Difference between revisions of "Scale calculus problems"

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Show that the shift action <math>\sigma</math> is a scale-smooth map by the following steps:
 
Show that the shift action <math>\sigma</math> is a scale-smooth map by the following steps:
  
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a) find a scale-Banach space <math>\mathbb{E} = (E_\ell)_{\ell\geq 0}</math> with <math>E_0=C^1(S^1,\mathbb R^2)</math>
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a) Find a scale-Banach space <math>\mathbb{E} = (E_\ell)_{\ell\geq 0}</math> with <math>E_0=C^1(S^1,\mathbb R^2)</math>.
  
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b) show that <math>\widetilde\sigma: S^1 \times \mathbb{E}\to\mathbb{E}</math> is <math>sc^1</math> and calculate <math>T\widetilde\sigma</math>
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b) Show that <math>\widetilde\sigma: S^1 \times \mathbb{E}\to\mathbb{E}</math> is <math>sc^1</math> and calculate the linearization <math>T\widetilde\sigma : (\tau,u, t , \xi) \mapsto \bigl(\widetilde\sigma(\tau,u), d_{(\tau,u)}\widetilde\sigma (T,u)\bigr)</math>.
  
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c) iteratively show that <math>\widetilde\sigma</math> is <math>sc^k</math> for all k
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c) Iteratively show that <math>\widetilde\sigma</math> is <math>sc^k</math> for all k.
  
 
Note: The actual map <math>\sigma</math> above is a restriction of the map <math>\widetilde\sigma</math> to the subset <math>\mathcal{C}^1_{\rm id} \subset \mathbb{E}</math>, where we view <math>S^1=\{|z|=1\}\subset\R^2</math>.
 
Note: The actual map <math>\sigma</math> above is a restriction of the map <math>\widetilde\sigma</math> to the subset <math>\mathcal{C}^1_{\rm id} \subset \mathbb{E}</math>, where we view <math>S^1=\{|z|=1\}\subset\R^2</math>.
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'''YOUR SOLUTION WANTS TO BE HERE'''
 
'''YOUR SOLUTION WANTS TO BE HERE'''
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== scale-smooth structure on <math>\mathcal{B}</math>  ==  
 
== scale-smooth structure on <math>\mathcal{B}</math>  ==  
  
  
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Now use the charts and transition maps discussed above to equip <math>\mathcal{B}</math> with the structure of a scale-manifold (i.e.\ modeled on a scale-Banach space, with scale-smooth transition maps).  
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Now use the charts and transition maps discussed above to equip <math>\mathcal{B}</math> with the structure of a scale-manifold (i.e. modeled on a scale-Banach space, with scale-smooth transition maps).  
  
  
  
 
'''YOUR SOLUTION WANTS TO BE HERE'''
 
'''YOUR SOLUTION WANTS TO BE HERE'''

Latest revision as of 10:41, 26 May 2017

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).


YOUR SOLUTION WANTS TO BE HERE


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 the map

\phi _{{u_{0}}}:\{\xi \in E\,|\,\|\xi \|_{{{\mathcal {C}}^{1}}}<\epsilon \}\to {\mathcal {B}},\qquad \xi \mapsto [u_{0}+\xi ].


YOUR SOLUTION WANTS TO BE HERE


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.


YOUR SOLUTION WANTS TO BE HERE


alternative Banach norms

Explain why working with {\mathcal {C}}^{k}, H\"older, or Sobolev spaces W^{{k,p}} does not resolve these differentiabliity issues. Possibly come up with other ideas for a Banach manifold structure on {\mathcal {B}}.


YOUR SOLUTION WANTS TO BE HERE


reparametrization on Riemann surfaces

Explore the same questions for a space of nonconstant functions u:\Sigma \to \mathbb{R} on a Riemann surface (\Sigma ,j) modulo the reparametrization action (\psi ,u)\mapsto u\circ \psi of a nontrivial automorphism group {{\rm {Aut}}}(\Sigma ,j)=\{\psi :\Sigma \to \Sigma \,|\,\psi ^{*}j=j\} (e.g. \Sigma =S^{2}{\text{or}}\;T^{2}).


YOUR SOLUTION WANTS TO BE HERE


scale smoothness of the shift map

Show that the shift action \sigma is a scale-smooth map by the following steps:

a) Find a scale-Banach space {\mathbb {E}}=(E_{\ell })_{{\ell \geq 0}} with E_{0}=C^{1}(S^{1},{\mathbb R}^{2}).

b) Show that \widetilde \sigma :S^{1}\times {\mathbb {E}}\to {\mathbb {E}} is sc^{1} and calculate the linearization T\widetilde \sigma :(\tau ,u,t,\xi )\mapsto {\bigl (}\widetilde \sigma (\tau ,u),d_{{(\tau ,u)}}\widetilde \sigma (T,u){\bigr )}.

c) Iteratively show that \widetilde \sigma is sc^{k} for all k.

Note: The actual map \sigma above is a restriction of the map \widetilde \sigma to the subset {\mathcal {C}}_{{{\rm {id}}}}^{1}\subset {\mathbb {E}}, where we view S^{1}=\{|z|=1\}\subset \mathbb{R} ^{2}.


YOUR SOLUTION WANTS TO BE HERE

scale-smooth structure on {\mathcal {B}}

Now use the charts and transition maps discussed above to equip {\mathcal {B}} with the structure of a scale-manifold (i.e. modeled on a scale-Banach space, with scale-smooth transition maps).


YOUR SOLUTION WANTS TO BE HERE

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