CENTRAL LIMIT THEOREM FOR THE MODULUS OF CONTINUITY OF AVERAGES OF OBSERVABLES ON TRANSVERSAL FAMILIES OF PIECEWISE EXPANDING UNIMODAL MAPS
release_6yyqofw4ljd6begeubsrbzlavi
by
Amanda de Lima,
Daniel Smania
Entity Metadata (schema)
abstracts[] |
{'sha1': '354bbe68485dbd89028e55e719a0966bff9a8d25', 'content': 'Consider a <jats:inline-formula>\n <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>\n </jats:inline-formula> family of mixing <jats:inline-formula>\n <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>\n </jats:inline-formula> piecewise expanding unimodal maps <jats:inline-formula>\n <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>\n </jats:inline-formula>, with a critical point <jats:inline-formula>\n <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>\n </jats:inline-formula>, that is transversal to the topological classes of such maps. Given a Lipchitz observable <jats:inline-formula>\n <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>\n </jats:inline-formula> consider the function <jats:disp-formula id="S1474748016000177_eqnU1">\n <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>\n </jats:disp-formula> where <jats:inline-formula>\n <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>\n </jats:inline-formula> is the unique absolutely continuous invariant probability of <jats:inline-formula>\n <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>\n </jats:inline-formula>. Suppose that <jats:inline-formula>\n <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>$\\unicode[STIX]{x1D70E}_{t}>0$</jats:tex-math></jats:alternatives>\n </jats:inline-formula> for every <jats:inline-formula>\n <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>\n </jats:inline-formula>, where <jats:disp-formula id="S1474748016000177_eqnU2">\n <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>\n </jats:disp-formula>\n We show that <jats:disp-formula id="S1474748016000177_eqnU3">\n <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>\n </jats:disp-formula> converges to <jats:disp-formula id="S1474748016000177_eqnU4">\n <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>\n </jats:disp-formula> where <jats:inline-formula>\n <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>\n </jats:inline-formula> is a dynamically defined function and <jats:inline-formula>\n <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>\n </jats:inline-formula> is the Lebesgue measure on <jats:inline-formula>\n <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>\n </jats:inline-formula>, normalized in such way that <jats:inline-formula>\n <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>\n </jats:inline-formula>. As a consequence, we show that <jats:inline-formula>\n <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>\n </jats:inline-formula> is not a Lipchitz function on any subset of <jats:inline-formula>\n <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>\n </jats:inline-formula> with positive Lebesgue measure.', 'mimetype': 'application/xml+jats', 'lang': None}
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filesets |
[]
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issue |
03
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language |
en
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license_slug |
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number |
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original_title |
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pages |
673-733
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publisher |
Cambridge University Press (CUP)
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refs[] |
{'index': 0, 'target_release_id': None, 'extra': {'doi': '10.1090/s0002-9947-1973-0335758-1'}, 'key': 'S1474748016000177_r12', 'year': None, 'container_name': None, 'title': None, 'locator': None}
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release_date |
2016-07-13
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release_stage |
published
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release_type |
article-journal
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release_year |
2016
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subtitle |
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title |
CENTRAL LIMIT THEOREM FOR THE MODULUS OF CONTINUITY OF AVERAGES OF OBSERVABLES ON TRANSVERSAL FAMILIES OF PIECEWISE EXPANDING UNIMODAL MAPS
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version |
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volume |
17
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webcaptures |
[]
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withdrawn_date |
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withdrawn_status |
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withdrawn_year |
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work_id |
ptazqjmidzfizobayino6x4scq
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['S1474748016000177']
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journal-article
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