Is space time? A spatiotemporal tiling of turbulence

We address the long standing problem of how to describe, by means
of discrete symbolic dynamics, the spatiotemporal chaos (or
turbulence) in spatially extended, strongly nonlinear field
theories.

One way to capture the essential features of turbulent motions is
offered by coupled map lattice models, in which the spacetime is
discretized, with the dynamics of small-scale spatial structures
modeled by maps attached to lattice sites. The discretization that
we study, the "spatiotemporal cat," has a remarkable feature that
its every solution is uniquely encoded by a linear transformation
from the corresponding finite alphabet symbol lattice.
A spatiotemporal window into system dynamics is provided by a
finite block of symbols, and the central question is to determine
the likelihood of a given block's occurrence. As spatiotemporal
states that share the same sub-blocks shadow each other
exponentially well within the corresponding spatiotemporal windows,
the dynamical zeta functions are now sums over spacetime tori,
rather than time-periodic orbits.

In the spatiotemporal formulation of turbulence there is no
evolution in time, there are only a repertoires of admissible
spatiotemporal patterns. In other words: throw away your
integrators, and look for guidance in clouds' repeating patterns.