  This paper investigates the dynamics of time-periodic Euler flows in
multi-connected, planar fluid regions which are ``stirred'' by the moving
boundaries. The classical Helmholtz theorem on the transport of vorticity
implies that if the initial vorticity of such a flow is generic among
real-valued functions in the $C^k$-topology ($k \geq 2$) or is $C^\omega$ and
nonconstant, then the flow has zero topological entropy. On the other hand, it
is shown that for constant initial vorticity there are stirring protocols which
always yield time-periodic Euler flows with positive entropy. These protocols
are those that generate flow maps in pseudoAnosov isotopy classes. These
classes are a basic ingredient of the Thurston-Nielsen theory and a further
application of that theory shows that pseudoAnosov stirring protocols with
generic initial vorticity always yield solutions to Euler's equations for which
the sup norm of the gradient of the vorticity grows exponentially in time. In
particular, such Euler flows are never time-periodic.
