Oscillating overlap-statistics conjecture for parameterised iterated function systems

Let $U\subset\mathbb R^k$ be an open parameter space, let $\Phi_u$ be the parameterised iterated function systems described above, and let $X_u$ be their attractors. For a slowly decaying $\sigma$-invariant ergodic probability measure $\mathfrak m$, let $\lambda_1(\mathfrak m,u),\ldots,\lambda_d(\mathfrak m,u)$ be the Lyapunov exponents, and let $T$, $L_{\mathfrak m,n}$, $R_{\mathfrak m,n}$, and $\phi_{\mathbf a}$ have the meanings used in the source. Oscillating overlap-statistics conjecture. If

$$\mathfrak h(\mathfrak m)>-\bigl(\lambda_1(\mathfrak m,u)+\cdots+\lambda_d(\mathfrak m,u)\bigr)$$

for Lebesgue almost every $u\in U$, then, under some weak assumptions on $A_i$ and $t_i$, for almost every $u\in U$, every $z\in X_u$, and sufficiently small $s$, one has

$$0=\liminf_{n\to\infty}\frac{T\left(\{\phi_{\mathbf a}(z)\}_{\mathbf a\in L_{\mathfrak m,n}},\frac{s}{R_{\mathfrak m,n}^{1/d}}\right)}{R_{\mathfrak m,n}}<\limsup_{n\to\infty}\frac{T\left(\{\phi_{\mathbf a}(z)\}_{\mathbf a\in L_{\mathfrak m,n}},\frac{s}{R_{\mathfrak m,n}^{1/d}}\right)}{R_{\mathfrak m,n}}=1.$$

This conjecture is intended to describe oscillation in a statistic measuring overlap among cylinder images; the supplied text does not establish it or specify the weak assumptions.