For $\beta \in (1,2]$ and $\alpha \in [0,2-\beta]$ consider the generalised $\beta$-transformation $T_{\alpha,\beta}:[0,1] \to [0,1]$ to be $$T_{\alpha, \beta}(x) = \beta x + \alpha \mod 1.$$ It is a well known fact that for each $\beta \in (1,2]$ and $\alpha \in [0,2-\beta]$ there exists a pair of sequences $a = (a_i), b = (b_i) \in \{0,1\}^\mathbb{N}$ satisfiying that $a_1 = 1$, $b_1 = 0$ and \begin{equation}\label{lex} \begin{split} &\sigma(a) \preccurlyeq \sigma^n(a) \prec \sigma(b); \\ &\sigma(a) \prec \sigma^n(b) \preccurlyeq \sigma(b) \quad \text{for all} \quad n \in \mathbb{N}, \end{split} \end{equation} where $\prec$ stands for the lexicographic order in $\left\{0,1\right\}^{\mathbb{N}}$. A pair $(a,b)$ satisfying the pevious condition is called lex-admissible. Then it is possible to define a map $K:\Delta \to \mathcal{L}$ by $K(\alpha,\beta) = (a,b)$ where $$\Delta = \left\{(\beta,\alpha): \beta \in (1,2] \quad \text{and} \quad \alpha \in [0,2-\beta]\right\}$$ and $$\mathcal{L} = \left\{(a,b) \in \left\{0,1\right\}^{\mathbb{N}} \times \left\{0,1\right\}^{\mathbb{N}}:(a,b) \quad \text{is lex admissible} \right\}.$$ Following the works of Glendinning in Topological conjugation of Lorenz maps by $\beta$-transformations. Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 2, 401–413, and Barnsley, Steiner, and Vince in Critical itineraries of maps with constant slope and one discontinuity. Math. Proc. Cambridge Philos. Soc. 157 (2014), no. 3, 547–565. it seems that the map is injective but not surjective. I would like to know if there is a reference adressing this.
Also, I want to ask is if there is a reference where it is studied the continuity of the map $K^{-1}: K(\Delta) \to \Delta$. Following the results by Parry contained in On the $\beta$-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416. it is possible to show the continuity of the map $K^{-1}$ when it is restricted to the elements of $\Delta$ where $a = 0^\infty$. However, I haven't found a reference addressing the general case yet.
EDIT: The inverse map $K^{-1}: K(\mathcal{\Delta}) \to \Delta$ can be defined implicitly by following the results of Li, Sahlsten and Samuel Intermediate $\beta$-shifts of finite type. Discrete Contin. Dyn. Syst. 36 (2016), no. 1, 323–344. as follows:
Consider $(a,b) = K(\alpha,\beta)$ and the symbolic space $$\Sigma_{a,b} = \left\{(x_i) \in \left\{0,1\right\}^{\mathbb{N}}: \sigma(a) \preccurlyeq \sigma^n((x_i)) \preccurlyeq \sigma(b) \text{ for all }n \geq 0\right\}.$$ Then $$\pi_{\alpha,\beta}((x_i)) = \dfrac{\alpha}{1-\beta} + \mathop{\sum}\limits_{i=1}^\infty \dfrac{x_i}{\beta^i}.$$ As $\pi_{a,b}(\sigma(a)) = 0$ and $\pi_{a,b}(\sigma(b)) = 1$ then the system $$0 = \alpha/(1-\beta) + \mathop{\sum}\limits_{i=1}^\infty a_{i+1} \cdot \beta^{-i} \quad \text{and} \quad 1 = \alpha/(1-\beta) + \mathop{\sum}\limits_{i=1}^\infty b_{i+1} \cdot \beta^{-i}$$ defines $K^{-1}$.
Thanks in advance for your help.
