Saturday, 21 August 2010

set theory - maximum with respect inclusion of a function whose output are sets

If W admits a periodic point, then, of course, S(x) is constant along its orbit. But, on the contrary, whatever nice and well-structured A and W you have, if W has no periodic points, there is always the bad map S(x):= the negative invariant set generated by x, that is cupkinmathbbNWk(x), and this map fails to satisfy the thesis while S(x)subsetS(W(x)) holds for all x.



This is to convince you that some assumption on the map S(x) is in order. Here, a mild and natural assumption, to be coupled with the compactness of Aneqemptyset, is weak upper semicontinuity, that is,
S:Ato2X


is continuous w.r.to the product topology on 2X where the two-point space 2:=0,1 is endowed with the left-order topology (whose only proper open subset is 0). As a consequence, the continuous image of A is a compact subset of 2X, therefore it has a maximal element with respect to inclusion. Due to your assumption on W, the equality necessarily holds for any maximal set S(x), proving your thesis (incidentally, note that no further assumption on W is needed).



Rmk 1. The upper semicontinuity of S introduced above may be equivalently stated as:



  • operatornamegraph(S):=(a,x)inAtimesX :ainS(x) is closed in AtimesX, where X has the discrete topology;


  • S:Xto2A is a closed map, that is, for any ainA the set S(x):=ainA:xinS(a) is a closed subset of A;


  • for any ainA (denoting mathcalNa the family of nbd's of a), there holds
    limsupbtoaS(b):=cupUinmathcalNacapbinUS(b)subsetS(a).


Rmk 2. The fact that a compact subset K of 2X, where 2 has the left-order topology, admits a maximal element, is, of course, a consequence of the Zorn lemma. Indeed, if Gamma is an infinite chain in K, it has a limit point in K, which turns out to be an upper bound of Gamma.

No comments:

Post a Comment