Embedding flows and smooth conjugacy (Q1355904)

Let \(f\) be a diffeomorphism on a smooth manifold \(M\). A smooth flow \({f^t}\), \(t \in {\mathbb{R}}\) on \(M\) is called an embedding flow of \(f\) if \(f^1=f\). The corresponding vector field \(V(x)=\frac{\partial }{\partial t}f^t(x)|_{t=0}\) is called an embedding vector field of \(f\). The authors use the functional equations for embedding vector fields to obtain the existence and uniqueness for smooth embedding flows and vector fields of local and global 1-dimensional diffeomorphisms. As an application of embedding flows, in the last two sections, the smooth conjugacy of local and global 1-dimensional diffeomorphisms is studied. Recall that two diffeomorphisms \(f\), \(g\) are called \(C^s\) conjugate if there is a \(C^s\) diffeomorphism \(h\) such that \(h\circ f=g\circ h\). In this way, some new results about smooth classification of diffeomorphisms are obtained (cf. Theorem 4.1 and 4.3).