A note on uniform convergence and transitivity

摘要

In this paper, let (X,d) be a metric space. Let f(n): X -> X be a sequence of continuous and topologically transitive functions such that (f(n)) converges uniformly to a function f. It is shown that if (X,d) is compact and perfect, lim(n-infinity)d(infinity)(f(n)(n),f(n)) = 0 and {f(n)(n)(x)} is dense in X for some x is an element of X, then f is totally transitive. We also present a sufficient condition for f to be topologically transitive (resp. syndetically transitive). Furthermore, we give a sufficient condition for f to be topologically weak mixing (resp. topologically mixing). In addition, for a compact metric space (X,d), suppose that the f(n): X -> X are continuous and converge uniformly to f. If for a given epsilon > 0, there exists a positive integer no such that for all n > n(0) and all l > 0, d(f(n)(l)(x), f(l)(x)) < epsilon for all x is an element of X, then the following statements hold: (1) f is syndetically sensitive (resp. cofinitely sensitive) whenever the f(n) are syndetically sensitive (resp. cofinitely sensitive). (2) f is multi-sensitive whenever the f(n) are multi-sensitive. (3) If f is sensitive (resp. cofinitely sensitive) with delta as a constant of sensitivity, then there exists an integer N > 0 such that f(n) is sensitive (resp. cofinitely sensitive) with 1/3 delta as a constant of sensitivity for any n >= N. (4) If f is multi-sensitive (resp. syndetically sensitive) with 6 as a constant of sensitivity, then there exists an integer N > 0 such that f(n) is multi-sensitive (resp. syndetically sensitive) with 1/9 delta as a constant of sensitivity for any n >= N.

摘要

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