  This paper presents a new foundational approach to information theory based
on the concept of the information efficiency of a recursive function, which is
defined as the difference between the information in the input and the output.
The theory allows us to study planar representations of various infinite
domains. Dilation theory studies the information effects of recursive
operations in terms of topological deformations of the plane. I show that the
well-known class of finite sets of natural numbers behaves erratically under
such transformations. It is subject to phase transitions that in some cases
have a fractal nature. The class is \emph{semi-countable}: there is no
intrinsic information theory for this class and there are no efficient methods
for systematic search.
  There is a relation between the information efficiency of the function and
the time needed to compute it: a deterministic computational process can
destroy information in linear time, but it can only generate information at
logarithmic speed. Checking functions for problems in $NP$ are information
discarding. Consequently, when we try to solve a decision problem based on an
efficiently computable checking function, we need exponential time to
reconstruct the information destroyed by such a function. At the end of the
paper I sketch a systematic taxonomy for problems in $NP$.
