CENTRAL LIMIT THEOREM FOR THE MODULUS OF CONTINUITY OF AVERAGES OF OBSERVABLES ON TRANSVERSAL FAMILIES OF PIECEWISE EXPANDING UNIMODAL MAPS release_6yyqofw4ljd6begeubsrbzlavi

Abstract

Consider a <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline1" xlink:type="simple" /><jats:tex-math>$C^{2}$</jats:tex-math></jats:alternatives> </jats:inline-formula> family of mixing <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline2" xlink:type="simple" /><jats:tex-math>$C^{4}$</jats:tex-math></jats:alternatives> </jats:inline-formula> piecewise expanding unimodal maps <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline3" xlink:type="simple" /><jats:tex-math>$t\in [a,b]\mapsto f_{t}$</jats:tex-math></jats:alternatives> </jats:inline-formula>, with a critical point <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline4" xlink:type="simple" /><jats:tex-math>$c$</jats:tex-math></jats:alternatives> </jats:inline-formula>, that is transversal to the topological classes of such maps. Given a Lipchitz observable <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline5" xlink:type="simple" /><jats:tex-math>$\unicode[STIX]{x1D719}$</jats:tex-math></jats:alternatives> </jats:inline-formula> consider the function <jats:disp-formula id="S1474748016000177_eqnU1"> <jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:href="S1474748016000177_eqnU1" xlink:type="simple" /><jats:tex-math>$$\begin{eqnarray}{\mathcal{R}}_{\unicode[STIX]{x1D719}}(t)=\int \unicode[STIX]{x1D719}\,d\unicode[STIX]{x1D707}_{t},\end{eqnarray}$$</jats:tex-math></jats:alternatives> </jats:disp-formula> where <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline6" xlink:type="simple" /><jats:tex-math>$\unicode[STIX]{x1D707}_{t}$</jats:tex-math></jats:alternatives> </jats:inline-formula> is the unique absolutely continuous invariant probability of <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline7" xlink:type="simple" /><jats:tex-math>$f_{t}$</jats:tex-math></jats:alternatives> </jats:inline-formula>. Suppose that <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline8" xlink:type="simple" /><jats:tex-math>$\unicode[STIX]{x1D70E}_{t}&gt;0$</jats:tex-math></jats:alternatives> </jats:inline-formula> for every <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline9" xlink:type="simple" /><jats:tex-math>$t\in [a,b]$</jats:tex-math></jats:alternatives> </jats:inline-formula>, where <jats:disp-formula id="S1474748016000177_eqnU2"> <jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:href="S1474748016000177_eqnU2" xlink:type="simple" /><jats:tex-math>$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{t}^{2}=\unicode[STIX]{x1D70E}_{t}^{2}(\unicode[STIX]{x1D719})=\lim _{n\rightarrow \infty }\int \left(\frac{\mathop{\sum }_{j=0}^{n-1}\left(\unicode[STIX]{x1D719}\circ f_{t}^{j}-\int \unicode[STIX]{x1D719}\,d\unicode[STIX]{x1D707}_{t}\right)}{\sqrt{n}}\right)^{2}\,d\unicode[STIX]{x1D707}_{t}.\end{eqnarray}$$</jats:tex-math></jats:alternatives> </jats:disp-formula> We show that <jats:disp-formula id="S1474748016000177_eqnU3"> <jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:href="S1474748016000177_eqnU3" xlink:type="simple" /><jats:tex-math>$$\begin{eqnarray}m\left\{t\in [a,b]:t+h\in [a,b]\text{ and }\frac{1}{\unicode[STIX]{x1D6F9}(t)\sqrt{-\log |h|}}\left(\frac{{\mathcal{R}}_{\unicode[STIX]{x1D719}}(t+h)-{\mathcal{R}}_{\unicode[STIX]{x1D719}}(t)}{h}\right)\leqslant y\right\}\end{eqnarray}$$</jats:tex-math></jats:alternatives> </jats:disp-formula> converges to <jats:disp-formula id="S1474748016000177_eqnU4"> <jats:alternatives><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:href="S1474748016000177_eqnU4" xlink:type="simple" /><jats:tex-math>$$\begin{eqnarray}\frac{1}{\sqrt{2\unicode[STIX]{x1D70B}}}\int _{-\infty }^{y}e^{-\frac{s^{2}}{2}}\,ds,\end{eqnarray}$$</jats:tex-math></jats:alternatives> </jats:disp-formula> where <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline10" xlink:type="simple" /><jats:tex-math>$\unicode[STIX]{x1D6F9}(t)$</jats:tex-math></jats:alternatives> </jats:inline-formula> is a dynamically defined function and <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline11" xlink:type="simple" /><jats:tex-math>$m$</jats:tex-math></jats:alternatives> </jats:inline-formula> is the Lebesgue measure on <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline12" xlink:type="simple" /><jats:tex-math>$[a,b]$</jats:tex-math></jats:alternatives> </jats:inline-formula>, normalized in such way that <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline13" xlink:type="simple" /><jats:tex-math>$m([a,b])=1$</jats:tex-math></jats:alternatives> </jats:inline-formula>. As a consequence, we show that <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline14" xlink:type="simple" /><jats:tex-math>${\mathcal{R}}_{\unicode[STIX]{x1D719}}$</jats:tex-math></jats:alternatives> </jats:inline-formula> is not a Lipchitz function on any subset of <jats:inline-formula> <jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:href="S1474748016000177_inline15" xlink:type="simple" /><jats:tex-math>$[a,b]$</jats:tex-math></jats:alternatives> </jats:inline-formula> with positive Lebesgue measure.
In application/xml+jats format

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