The order of coexistence of homoclinic trajectories for interval maps - Kuznietsov, M. V. (Ukrains’kyi Matematychnyi Zhurnal, 2019)
UDC 517.9А 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 formulated in that the ordering $1 \triangleright 3 \triangleright 5 \triangleright 7 \triangleright … \triangleright 2 · 1 \triangleright 2 · 3\triangleright 2 · 5 \triangleright … \triangleright 2^2 · 1 \triangleright 2^2 · 3 \triangleright 2^2 · 5 \triangleright … $ determines the coexistence of homoclinic trajectories of one-dimensional systems: If a one-dimensional dynamical system possesses an $n$-homoclinic trajectory, then it also has an $m$-homoclinic trajectory for each $m$ such that $ n \triangleright m .$ It is also proved that every one-dimensional dynamical system with a cycle of period $ n eq 2^i $ also possesses an $n$-homoclinic trajectory.