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In this paper, we prove the continuity of iteration operators 
$\mathcal {J}_n$ on the space of all continuous self-maps of a locally compact Hausdorff space X and generally discuss dynamical behaviors of them. We characterize their fixed points and periodic points for 
$X=\mathbb {R}$ and the unit circle 
$S^1$. Then we indicate that all orbits of 
$\mathcal {J}_n$ are bounded; however, we prove that for 
$X=\mathbb {R}$ and 
$S^1$, every fixed point of 
$\mathcal {J}_n$ which is non-constant and equals the identity on its range is not Lyapunov stable. The boundedness and the instability exhibit the complexity of the system, but we show that the complicated behavior is not Devaney chaotic. We give a sufficient condition to classify the systems generated by iteration operators up to topological conjugacy.
