In a multifractal paradigm of motion, Shannon&rsquo
s information functionality of a minimization principle induces multifractal&ndash
type Newtonian behaviors. The analysis of these behaviors through motion geodesics shows the fact that the center of the Newtonian-type multifractal force is different from the center of the multifractal trajectory. The measure of this difference is given by the eccentricity, which depends on the initial conditions. In such a context, the eccentricities&rsquo
geometry becomes, through the Cayley&ndash
Klein metric principle, the Lobachevsky plane geometry. Then, harmonic mappings between the usual space and the Lobachevsky plane in a Poincaré
metric can become operational, a situation in which the Ernst potential of general relativity acquires a classical nature. Moreover, the Newtonian-type multifractal dynamics, perceived and described in a multifractal paradigm of motion, becomes a local manifestation of the gravitational field of general relativity.