  Holographic functional methods are introduced as probes of discrete
time-stepped maps that lead to chaotic behavior. The methods provide continuous
time interpolation between the time steps, thereby revealing the maps to be
quasi-Hamiltonian systems underlain by novel potentials that govern the motion
of a perceived point particle. Between turning points, the particle is strictly
driven by Hamiltonian dynamics, but at each encounter with a turning point the
potential changes abruptly, loosely analogous to the switchbacks on a mountain
road. A sequence of successively deepening switchback potentials explains, in
physical terms, the frequency cascade and trajectory folding that occur on the
particular route to chaos revealed by the logistic map.
