What is this one parameter subgroup of the Lie group Diff(R)?

Question

I am interested in the time reparametrization. t is replaced by $\tau$. See how the infinite dimentional lie group Diff(R) is introcuced. t is a real number of R. and $\tau$ a function on R. To have them on an equal footing we will identify the variable t to the function t -> t (the identity on R). We considere $\tau (t(.))$ it is the composition of $\tau$ and Id. the composition is the law group. the identity is a bijection on R. the composition is associative , has a neutral element and we can choose the elements of the group so that each element has a symmetric element. it is the case with the Lie group Diff(R). the function sinh() belongs to it and has a reciprocal element asinh such that asin (sinh(x)) = id(x) = x. i read that for each $g \in Diff(R)$ there is a unique subgroup containing Id and g. What are the elements of this subgroup? Of course i have in mind to do the same thing for other functions than sinh, but it would be a concrete firs step in the time reparametrization machinery.

Answer 1

I found a useful definition of a one parameter subgroup of a Lie group G. It is a map f from R to G so that f(s)*f(t) = f(s+t) for all s and t. here * is the law in G. We have f(0)*f(t) = f(t) so f(0) is the identity. f(-t)*f(0) = Id. so f(-t) is the inverse of f(t). In Diff(R) the elements are differentiable bijective maps on R. The binary law * is the composition law $\circ$ .The inverse with this law is the reciprocal function. I choose an element in Diff(R) here the function sinh, and the aim is to build the one parameter subgroup that maps 0 to Id and 1 to sinh. I will use the one parameter definition to define f(s) first on Z tben on Q and next on R. As $f(1)\circ f(1)$ = sinh(sinh) = f(2) we get the value on 2 then on integer values of t. i introduce the notation $sinh^{(n)}$ for sinh(sinh(...))) n times and in the same way the notation with n negative by using arsinh. we can consider the function g so that $g^{(q)} = sinh$ so $sinh^{(1/q)}$ is well defined and $sinh^{(p/q)}$ also. As real numbers are limits of fractions we get a way to define $sinh^{(r)}$ for all real numbers r. for any z in Diff(R) the one parameter mapping 0 to Id and 1 to z may be constructed in the same way. a usual notation is exp(tz)