TITLE: Measures of full Hausdorff dimension for self-affine sets
QUESTION [6 upvotes]: 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 $\Lambda_{\beta,\tau}$.

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).

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). 

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?

REPLY [4 votes]: 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 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 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.