Scale calculus problems

Scale calculus was developed to make reparametrization actions on function spaces smooth. All the key issues and ideas can already be seen at the example of $S^{1}=\mathbb{R} /\mathbb{Z }$ acting on a space of nonconstant functions on $S^{1}$, for example the shift action

$\sigma :S^{1}\times {\mathcal {C}}_{{{\rm {id}}}}^{1}\;\to \;{\mathcal {C}}_{{{\rm {id}}}}^{1},\quad (\tau ,u)\mapsto u(\tau +\cdot )$

${\text{on}}\qquad \qquad \qquad \qquad {\mathcal {C}}_{{{\rm {id}}}}^{1}:=\{u:S^{1}\to S^{1}\;|\;u\in {\mathcal {C}}^{1}\;{\text{and homotopic to the identity}}\}.\qquad \qquad \qquad \qquad .$

The purpose of the following exercises is to give the quotient of S^1-valued functions homotopic to the identity, modulo reparametrization by shifts, ${\mathcal {B}}:={\mathcal {C}}_{{{\rm {id}}}}^{1}/S^{1}$ the structure of a scale-smooth manifold ... and to see that classical Banach space topologies fail at giving it a differentiable Banach manifold structure.

differentiability of shift map

Compute the directional derivatives of the shift map $\sigma$, first at $(\tau ,u)=(0,0)$, then at $(\tau ,u)\in S^{1}\times {\mathcal {C}}_{{{\rm {id}}}}^{2}$.

Next, understand how to identify $T{\mathcal {C}}_{{{\rm {id}}}}^{1}\simeq {\mathcal {C}}^{1}(S^{1},\mathbb{R} )$, and write down a conjectural formula for the differential $d\sigma :\mathbb{R} \times {\mathcal {C}}^{1}(S^{1},\mathbb{R} )\to {\mathcal {C}}^{1}(S^{1},\mathbb{R} )$.

Then recall the definition of differentiability in terms of uniform linear approximation and see whether it can be satisfied at any base point $(\tau ,u)$.

local charts

Verify that ${\mathcal {B}}$ has local Banach manifold charts modeled on $E:=\{\xi \in {\mathcal {C}}^{1}(S^{1},\mathbb{R} )\,|\,\xi (0)=0\}$. To set up a chart near a given point, pick a representative $[u_{0}]\in {\mathcal {B}}$ with ${\dot u}_{0}(0)\neq 0$ and consider

Failed to parse (syntax error): \phi_{u_0}: \{\xi\in E\,|\, \|\xi\|_{\mathcal{C}^1<\epsilon \} \to \mathcal{B} , \qquad \xi \mapsto [u_0+\xi] .

transition maps

For $[u_{0}]=[u_{1}]\in {\mathcal {B}}$ with representatives $u_{0}\neq u_{1}$ but ${\dot u}_{i}(0)\neq 0$ determine the transition map $\phi _{{u_{1}}}^{{-1}}\circ \phi _{{u_{0}}}$.

Then find an example of a point and direction in which this transition map is not differentiable.

solution

Show that the shift action $\sigma$ of #homework 7 is a scale-smooth map by the following steps:

a) find a scale-Banach space ${\mathcal C}=(C_{\ell })_{{\ell \geq 1}}$ with $C_{1}=C^{1}(S^{1},{\mathbb R}^{2})$

b) show that $\sigma :S^{1}\times {\mathcal C}\to {\mathcal C}$ is $sc^{1}$ and calculate $T\sigma$

c) iteratively show that $\sigma$ is $sc^{k}$ for all k

Note: The actual map in #homework 7 is a restriction of this map here to a submanifold of ${\mathcal C}$ (taking values in $S^{1}$, of fixed degree).