In this paper, we prove that the ODE system, $ \dot{x}=sinz+cosy$ $ \dot{y}=sinx+cosz$ $ \dot{z}=siny+cosx$ whose right-hand side is the Arnold-Beltrami-Childress (ABC) flow with parameters A = B = C = 1, has periodic orbits on $(2 \pi T)^3$ with rotation vectors parallel to (1, 0, 0), (0, 1, 0), and (0, 0, 1). An application of this result is that the well-known G-equation model for turbulent combustion with this ABC flow on $R^3$ has a linear (i.e., maximal possible) flame speed enhancement rate as the amplitude of the flow grows.

We show that for any amenable group \Gamma and any Z\Gamma-module M of type FL with vanishing Euler characteristic, the entropy of the natural \Gamma-action on the Pontryagin dual of M is equal to the L^2-torsion of M. As a particular case, the entropy of the principal algebraic action associated with the module Z\Gamma / Z\Gamma f is equal to the logarithm of the Fuglede-Kadison determinant of f whenever f is a non-zero-divisor in Z\Gamma. This confirms a conjecture of Deninger. As a key step in the proof we provide a general Szeg\H{o}-type approximation theorem for the Fuglede-Kadison determinant on the group von Neumann algebra of an amenable group. As a consequence of the equality between L^2-torsion and entropy, we show that the L^2-torsion of a non-trivial amenable group with finite classifying space vanishes. This was conjectured by Lueck. Finally, we establish a Milnor-Turaev formula for the L^2-torsion of a finite \Delta-acyclic chain complex.

In this paper, we study time-periodic perturbation of classical systems with two degrees of freedom. A transition chain is established, by passing through small neighborhood of double resonant point, to connect any two cohomology classes corresponding to resonant frequencies. Applying the result to nearly integrable Hamiltonian systems with three degrees of freedom, one obtains a transition chain along which one is able to construct diffusion orbits suggested by Arnold in [A66].