  We study the dynamic behavior at high energies of a chain of anharmonic
oscillators coupled at its ends to heat baths at possibly different
temperatures. In our setup, each oscillator is subject to a homogeneous
anharmonic pinning potential $V_1(q_i) =|q_i|^{2k}/2k$ and harmonic coupling
potentials $V_2(q_i- q_{i-1}) = (q_i- q_{i-1})^2/2$ between itself and its
nearest neighbors. We consider the case $k > 1$ when the pinning potential is
stronger then the coupling potential. At high energy, when a large fraction of
the energy is located in the bulk of the chain, breathers appear and block the
transport of energy through the system, thus slowing its convergence to
equilibrium.
  In such a regime, we obtain equations for an effective dynamics by averaging
out the fast oscillation of the breather. Using this representation and related
ideas, we can prove a number of results. When the chain is of length three and
$k> 3/2$ we show that there exists a unique invariant measure. If $k > 2$ we
further show that the system does not relax exponentially fast to this
equilibrium by demonstrating that zero is in the essential spectrum of the
generator of the dynamics. When the chain has five or more oscillators and $k>
3/2$ we show that the generator again has zero in its essential spectrum.
  In addition to these rigorous results, a theory is given for the rate of
decrease of the energy when it is concentrated in one of the oscillators
without dissipation. Numerical simulations are included which confirm the
theory.
