinic if its α- and ω-limit sets coincide and form the same cycle of period n. We prove the statement that the ordering 1 ⊳ 3 ⊳ 5 ⊳ 7 ⊳ ... ⊳ 2 ⋅ 1 ⊳ 2 ⋅ 3 ⊳ 2 ⋅ 5 ⊳ ... ⊳ 22 ⋅ 1 ⊳ 22 ⋅ 3 ⊳ 22 ⋅ 5 ⊳ ... determines the coexistence of homoclinic trajectories of one-dimensional systems in a sense that if a onedimensional dynamical system possesses an n-homoclinic trajectory, then it also has an m-homoclinic trajectory for each m such that n ⊳ m. It is also proved that every one-dimensional dynamical system with a cycle of period n ≠ 2i also possesses an n-homoclinic trajectory.