  This paper will study topological, geometrical and measure-theoretical
properties of the real Fibonacci map. Our goal was to figure out if this type
of recurrence really gives any pathological examples and to compare it with the
infinitely renormalizable patterns of recurrence studied by Sullivan. It turns
out that the situation can be understood completely and is of quite regular
nature. In particular, any Fibonacci map (with negative Schwarzian and
non-degenerate critical point) has an absolutely continuous invariant measure
(so, we deal with a ``regular'' type of chaotic dynamics). It turns out also
that geometrical properties of the closure of the critical orbit are quite
different from those of the Feigenbaum map: its Hausdorff dimension is equal to
zero and its geometry is not rigid but depends on one parameter.
