FIG. 1: Comparison of the total extended Hamiltonian versus time (normalized with respect to its value at t = 0) for the harmonic oscillator undergoing simple Bulgac-Kusnezov dynamics ( H $^{BK}$), NH controlled Bulgac-Kusnezov dynamics ( H $^{BKNH}$), and Bulgac-Kusnezov-Nos´ e-Hoover chain dy- namics ( H $^{BKNHC}$). Two curves have been displaced vertically for clarity. The time-reversible measure-preserving algorithms developed in this paper conserve the extended Hamiltonian to high accuracy in all three cases. 30  1  1.01  1.02  1.03  1.04  0  500  1000 1500 2000 2500 H t H BK H BKNH H BKNHC