View source for Talk:Expansive systems

This is a very good article, but it needs some clarifications. Moreover,
a section on non-invertible discrete dynamical systems is necessary.

Typing has been done apparently without checking its effects; there are
many small corrections to be done (I tried to make some of them).



General remarks:

A) Please do not start sentences with a symbol (<math>f, x</math> etc.). This is
often confusing.

B) Please unify the way names like "quasi-Anosov", "pseudo-Anosov",
etc. are written. Should the first letter be capital? Should there be
a hyphen?

C) Expressions like <math>f</math>-invariant, <math>f</math>-persistent and similar should be
written with a hyphen. Moreover, hyphens should be outside the math
environment; they are not minus signs. There should be no spaces to
the left or right of a hyphen.



Concrete remarks:

1) What do you mean by a "discrete dynamical system"? Apparently only
an invertible one. However, a large part of the theory of dynamical
systems is about non-invertible ones (continuous maps from a space to
itself). In particular, there should be a section about non-invertible
maps and expansivity (called sometimes positive expansivity) similar
to the section on the flows.

2) When you say "chaotic", what do you mean? Which type of chaos?

3) I do not understand the phrase "However, the motion, the
trajectories...".

4) In the same sentence: this is true only if there are infinitely
many points in the <math>\alpha-ball</math> around <math>x</math> (in principle, <math>x can be
isolated).

5) Why is <math>M/R_\delta</math> compact metrizable?

6) The reader may know what a conjugacy is. Maybe just refer to
some other article in Scholarpedia?

7) Similarly with the 2-shift.

8) Notation <math>\{a_n\}</math> for a sequence is bad, it does not distinguish
between a sequence and a set. As a set, <math>\{a_n\}=\{0,1\}</math>. It is better to
denote sequences as <math>(a_n)</math>.

9) It seems that if <math>a_K\neq b_K</math> then to get distance at least 1 one
should take <math>K</math>-th iterate, not <math>(-K)</math>-th.

10) Can a set be expansive, or only a map?

11) Do points evolve?

12) In "Another example" formula, some parentheses after 1/... are
necessary.

13) In Corollary after Theorem [L2], do we assume that <math>M</math> is locally
connected?

14) The proof of the following Application is not full. In particular,
it omits the rational rotation number case.

15) In the paragraph after Theorem [L1], I do not understand the
grammatical structure of teh sentence.

16) In the same paragraph, a reference is missing.

17) In the same paragraph, what is e)?

18) In Theorem in the "On surfaces" section, it should be explained
what <math>r</math> is.

19) The last 2 sentences in this section are unclear. This is a proof
of something that is not stated explicitly.

20) The phrase "because of the classification theorem, every expansive
homeomorphism of <math>T^{2}</math>" is not understandable.

21) In the same paragraph: Figure 3 illustrates "no c)", not "no b)".

22) Is "Are all the semi-trajectories of an expansive system
persistent?" an open problem?






Dear reviewer:
We have taken account of all your suggestions and concerns. 
best regards, 
Marcelo Cerminara and Jorge Lewowicz




Dear Marcelo and Jorge,

I made some minor changes. The largest one was to remove the definition of the expansiveness of a non-invertible system (which was basically: the system is expansive if its inverse limit is expansive); I am not sure about its usefulness.

But of course you as authors will decide what is the best form and contents of your article.

Best regards,
Reviewer B

== User 3: Expansiveness of Denjoy map ==

it might be good to specify and stress that, in the Denjoy case, that the expansiveness holds on the subsystem of nonwandering set.