Chaotic attractors exhibit fascinating orbital structures in which complexity and order are locally interwoven in a nested manner. Recently it has turned out that their structures can be characterized by the generalized dimensions D(q),(-co<q<co) and the singularity spectra I(a) of the natural invariant measures.1),2) It has also turned out for two-dimensional maps that D(q) and I(a) are closely related to the q-weighted average A(q) of the coarse-grained local expansion rates A of nearby orbits along the local unstable manifold and their spectrum h(A) so that the structures of chaotic attractors can also be characterized by A(q) and h(A).3H) Indeed, it has been shown that the singular local structures at the bifurcation points of the band mergings and crises bring about discontinuous transitions of A(q) as q is varied, so that their different phases represent different local structures of chaotic attractors.)-9) These q-phase transitions are caused by linear parts in h(A) created by singular local structures. Such a linear part is produced by a collision of the chaotic attractor with an unstable periodic orbit which gives a new maximum Amax of A. In twodimensional maps, such as the Henon and dissipative standard maps, s1,1ch a collision is accompanied by the accumulation of homoclinic or heteroclinic tangency points to the periodic orbie),9)-1l) In this paper we shall formulate the scaling exponent a ) of the natural invariant measure at the periodic orbit and the slope qp= h'(A) of the linear part in terms of the eigenvalues of certain periodic orbits and the order z of the tangency. This will lead to a new kind of relations between two exponents a and A on the periodic orbits. For a chaotic orbit {Xm}, (m=O, 1, 2, ... ) generated by a twoodimensional dissipative map Xm+l=F(Xm), let ,h(Xm) be the local expansion rate of nearby orbits at Xm along the local unstable manifold, i.e., ,h(Xm)= 10gIDF(Xm)ul(Xm)1 with Ul(Xm) being the ,unit vector tangent to the local unstable manifold at X m,3) and, using the temporal coarse-graIning first introduced by Fujisaka,12) define the coarse-grained local expansion rate3H)