If $\beta$ \beta^{-1}$is Pisot, then there is an ergodic measure of maximal dimension. This is a special case of the rather difficult Theorem 2.15 in the paper Dimension Theory of iterated function systems by De-Jun Feng and Huyi Hu. Very roughly speaking, Feng and Hu adapt Ledrappier-Young theory to the IFS setting. Note that the assumptions that$\beta$\beta^{-1}$ is Pisot and $\tau<1/2$ are crucial since they ensure that the weak separation condition holds (this is a key assumption in their theorem). As far as I know it is still not known if an ergodic measure of maximal dimension exists for all $\beta\in (1/2,1)$ (of course, this would follow from the above and the extremely difficult conjecture that the only singular Bernoulli convolutions come from Pisot numbers).
I don't know of any explicit formulas for the Hausdorff dimension of the attractor in the case that $\beta$ \beta^{-1}$is Pisot. In my paper Overlapping self-affine sets I gave a fairly explicit upper bound: $$\dim_H(\Lambda_{\beta,\tau})\le 1-\frac{\log (2\beta)}{\log\tau} + \tau_\beta(q)-(1-q),$$ where$\tau_\beta$is the (negative of the)$L^q$spectrum of the (uniform) Bernoulli convolution of parameter$\beta$and$q=\log\beta/\log\tau$(See Theorem 15 and the remark afterward). It is well known that the multifractality of the BC for Pisot parameters ensures that$\tau_\beta(q)<1-q$. I used to believe that this upper bound is in fact the Hausdorff dimension, but I don't have a proof. 1 For Pisot$\beta$there is an ergodic measure of maximal dimension. This is a special case of the rather difficult Theorem 2.15 in the paper Dimension Theory of iterated function systems by De-Jun Feng and Huyi Hu. Note that the assumptions that$\beta$is Pisot and$\tau<1/2$are crucial since they ensure that the weak separation condition holds. As far as I know it is still not known if an ergodic measure of maximal dimension exists for all$\beta\in (1/2,1)$(of course, this would follow from the above and the extremely difficult conjecture that the only singular Bernoulli convolutions come from Pisot numbers). I don't know of any explicit formulas for the Hausdorff dimension of the attractor in the case that$\beta$is Pisot. In my paper Overlapping self-affine sets I gave a fairly explicit upper bound: $$\dim_H(\Lambda_{\beta,\tau})\le 1-\frac{\log (2\beta)}{\log\tau} + \tau_\beta(q)-(1-q),$$ where$\tau_\beta$is the (negative of the)$L^q$spectrum of the (uniform) Bernoulli convolution of parameter$\beta$and$q=\log\beta/\log\tau$(See Theorem 15 and the remark afterward). It is well known that the multifractality of the BC for Pisot parameters ensures that$\tau_\beta(q)<1-q\$.