We provide a detailed study of the quantitative
behavior of first return times
of points to small neighborhoods of themselves.
Let $K$ be a self-conformal set
(satisfying a certain separation condition)
and let $S:K\to K$ be the natural self-map induced by the shift.
We study the quantitative
behavior of the first return time,
$ \tau_{B(x,r)}(x)=$inf{$1\le k \le n | S^k x \in B(x,r)},$
of a point $x$ to the ball $B(x,r)$
as $r$ tends to $0$.
For a function $\varphi:(0,\infty)\to\mathbb R$,
let
A$(\varphi(r))$ denote the set of accumulation points of
$\varphi(r)$ as $r\to 0$.
We show that
the first return time exponent,
$\frac{\log\tau_{B(x,r)}(x)} {-\log r}$,
has an extremely complicated and surprisingly intricate structure:
for any
compact subinterval $I$ of $(0,\infty)$,
the set of points $x$
such that for each $t\in I$
there exists arbitrarily small $r>0$
for which
the first return time $\tau_{B(x,r)}(x)$ of
$x$ to the neighborhood $B(x,r)$
behaves like
$1/r^t$,
has full Hausdorff dimension on any
open set, i. e.
dim$(G\cap ${$x\in K| $A ($\frac
{\log\tau_{B(x,r)}(x)}{-\log r}) =I$})
$=$dim $K$
for any open set $G$ with $G\cap K$≠$\emptyset$.
As a consequence we deduce that the so-called multifractal formalism
fails comprehensively for the
first return time
multifractal spectrum.
Another application of our results concerns the construction
of a certain class of Darboux functions.
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