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On the iteration of transformations in noncompact minimal dynamical systems

1958; American Mathematical Society; Volume: 9; Issue: 5 Linguagem: Inglês

10.1090/s0002-9939-1958-0096975-9

1088-6826

Felix E. Browder,

Functional Equations Stability Results

Let A be a Hausdorff space, 4 a continuous mapping of A into itself. It is the purpose of the present paper to discuss various topics centering around the following question: If g is a bounded continuous function on A, does there exist a bounded continuous function f on A such that f(ba) -f(a) =g(a) for all a in A? Suppose that for each ao in A, the set {Ionao, nO} is dense in A. Theorem 1 asserts that a necessary and sufficient condition for the existence of such an f is that Z k0 g(40ka) I should be uniformly bounded for all positive j and all points a of A. For homeomorphisms of compact spaces, this result was previously obtained by Gottschalk and Hedlund [5, Theorem 14.11, p. 135].1 A related problem for linear operators in a Banach space is obtained by letting X be the Banach space of bounded continuous functions on A with the uniform norm, T the linear transformation of X into itself defined by (Tf)(a) =f(Oa), aEEA. In terms of X and T, Theorem 1 states that g will lie in the range of (IT) if and only if the sequence of norms || Dk=o Tkglj is uniformly bounded for all positive j. In a reflexive Banach space, this characterization of the range of (IT) is valid for any linear transformation T for which || Tnj1 is bounded for all n. A sufficient condition in a general Banach space would seem to require an assumption that the elements EJ=o Tkg } lie for all j in a fixed weakly compact subset K of X. It would be interesting to obtain a proof of Theorem 1 along these lines. We shall content ourselves with showing by these mnethods that if m is a totally-finite measure on a a--algebra on A, Lo(m) the space of m-essentially bounded measurable functions, 4 a measure preserving mapping of A into A, then in order that for an element g in Lo(m), there should exist anf in Lc(m) such thatf(ka) -f(a) =g(a) a.e. in m, it is necessary and sufficient that m-ess. sup. | =0 g(Qka) | should be uniformly bounded for all positive j.2 Such a result raises another sort of question. For a topological space A if g is continuous and f is a solution of the equation f(ba)