Consider the iterated function system $T\_{1}(x)=(\beta x,\tau y)$, $T\_{2}(x,y)=(\beta x+(1-\beta),\tau y+ (1-\tau))$ for $\beta\in(1/2,1)$ and $\tau\in (0,1/2)$ with self affine set $\Lambda\_{\beta,\tau}$.




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It is known that for almost all $\beta$ and all $\tau$ $\dim\_{H}\Lambda\_{\beta,\tau}=\dim b=1-\log(2\beta)/\log(\tau)$ where $b=(1/2,1/2)$ is the Bernoulli measure on $\Lambda$. Essential this is due to the fact that the projection of $b$ to the $x$-axis is an infinite Bernoulli convolution which is know to be generic absolutely continues (Solomyak theorem).




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On the other hand we have shown that if $\beta^{-1}$ is a Pisot number we have
 $\dim b<\dim\_{H}\Lambda\_{\beta,\tau}<1-\log(2\beta)/\log(\tau)$ for all Bernoulli measures $b$. The projection of all $b$ in this case is singular with a dimension drop (Erdös theorem). 




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Now here comes the question: Is there a (ergodic) measure of full dimension and what is $\dim\_{H}\Lambda\_{\beta,\tau}$ in the later case?



Measures of full Hausdorff dimension for self-affine sets
If $\beta^{-1}$ 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](http://arxiv.org/abs/1002.2036) 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 $\beta^{-1}$ 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 $\beta^{-1}$ is Pisot. In my paper [Overlapping self-affine sets](http://arxiv.org/abs/math/0408203) 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. 


