For a compact metric space ( X, d) and a. ( 0, 1), letLipa( X) be the linear space of all complex- valued functions f on X satisfying L( f) = supx= y | f ( x) - f ( y)| da( x, y) < 8 and lipa( X) be the subspace of Lipa( X) consisting of functions f with lim f ( x)- f ( y) da( x, y) = 0 as d( x, y). 0. In this paper, we give a characterization of a bijective map T : lipa( X) -. lipa( Y), not necessarily linear, which is an isometry with respect to the Holder seminorm L( center dot). It is shown that there exist K0 > 0, a surjective map : Y -. X with da( y, z) = K0 da(( y),( z)) for all y, z. Y, and a function : lipa( X) -. C ( which is linear or real- linear if T is so) such that either T f ( y) = T 0( y) + t K0 f (( y)) +( f) ( f. lipa ( X), y. Y) T f ( y) = T 0( y) + t K0 f ( ( y)) + ( f) ( f. lipa ( X), y. Y), where t = ei. for some.. [ 0, p).