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 (\(f, x\) 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 \(f\)-invariant, \(f\)-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 \(\alpha-ball\) around \(x\) (in principle, \(x can be isolated). 5) Why is <math>M/R_\delta\) 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 \(\{a_n\}\) for a sequence is bad, it does not distinguish between a sequence and a set. As a set, \(\{a_n\}=\{0,1\}\). It is better to denote sequences as \((a_n)\).

9) It seems that if \(a_K\neq b_K\) then to get distance at least 1 one should take \(K\)-th iterate, not \((-K)\)-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 \(M\) 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 \(r\) 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 \(T^{2}\)" 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.