Topological transformation groups with a fixed end point

Gray, William J.

doi:10.1090/s0002-9939-1967-0207890-7

Cited by 3 publications

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“…Let (X, T, n) be a topological transformation group, where X is a non-trivial locally connected Hausdorff continuum and T is generated by a compact subgroup C and a connected subgroup K. If T leaves an end point e of X fixed, then T has another fixed point. [3] PROOF. Let z be a non-cut point of X other than e and let X-x = U u V be a separation of Jf-x such that e e U and C2 C V. If /> e C2, then Kp is a connected set of non-cut points of X and it follows that If KCz is a point, K and C have a fixed point in common, and we are through.…”

Section: If X Is Irreducibly T-invariant Then X Contains No Cut Pointmentioning

confidence: 98%

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A note on group invariant continua

Gray

1968

J. Aust. Math. Soc.

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Let (X, T, n) be a topological transformation group, where X is a Hausdorff continuum. We will say that X is irreducibly T-invariant if no proper subcontinuum of X is ^-invariant. Wallace, [6], has shown that if T is abelian and X is irreducibly ^-invariant, then X has no cut point; he then asked if this statement remains true if "abelian" is replaced by "compact". In this paper we answer this question in the affirmative, and prove a related result when T satisfies a recursive property.A general problem due to Wallace is as follows: Assume T leaves an endpoint of X fixed. Under what conditions on X and T does T have another fixed point? This problem has been investigated by Wallace [8], Wang, [5], Chu, [1], and Gray, [3,4]. In Theorem 2, we show that if X is locally connected, and T is generated by a compact subgroup and a connected subgroup, then T has another fixed point.Departing from [7] slightly, we call a subcontinuum C of X a universal subcontinuum (USC) if given a subcontinuum D of X, D n C is a continuum. The intersection of arbitrarily many USC is again a USC. If X-x = U u V, where U and V are non-empty separated subsets of X (hereafter referred to as a "separation of X-x"), then U u {x} is a USC. The property of being a USC is topological. The proofs of these statements are to be found in [7].The terminology pertaining to transformation groups is taken from [2]. THEOREM 1. Let (X, T, n) be a topological transformation group where X is a Hausdorff continuum and one of the following conditions is satisfied:(i) T is compact, (ii) X is locally connected and T is pointwise regularly almost periodic. If X is irreducibly T-invariant, then X contains no cut point.PROOF. Assume that X is irreducibly T-invariant. We make the following observations:1. If x e X, no proper USC of X contains the orbit Tx of x.For otherwise the intersection, D, of all USC which contains Tx is a proper subcontinuum of X containing Tx. We easily verify that that D is T-invariant, and hence X is not irreducibly T-invariant. 310

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Section: If X Is Irreducibly T-invariant Then X Contains No Cut Pointmentioning

confidence: 98%

“…Under what conditions on X and T does T have another fixed point? This problem has been investigated by Wallace [8], Wang, [5], Chu, [1], and Gray, [3,4]. In Theorem 2, we show that if X is locally connected, and T is generated by a compact subgroup and a connected subgroup, then T has another fixed point.…”

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confidence: 89%