Central limit theorem for the modulus of continuity of averages of observables on transversal families of piecewise expanding unimodal maps
| Autor: | Amanda de Lima, Daniel Smania |
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| Rok vydání: | 2018 |
| Předmět: |
Pure mathematics
Mathematics::Dynamical Systems Lebesgue measure General Mathematics 010102 general mathematics Function (mathematics) Dynamical Systems (math.DS) Absolute continuity TEORIA ERGÓDICA 01 natural sciences Modulus of continuity 010101 applied mathematics Mixing (mathematics) Transversal (combinatorics) Piecewise FOS: Mathematics 0101 mathematics Mathematics - Dynamical Systems 37C30 37C40 37E05 37A05 Central limit theorem Mathematics |
| Zdroj: | Repositório Institucional da USP (Biblioteca Digital da Produção Intelectual) Universidade de São Paulo (USP) instacron:USP |
| ISSN: | 1474-7480 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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