The order of coexistence of homoclinic trajectories for interval maps

Abstract

UDC 517.9
А nonperiodic trajectory of a discrete dynamical system is called $n$-homoclinic if its $\alpha$- and $\omega$-limit sets coincide and form the same cycle of period $n.$ We prove the statement formulated in that the ordering $1 \triangleright 3 \triangleright 5 \triangleright 7 \triangleright \ldots \triangleright 2 \cdot 1 \triangleright 2 \cdot 3\triangleright 2 \cdot 5 \triangleright \ldots \triangleright 2^2 \cdot 1 \triangleright 2^2 \cdot 3 \triangleright 2^2 \cdot 5 \triangleright \ldots$ 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 $n \neq 2^i$ also possesses an $n$-homoclinic trajectory.