Assume that every [[PHI].sub.n] and [[PHI].sup.-1.sub.n], n = 1,..., N, are sufficiently smooth ([C.sup.1]

diffeomorphisms) and that there exist constants 0 < c < C such that c [less than or equal to] I det [mathematical expression not reproducible], where [mathematical expression not reproducible] is the Jacobian matrix of [mathematical expression not reproducible]

Wan, "Computation of the stability condition for the Hopf bifurcation of

diffeomorphisms on [R.sub.2]," SIAM Journal onApplied Mathematics, vol.

Mather, "More Denjoy minimal sets for area preserving

diffeomorphisms," Commentarii Mathematici Helvetici, vol.

In all these cases, the transformation is singular at the singularity, so it is not a

diffeomorphism. The atlas, the differential structure, is changed, and in the new atlas, with its new differential structure, the

diffeomorphisms preserve, of course, the semiregularity of the metric.

All homeomorphisms and

diffeomorphisms considered are orientation-preserving.

Among their topics are on deformations and rigidity of the Godbillion Vey class, Thurston's h-principle for two-dimensional foliations of co-dimension greater than one, a few open problems on characteristic classes of foliations, on the global estimate of the Diederich-Fornaess index of Levi-flat real hyperspaces, homomorphisms on groups of volume-preserving

diffeomorphisms via fundamental groups, and rotation number and lifts of a Fuchsian action of the modular group on the circle.

The equivalence [theta] provides a family of

diffeomorphismsThen the group G = O(p + 1, q + 1) acts as conformal

diffeomorphisms on [S.sup.p] x [S.sup.q], and also on its quotient space X = ([S.sup.p] x [S.sup.q])/[Z.sub.2] by identifying the direct product of antipodal points.

However, diffeomorphic transformation is required for a geodesic connecting two images, F and M, in the space of diffeomorphic transformations, and the computational and memory costs are significant because of the dense-in-time velocity field calculations and requisite reintegration of the

diffeomorphisms after each iterative update [12].

Chazottes, Equilibrium States and the Ergodic Theory of Anosov

Diffeomorphisms, vol.

are

diffeomorphisms [R.sup.d] [right arrow] []0; 1[.sup.d] [subset] [[0; 1].sup.d] and we get