Orbits of Higher-Dimensional Hereditarily Indecomposable Continua

Abstract

ABSTRACT. Let X be a continuum.The following theorems are proved.THEOREM. i/dimX > 1, tkenX contains uncountably many nonhomeomorphic continua.THEOREM. If dim X > 1 and X is hereditarily indécomposable, then X has uncountably many orbits under the action of its hcrmex/rnorphisrn group.In 1970, Howard Cook [3] gave a beautiful proof that hereditarily equivalent continua are tree-like. In 1982, the author [9] proved that homogeneous, hereditarily indecomposable continua are tree-like. A step in both these proofs was to show that such continua must be one-dimensional.In this paper, we give a simple and independent proof that these continua must be one-dimensional.Moreover, the results are even stronger: namely, if X is a continuum and dim AT > 1, then X contains uncountably many nonhomeomorphic subcontinua. If, in addition, X is hereditarily indecomposable, then X has uncountably many orbits under the action of its homeomorphism group.A continuum is a compact, connected, nonvoid metric space. A continuum is hereditarily equivalent if it is homeomorphic to each of its proper nondegenerate subcontinua.A continuum is indecomposable if it is not the union of two of its proper subcontinua.A continuum is hereditarily indecomposable if each of its subcontinua is indecomposable.R. H. Bing [2] constructed examples of hereditarily indecomposable continua of dimension n, 1 < n < oo. Bing [2] showed that, for 1 < n < oo, no such continuum is homogeneous.

Maps Which Preserve Graphs

“…Theorem II.7 is a generalization of Theorem 3 of [9] for weakly confluent maps. Again, the proof is almost the same.…”

On hereditarily indecomposable continua, Henderson compacta and a question of Yohe

“…We shall outline the idea of the referee, mentioned in the introduction. It is based on Rogers' approach [24]. Fix a Waraszkiewicz spiral W and let S be the circle in W .…”

“…Fix a Waraszkiewicz spiral W and let S be the circle in W . By Theorem 1 of [24] there is an HI hereditarily SID continuum A admitting a map f onto W . One can also assume that in the complement of the preimage of S there is an increasing sequence of subcontinua whose union is dense in A.…”

“…It is possible to construct a continuum X that is a disjoint union of A and B such that A is terminal in X and the map f extends over X. Then X is an HI hereditarily SID Cantor manifold, and Theorem 2 of [24] guarantees that by using different Waraszkiewicz spirals one gets the desired collection of continua.…”

Atriodic homogeneous nondegenerate continua are one-dimensional

“…The first was proved by Mackowiak and Tymchatyn [4, p. 31] in their investigation of atriodic, homogeneous continua, the second is due to Bing [1], and the last two were proved by the author in [5]. PROPOSITION l. Each indecomposable subcontinuum of an atriodic, homogeneous continuum X is terminal in X.…”