From: clark@rainier.math.nwu.edu (Clark Robinson)
Subject: Re: Structural stability
Date: 6 Feb 2001 23:00:03 -0600
Newsgroups: sci.math.research,sci.nonlinear
Summary: Two close Anosov diffeomorphisms are coninuously conjugate

On 6 Feb 2001 08:28:03 -0600, replayer-return@usenet-replayer.com 
<replayer-return@usenet-replayer.com> wrote:

>Dear all,
>structural stability of Anosov systems grants us that if f is 
>an Anosov diffeomorphism (of compact manifold M into itself) and g 
>is a diffeomorphism (again of M into itself) sufficiently close to f, 
>then one can find homeomorphism h such that
>
>			hg = fh.
>
>Moreover, the map g -> h(g) is continuous. (see, for example, 
>[S. Smale, Differentiable Dynamical Systems,
>Bull. Am. Math. Soc., nov 1967, 747--817])

>I am interested in an upper bound of the "distorsion" introduced 
>by h as a function of the distance between f and g.  More precisely,
>if A=D(f,g)=sup_{x \in M} d(f(x), g(x)) I would like to find an upper
>bound
>
>		D(h, Id) <= F(A)
>
>to the distance between h and the identity on M Id.  Is anything known
>about this?

The correct bound would be 
         D(h, Id) <= C_f D(f,g)
where C_f is a constants which depends on the derivative of  f.  
The bound comes from the necesity to invert  (f_#-Id)  where
(f_#-Id)v = Df v f^{-1} -v, where v is a vector field on the manifold.  
This is most easily seen in the proofs which use the "Implicity Function
Theorem" rather than the more geometric proofs.  So, for example, it
follows more easily from the proof given by Moser ("On a Theorem of Anosov",
J.Diff.Equat. 5(1969)) than the original proof of Anosov.  

Clark Robinson 
Northwestern University


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