The entropy-rate dimension conjecture for self-similar measures on the line (including the exact overlaps conjecture)

Theorem 1.1: for every finite system of contracting similarities of $\mathbb R$ and every positive probability vector, $\dim_H\mu=\min\{1,h_{\mathrm{RW}}/\chi\}$, with no separation or arithmetic assumption; exact overlaps, repeated maps, negative and unequal ratios are allowed. Without exact overlaps this gives $\min\{1,H(p)/\chi\}$ (the exact overlaps conjecture in dimension one) and the attractor formula $\dim_HK=\min\{1,s\}$ with $\sum|r_i|^s=1$. It says nothing about absolute continuity (for instance of Bernoulli convolutions) and does not treat higher-dimensional or non-conformal systems.

Theorem 1.1: for every finite system of contracting similarities of $\mathbb R$ and every positive probability vector, $\dim_H\mu=\min\{1,h_{\mathrm{RW}}/\chi\}$, with no separation or arithmetic assumption; exact overlaps, repeated maps, negative and unequal ratios are allowed. Without exact overlaps this gives $\min\{1,H(p)/\chi\}$ (the exact overlaps conjecture in dimension one) and the attractor formula $\dim_HK=\min\{1,s\}$ with $\sum|r_i|^s=1$. It says nothing about absolute continuity (for instance of Bernoulli convolutions) and does not treat higher-dimensional or non-conformal systems.