Download e-book for kindle: Continua: With the Houston Problem Book by Howard Cook, William T. Ingram, Krystyna Kuperberg, Andrew

By Howard Cook, William T. Ingram, Krystyna Kuperberg, Andrew Lelek, Piotr Minc

ISBN-10: 0824796500

ISBN-13: 9780824796501

This quantity includes the lawsuits of the certain consultation on sleek equipment in Continuum thought offered on the a hundredth Annual Joint arithmetic conferences held in Cincinnati, Ohio. It additionally beneficial properties the Houston challenge ebook which incorporates a lately up-to-date set of 2 hundred difficulties gathered over numerous years on the collage of Houston.;These lawsuits and difficulties are aimed toward natural and utilized mathematicians, topologists, geometers, physicists and graduate-level scholars in those disciplines.

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Extra resources for Continua: With the Houston Problem Book

Example text

Barge and R. Gillette [5]: Let / be an orientation-preserving homeomorphism of the plane that leaves invariant a continuum A which irreducibly separates the plane into exactly two domains. Then the rotation set of A contains all rational numbers in its convex hull, and each such rational number is the rotation number of a periodic orbit in A. This theorem, together with a result of M. Handel 20 ALLIGOOD [12] that the rotation set of an annulus homeomorphism is closed, shows that the rotation set of an invariant continuum which irreducibly separates the plane (or annulus) is a closed interval.

Then X has the fixed point property. Proof. Suppose X is such a continuum. Notice that X contains no simple closed curves and so is uniquely arcwise connected. Then suppose f : X X is a. fixed point free continuous map. Let ao G X . Then / {[ao^ f{ao)]) n [aoj/(<^o)] ^ 0 since /(ao), at least, belongs to the intersection. Applying Zorn’s lemma to the collection {\P)Q] £ [(iOi /(<^o)] I b) q] /b ) q] 0 } we obtain an arc [a, b] minimal with respect to the property that [a, 6] fi /[a, 6] ^

But [a, g] is a proper subset of [a,r], so this is a contradiction. In the alternate case, if [a,g] ^ we also have [a, g] ^ £ by maximality of There are two subcases. Either /(g) g [a,g] or /(g) ^ [a,gj. In the latter case, [g>/(g)] C [a, g] = [p, g] for some p ^ q. Let U, F, and W be disjoint open sets with p e W , q e U, /(g) G V, and /([/) C y . Now, by choice of g, there exists r G [p,g] such that [a,r] G C, and [r, g] C [7. Then /[r, g] is an arcwise connected continuum containing [/(r),/(g)] and missing p.

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Continua: With the Houston Problem Book by Howard Cook, William T. Ingram, Krystyna Kuperberg, Andrew Lelek, Piotr Minc


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