CENTRAL LIMIT THEOREM FOR THE MODULUS OF CONTINUITY OF AVERAGES OF OBSERVABLES ON TRANSVERSAL FAMILIES OF PIECEWISE EXPANDING UNIMODAL MAPS

Abstract

Consider a$C^{2}$family of mixing$C^{4}$piecewise expanding unimodal maps$t\in [a,b]\mapsto f_{t}$, with a critical point$c$, that is transversal to the topological classes of such maps. Given a Lipchitz observable$\unicode[STIX]{x1D719}$consider the function$$\begin{eqnarray}{\mathcal{R}}_{\unicode[STIX]{x1D719}}(t)=\int \unicode[STIX]{x1D719}\,d\unicode[STIX]{x1D707}_{t},\end{eqnarray}$$where$\unicode[STIX]{x1D707}_{t}$is the unique absolutely continuous invariant probability of$f_{t}$. Suppose that$\unicode[STIX]{x1D70E}_{t}>0$for every$t\in [a,b]$, where$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{t}^{2}=\unicode[STIX]{x1D70E}_{t}^{2}(\unicode[STIX]{x1D719})=\lim _{n\rightarrow \infty }\int \left(\frac{\mathop{\sum }_{j=0}^{n-1}\left(\unicode[STIX]{x1D719}\circ f_{t}^{j}-\int \unicode[STIX]{x1D719}\,d\unicode[STIX]{x1D707}_{t}\right)}{\sqrt{n}}\right)^{2}\,d\unicode[STIX]{x1D707}_{t}.\end{eqnarray}$$We show that$$\begin{eqnarray}m\left\{t\in [a,b]:t+h\in [a,b]\text{ and }\frac{1}{\unicode[STIX]{x1D6F9}(t)\sqrt{-\log |h|}}\left(\frac{{\mathcal{R}}_{\unicode[STIX]{x1D719}}(t+h)-{\mathcal{R}}_{\unicode[STIX]{x1D719}}(t)}{h}\right)\leqslant y\right\}\end{eqnarray}$$converges to$$\begin{eqnarray}\frac{1}{\sqrt{2\unicode[STIX]{x1D70B}}}\int _{-\infty }^{y}e^{-\frac{s^{2}}{2}}\,ds,\end{eqnarray}$$where$\unicode[STIX]{x1D6F9}(t)$is a dynamically defined function and$m$is the Lebesgue measure on$[a,b]$, normalized in such way that$m([a,b])=1$. As a consequence, we show that${\mathcal{R}}_{\unicode[STIX]{x1D719}}$is not a Lipchitz function on any subset of$[a,b]$with positive Lebesgue measure.