Abstract
Let <TEX>$f:X{\rightarrow}X$</TEX> be a continuous surjection of a compact metric space and let (<TEX>$X_f,{\tilde{f}}$</TEX>) be the inverse limit of a continuous surjection f on X. We show that for a continuous surjective map f, if f has the asymptotic average, the average shadowing, the ergodic shadowing property then <TEX>${\tilde{f}}$</TEX> is topologically transitive.
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