  Recurrence plots provide a graphical representation of the recurrent patterns
in a timeseries, the quantification of which is a relatively new field. Here we
derive analytical expressions which relate the values of key statistics,
notably determinism and entropy of line length distribution, to the correlation
sum as a function of embedding dimension. These expressions are obtained by
deriving the transformation which generates an embedded recurrence plot from an
unembedded plot. A single unembedded recurrence plot thus provides the
statistics of all possible embedded recurrence plots. If the correlation sum
scales exponentially with embedding dimension, we show that these statistics
are determined entirely by the exponent of the exponential. This explains the
results of Iwanski and Bradley (Chaos 8 [1998] 861-871) who found that certain
recurrence plot statistics are apparently invariant to embedding dimension for
certain low-dimensional systems. We also examine the relationship between the
mutual information content of two timeseries and the common recurrent structure
seen in their recurrence plots. This allows time-localized contributions to
mutual information to be visualized. This technique is demonstrated using
geomagnetic index data; we show that the AU and AL geomagnetic indices share
half their information, and find the timescale on which mutual features appear.
