Estimating the measure of a pre-image of a polynomial

This question was previously posted on MSE https://math.stackexchange.com/questions/3305781/estimating-the-measure-of-a-pre-image-of-a-polynomial

Let $$\sigma := 2/(3\sqrt{3})$$, be a real number. And consider the polynomial functions $$P_i:[-1,1]^{i+1}\to\mathbb{R}$$ defined in the following way:

• $$P_0 (x_0) = x_0$$;
• $$P_1(x_0,x_1) = x_0^3 + \sigma\cdot x_1$$;
• $$\vdots$$
• $$P_{i+1}(x_0,\ldots,x_{i+1}) = (P_{i}(x_0,...,x_{i}))^3 + \sigma\cdot x_{i+1}$$.

Note that there is only one real root $$x^*:= x^*(\sigma)$$ of the following cubic equation $$x^3 + \sigma = x.$$

Now, let $$\varepsilon>0$$ (small) and define $$Q_i(\varepsilon) := m(P_i^{-1}(x^* - \varepsilon,x^* +\varepsilon)),$$ where $$m$$ is the lebesgue measure (remember $$P_i^{-1}(x_* - \varepsilon,x_* +\varepsilon) \subset [-1,1]$$).

My Question: Is it possible estimate $$Q_i(\varepsilon)$$? Moreover, there exists $$i_0>0$$, such that $$Q_i(\varepsilon)\geq Q_{i+1}(\varepsilon)$$, $$\forall$$ $$i>i_0$$?