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We consider the distribution of the orbits of the number 1 under the$\beta$-transformations $T_\beta$ as $\beta$ varies. Mainly, the size of theset of $\beta>1$ for which a given point can be well approximated by the orbitof 1 is measured by its Hausdorff dimension. That is, the dimension of thefollowing set $$ E\big({\ell_n}_{n\ge 1}, x_0\big)=\Big{\beta>1:|T^n_{\beta}1-x_0|<\beta^{-\ell_n}, {for infinitely many} n\in \N\Big} $$ isdetermined, where $x_0$ is a given point in $[0,1]$ and ${\ell_n}_{n\ge 1}$ isa sequence of integers tending to infinity as $n\to \infty$. For the proof ofthis result, the notion of the recurrence time of a word in symbolic space isintroduced to characterize the lengths and the distribution of cylinders (theset of $\beta$ with a common prefix in the expansion of 1) in the parameterspace ${\beta\in \R: \beta>1}$.

Diophantine approximation of the orbit of 1 in the dynamical system of bete expansions