Possible heights of Alexandroff square transformation groups

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Abstract: In the following text we compute possible heights of

m a t h b b A

(Alexandroff square),

m a t h b b O

(unit square

[0, 1] i m e s [0, 1]

with lexicographic order topology) and

m a t h b b U

(unit square

[0, 1] i m e s [0, 1]

with induced topology of Euclidean plane). We prove

P_{h} (m a t h b b A) = n : n g e q 5 c u p + i n f t y

,

P_{h} (m a t h b b O) = n : n g e q 4 c u p + i n f t y

,

P_{h} (m a t h b b U) = n : n g e q 1 c u p + i n f t y

(where for topological space

X

, by

P_{h} (X)

we mean the collection of heights of transformation groups with phase space

X

. In this way we also prove that there is not any topological transitive (resp. Devaney chaotic) Alexandroff square transformation group.

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