Talk:Smale horseshoe

This is a very informative and interesting article. There are a few minor inaccuracies which I will mention below. I made some minor changes using the editing feature of this web site, but it looks as if they actually got posted into the article itself.

The authors perhaps should look at the article now to see if they accept the changes. If either the authors accept the changes or post their own concerning the topics below. The article should be accepted.

1. The definition of 'Horseshoe' gives rise to the full two shift.

The sentence beginning with "The utility of Smale's analysis..." is incorrect as stated. The correct statement is that a trasverse homoclinic point implies that some power $f^T$ of $f$ has a horseshoe containing $r$.

2. Uniformly hyperbolic systems are not specifically defined here. The authors obviously mean Axiom A + transversality. I suggested that this simply be stated more or less as

"... called uniformly hyperbolic systems. These are systems for which the non-wandering set is a hyperbolic set and the its stable and unstable manifolds only have transverse intersections. "

added 6 PM 11-27-07:

3. The reference to 'stability' as proved by Anosov and the link provided is misleading. The link refers to 'orbital stability' while the authors here mean 'structural stability.'

Suggested change:  replace  "stability" with "structural stability"

4. The authors have now introduced "non-wandering set" stability which is the original type of stability considered by Smale in his article of the same name. To correctly formulate the conditions for $\Omega-$stability, one needs several objects.

 a. the definition of the non-wandering set.
 b. the definition of 'cycle' -- which involves Smale's Spectral Decomposition Theorem.  

  Nowadays, in view of the work of many people (in particular that of Franke and Selgrade) these can be stated 

very succinctly as "The chain recurrent set is hyperbolic". Thus one only needs the definition of the chain recurrent set (and hyperbolicity, of course) to state the modern version of both the $\Omega-$stability theorem and the general structural stability theorem. Simply change the name "$\Omega-$stability to ${\cal R}-$stability where ${\cal R}$ is the chain recurrent set. Then

  1. $f$ is ${\cal R}-$stable if and only if ${\cal R}$ is (uniformly) hyperbolic. 

  2. $f$ is structurally stable if and only if ${\cal R}$ is hyperbolic

Of course, the authors are aware of these things and are trying to keep the concepts simple.  My original suggested changes were to correct some minor errors and to try to keep the re-typing minimal.

Now that the authors have brought up the concept of "non-wandering set," why not simply go to "chain recurrent set" instead. I think the latter definition (especially being clearly related to computer generated orbits) is easier to understand than that of non-wandering set anyway.

This is a matter of taste, and I leave the acceptance or rejection of this idea up to the authors.

If at least the current proposed editing changes are accepted by the author, I think the article should be accepted.

Another comment: I did not see chain recurrent set among the objects in Scholarpedia, so I suggested that it be added.

I accept the article in its current version. I'll stick with non-wandering set stability since the Omega stability theorem is used more in the references than chain recurrent set stability theorem. Things may change in the future.

I have not written a uniformly hyperbolic dynamics article, nor do I intend to. It would be a good idea to include chain recurrent sets in scholarpedia somewhere and perhaps in a treatment of uniform hyperbolicity.