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.