Let
$\unicode[STIX]{x1D703}$
be an irrational real number. The map
$T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$
from the unit interval
$\mathbf{I}= [\!0,1\![$
(endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map
$[T_{\unicode[STIX]{x1D703}},\text{Id}]$
and the unilateral dyadic Bernoulli shift when
$\unicode[STIX]{x1D703}$
is extremely well approximated by the rational numbers, namely, if
$$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$
A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map
$[T_{\unicode[STIX]{x1D703}},\text{Id}]$
is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on
$\unicode[STIX]{x1D703}$
: the explicit map provided by Parry’s method is an isomorphism between the map
$[T_{\unicode[STIX]{x1D703}},\text{Id}]$
and the unilateral dyadic Bernoulli shift whenever
$$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$
This condition can be relaxed again into
$$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}|<+\infty ,\end{eqnarray}$$
where
$[0;a_{1},a_{2},\ldots ]$
is the continued fraction expansion and
$(p_{n}/q_{n})_{n\geq 0}$
the sequence of convergents of
$\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$
. Whether Parry’s map is an isomorphism for every
$\unicode[STIX]{x1D703}$
or not is still an open question, although we expect a positive answer.
The bifurcation locus for numbers of bounded type
