  We study the dynamics of $L^p$-multipliers on non-compact simply connected
harmonic manifolds of purely exponential volume growth. These are linear
operators on the $L^p$-spaces which behave nicely on radial functions under
Fourier transformation. In the process we complement the results of Kingshook
Biswas and Rudra P. Sarkar by showing that if they are acting nicely on smooth
function with compact support under Fourier transform for $p\leq 2$ these can
not be chaotic. Furthermore, we use this to study the behaviour of the heat
semi-group, the resolvent and the convolution algebra arising from convolution
with radial functions. In the process, we obtain a Young inequality for the
convolution on non-compact simply connected harmonic manifolds, an extension of
the Kunz-Stein and study the domain of holomorphicity of the Fourier transform.
