Forcing Relations for Homoclinic Orbits of the Reversible Horseshoe Map

Forcing relations for homoclinic orbits of the reversible horseshoe map are studied. First, a new concept named the homoclinic orbit listed in decreasing order is introduced. The linear ordering holds for these homoclinic orbits. Next, we prove that the linear ordering holds for the homoclinic orbits whose inner core length is less than or equal to seven. A collection including the homoclinic orbits and heteroclinic orbit, and another collection including the homoclinic orbit, periodic orbits, and heteroclinic orbit are introduced. For these collections, the distributary structured forcing relations hold. Finally, for a given length of inner cores, we determine the homoclinic orbit that forces all the other homoclinic orbits, and the one that is forced by all the other homoclinic orbits.