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>$\\unicode[STIX]{x1D70E}_{t}&gt;0$</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}
container {'state': 'active', 'ident': 'enxafoteq5dctg5kwrketuplim', 'revision': 'ccbd223c-4709-4fda-9f61-3fe1702d56a9', 'redirect': None, 'extra': {'country': 'gb', 'ezb': {'color': 'red', 'ezb_id': '30303'}, 'kbart': {'clockss': {'year_spans': [[2008, 2022]]}, 'portico': {'year_spans': [[2008, 2022]]}, 'scholarsportal': {'year_spans': [[2002, 2020]]}}, 'publisher_type': 'unipress', 'sherpa_romeo': {'color': 'green'}, 'urls': ['https://www.cambridge.org/core/journals/journal-of-the-institute-of-mathematics-of-jussieu', 'http://journals.cambridge.org/action/displayJournal?jid=JMJ']}, 'edit_extra': None, 'name': 'Journal of the Institute of Mathematics of Jussieu', 'container_type': 'journal', 'publication_status': None, 'publisher': 'Cambridge University Press', 'issnl': '1474-7480', 'issne': '1475-3030', 'issnp': '1474-7480', 'wikidata_qid': None}
container_id enxafoteq5dctg5kwrketuplim
contribs[] {'index': 0, 'creator_id': None, 'creator': None, 'raw_name': 'Amanda de Lima', 'given_name': None, 'surname': None, 'role': 'author', 'raw_affiliation': None, 'extra': {'seq': 'first'}}
{'index': 1, 'creator_id': None, 'creator': None, 'raw_name': 'Daniel Smania', 'given_name': None, 'surname': None, 'role': 'author', 'raw_affiliation': None, 'extra': None}
ext_ids {'doi': '10.1017/s1474748016000177', 'wikidata_qid': None, 'isbn13': None, 'pmid': None, 'pmcid': None, 'core': '29516374', 'arxiv': None, 'jstor': None, 'ark': None, 'mag': None, 'doaj': None, 'dblp': None, 'oai': None, 'hdl': None}
files[] {'state': 'active', 'ident': 'iv3vzfxzwvbt7bbtghgkeir5fm', 'revision': '2b5d1da4-3eae-4906-b207-3eb4d69b77db', 'redirect': None, 'extra': {}, 'edit_extra': None, 'size': 783491, 'md5': 'd8066717f4f13f6c337851469f138f0a', 'sha1': '9bc8d9374b8bc3b7e99c2ed7b1a3a2f2fae09541', 'sha256': '492bc9b630bb5801dd14fce3005bd50f7b358af7f169cb104ac576b7f1dca331', 'urls': [], 'mimetype': 'application/pdf', 'content_scope': None, 'release_ids': ['6yyqofw4ljd6begeubsrbzlavi'], 'releases': None}
filesets []
issue 03
language en
license_slug
number
original_title
pages 673-733
publisher Cambridge University Press (CUP)
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}
{'index': 1, 'target_release_id': None, 'extra': {'doi': '10.1002/9780470316962'}, 'key': 'S1474748016000177_r6', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 2, 'target_release_id': None, 'extra': {'doi': '10.1088/0951-7715/21/8/003'}, 'key': 'S1474748016000177_r11', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 3, 'target_release_id': None, 'extra': {'doi': '10.1017/s0143385708001077'}, 'key': 'S1474748016000177_r4', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 4, 'target_release_id': None, 'extra': {'doi': '10.1007/bf00532744'}, 'key': 'S1474748016000177_r10', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 5, 'target_release_id': None, 'extra': {'doi': '10.3934/dcds.2009.23.685'}, 'key': 'S1474748016000177_r3', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 6, 'target_release_id': None, 'extra': {'doi': '10.1088/0951-7715/21/4/003'}, 'key': 'S1474748016000177_r2', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 7, 'target_release_id': None, 'extra': {'doi': '10.1007/s00220-007-0320-5'}, 'key': 'S1474748016000177_r1', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 8, 'target_release_id': None, 'extra': {'doi': '10.1142/s0219493712003675'}, 'key': 'S1474748016000177_r13', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 9, 'target_release_id': None, 'extra': {'authors': ['Viana'], 'volume-title': 'Sthochastic Dynamics of Deterministic Systems'}, 'key': 'S1474748016000177_r21', 'year': 1997, 'container_name': 'Sthochastic Dynamics of Deterministic Systems', 'title': None, 'locator': None}
{'index': 10, 'target_release_id': None, 'extra': {'doi': '10.1017/s014338570000050x'}, 'key': 'S1474748016000177_r20', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 11, 'target_release_id': None, 'extra': {'doi': '10.1007/s00440-014-0575-7'}, 'key': 'S1474748016000177_r19', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 12, 'target_release_id': None, 'extra': {'doi': '10.3934/dcds.2011.31.877'}, 'key': 'S1474748016000177_r18', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 13, 'target_release_id': None, 'extra': {'doi': '10.1088/0951-7715/11/1/002'}, 'key': 'S1474748016000177_r17', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 14, 'target_release_id': None, 'extra': {'doi': '10.1007/bf01667385'}, 'key': 'S1474748016000177_r9', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 15, 'target_release_id': None, 'extra': {'doi': '10.1007/s002200050134'}, 'key': 'S1474748016000177_r16', 'year': None, 'container_name': None, 'title': None, 'locator': None}
{'index': 16, 'target_release_id': None, 'extra': {'authors': ['Karatzas'], 'volume-title': 'Brownian Motion and Stochastic Calculus'}, 'key': 'S1474748016000177_r8', 'year': 1991, 'container_name': 'Brownian Motion and Stochastic Calculus', 'title': None, 'locator': None}
{'index': 17, 'target_release_id': None, 'extra': {'authors': ['Philipp']}, 'key': 'S1474748016000177_r15', 'year': 1975, 'container_name': 'Mem. Amer. Math. Soc.', 'title': None, 'locator': 'no. 161, iv+140 pp'}
{'index': 18, 'target_release_id': None, 'extra': {'doi': '10.1088/0951-7715/25/7/2203'}, 'key': 'S1474748016000177_r5', 'year': None, 'container_name': None, 'title': None, 'locator': None}
release_date 2016-07-13
release_stage published
release_type article-journal
release_year 2016
subtitle
title CENTRAL LIMIT THEOREM FOR THE MODULUS OF CONTINUITY OF AVERAGES OF OBSERVABLES ON TRANSVERSAL FAMILIES OF PIECEWISE EXPANDING UNIMODAL MAPS
version
volume 17
webcaptures []
withdrawn_date
withdrawn_status
withdrawn_year
work_id ptazqjmidzfizobayino6x4scq
As JSON via API

Extra Metadata (raw JSON)

crossref.alternative-id ['S1474748016000177']
crossref.type journal-article