Dynamical systems of eternal inflation: A possible solution to the
  problems of entropy, measure, observables and initial conditions
  There are two main approaches to non-equlibrium statistical mechanics: one
using stochastic processes and the other using dynamical systems. To model the
dynamics during inflation one usually adopts a stochastic description, which is
known to suffer from serious conceptual problems. To overcome the problems
and/or to gain more insight, we develop a dynamical systems approach. A key
assumption that goes into analysis is the chaotic hypothesis, which is a
natural generalization of the ergodic hypothesis to non-Hamiltonian systems.
The unfamiliar feature for gravitational systems is that the local phase space
trajectories can either reproduce or escape due to the presence of cosmological
and black hole horizons. We argue that the effect of horizons can be studied
using dynamical systems and apply the so-called thermodynamic formalism to
derive the equilibrium (or Sinai-Ruelle-Bowen) measure given by a variational
principle. We show that the only physical measure is not the Liouville measure
(i.e. no entropy problem), but the equilibrium measure (i.e. no measure
problem) defined over local trajectories (i.e. no problem of observables) and
supported on only infinite trajectories (i.e. no problem of initial
conditions). Phenomenological aspects of the fluctuation theorem are discussed.
