  We study the asymptotic behavior of the homotopy groups of simply connected
finite $p$-local complexes, and define a space to be locally hyperbolic if its
homotopy groups have exponential growth. Under some certain conditions related
to the functorial decomposition of loop suspension, we prove that the suspended
finite complexes are locally hyperbolic if suitable but accessible information
of the homotopy groups is known. In particular, we prove that Moore spaces are
locally hyperbolic, and other candidates are also given.
