Expansive homeomorphisms and indecomposable subcontinua
Mouron, Christopher G.
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It is the goal of this dissertation to classify which continua admit expansive homeomorphisms and which do not. Plykin's attractors  and the dyadic solenoid are examples of continua that admit an expansive homeomorphism. Both of these continua have the property of being indecomposable. A continuum is decomposable if it is the union of two proper subcontinuum. A continuum is indecomposable if it is not decomposable. Indecomposable continua are created by stretching and bending arcs an infinite number of times back and forth. Intuitively, it appears that in order to have expansiveness, the subset of the continuum between points that are close to each other would have to be continually stretched in order to move points away from each other. However, because of compactness, some folding or wrapping must also occur. Every known continuum that admits an expansive homeomorphism has an indecomposable subcontinuum. Thus, one of the major questions in this-area is "If X admits an expansive homeomorphism, must X contain a nondegenerate indecomposable subcontinuum?". However, it is known that X being indecomposable is not necessary. For example, it will be shown later that there exists an expansive homeomorphism of the solenoid that contains a fixed point. If we attach two solenoids at that fixed point, the resulting space will admit an expansive homeomorphism and be decomposable. However, the resulting space will clearly contain subcontinua that are indecomposable.