  A space X has the local fixed point property LFPP, (local periodic point
property LPPP) if it has an open basis $\mathcal{B}$ such that, for each $B\in
\mathcal{B}$, the closure $\overline{B}$ has the fixed (periodic) point
property. Weaker versions wLFPP, wLPPP are also considered and examples of
metric continua that distinguish all these properties are constructed. We show
that for planar or one-dimensional locally connected metric continua the
properties are equivalent.

Dynamical Systems