  An explicit relation between the dimensional loss ($\Delta D$), entropy
production and transport is established under thermal gradients, relating the
microscopic and macroscopic behaviors of the system. The extensivity of $\Delta
D$ in systems with bulk behavior follows from the relation. The maximum
Lyapunov exponents in thermal equilibrium and $\Delta D$ in non-equilibrium
depend on the choice of heat-baths, while their product is unique and
macroscopic. Finite size corrections are also computed and all results are
verified numerically.
