15:{"metadata":[["$","title","0",{"children":"Note on the Transition to Intermittency for the exponential of the Square of a Steinhaus Series"}],["$","meta","1",{"name":"description","content":"Intermittency of $\\mathcal{E}_N(x,g)=\\exp\\lbrack g| S_N(x)|^2\\rbrack$ as $N\\to +\\infty$ is investigated on a $d$-dimensional torus $\\Lambda$, when $S_N(x)$ is a finite Steinhaus series of $(2N+1)^d$ terms normalized to $<| S_N(x)|^2> =1$. Assuming ergodicity of $\\mathcal{E}_N(x,g)$ as $N\\to +\\infty$ in the domain $g<1$, where $\\lim_{N\\to +\\infty}<\\mathcal{E}_N(g)>$ exists, transition to intermittency is proved as $g$ increases past the threshold $g_{th}=1$. This transition goes together with a transition from (assumed) ergodicity at $g\\rbrack^{-1}\\int_{\\Lambda}\\mathcal{E}_N(x,g) d^dx=0$ at $g>g_{th}$. In this asymptotic sense one can say that ergodicity is lost as $g$ increases past the value $g=1$."}],["$","meta","2",{"name":"author","content":"Philippe Mounaix"}],["$","meta","3",{"name":"author","content":"Pierre Collet"}],["$","meta","4",{"name":"keywords","content":"Mathematical Physics"}],["$","meta","5",{"name":"referrer","content":"origin"}],["$","meta","6",{"name":"robots","content":"index, follow"}],["$","link","7",{"rel":"canonical","href":"https://synthical.com/article/Note-on-the-Transition-to-Intermittency-for-the-exponential-of-the-Square-of-a-Steinhaus-Series-134f5db4-ffd3-11ed-90ce-72eb57fa10b3"}],["$","link","8",{"rel":"alternate","hrefLang":"de-DE","href":"https://synthical.com/article/Note-on-the-Transition-to-Intermittency-for-the-exponential-of-the-Square-of-a-Steinhaus-Series-134f5db4-ffd3-11ed-90ce-72eb57fa10b3/de"}],["$","link","9",{"rel":"alternate","hrefLang":"fr-FR","href":"https://synthical.com/article/Note-on-the-Transition-to-Intermittency-for-the-exponential-of-the-Square-of-a-Steinhaus-Series-134f5db4-ffd3-11ed-90ce-72eb57fa10b3/fr"}],["$","link","10",{"rel":"alternate","hrefLang":"es-ES","href":"https://synthical.com/article/Note-on-the-Transition-to-Intermittency-for-the-exponential-of-the-Square-of-a-Steinhaus-Series-134f5db4-ffd3-11ed-90ce-72eb57fa10b3/es"}],["$","link","11",{"rel":"alternate","hrefLang":"it-IT","href":"https://synthical.com/article/Note-on-the-Transition-to-Intermittency-for-the-exponential-of-the-Square-of-a-Steinhaus-Series-134f5db4-ffd3-11ed-90ce-72eb57fa10b3/it"}],["$","link","12",{"rel":"alternate","hrefLang":"ru-RU","href":"https://synthical.com/article/Note-on-the-Transition-to-Intermittency-for-the-exponential-of-the-Square-of-a-Steinhaus-Series-134f5db4-ffd3-11ed-90ce-72eb57fa10b3/ru"}],["$","meta","13",{"property":"og:title","content":"Note on the Transition to Intermittency for the exponential of the Square of a Steinhaus Series"}],["$","meta","14",{"property":"og:description","content":"Intermittency of $\\mathcal{E}_N(x,g)=\\exp\\lbrack g| S_N(x)|^2\\rbrack$ as $N\\to +\\infty$ is investigated on a $d$-dimensional torus $\\Lambda$, when $S_N(x)$ is a finite Steinhaus series of $(2N+1)^d$ terms normalized to $<| S_N(x)|^2> =1$. Assuming ergodicity of $\\mathcal{E}_N(x,g)$ as $N\\to +\\infty$ in the domain $g<1$, where $\\lim_{N\\to +\\infty}<\\mathcal{E}_N(g)>$ exists, transition to intermittency is proved as $g$ increases past the threshold $g_{th}=1$. This transition goes together with a transition from (assumed) ergodicity at $g\\rbrack^{-1}\\int_{\\Lambda}\\mathcal{E}_N(x,g) d^dx=0$ at $g>g_{th}$. In this asymptotic sense one can say that ergodicity is lost as $g$ increases past the value $g=1$."}],["$","meta","15",{"property":"og:url","content":"https://synthical.com/article/Note-on-the-Transition-to-Intermittency-for-the-exponential-of-the-Square-of-a-Steinhaus-Series-134f5db4-ffd3-11ed-90ce-72eb57fa10b3"}],["$","meta","16",{"property":"og:site_name","content":"Synthical"}],["$","meta","17",{"property":"og:locale","content":"en-US"}],["$","meta","18",{"property":"og:image","content":"https://synthical.com/preview/article/134f5db4-ffd3-11ed-90ce-72eb57fa10b3"}],["$","meta","19",{"property":"og:image:alt","content":"Note on the Transition to Intermittency for the exponential of the Square of a Steinhaus Series"}],["$","meta","20",{"property":"og:image:width","content":"1200"}],["$","meta","21",{"property":"og:image:height","content":"630"}],["$","meta","22",{"property":"og:type","content":"article"}],["$","meta","23",{"property":"article:published_time","content":"2009-01-13T14:19:03.000Z"}],["$","meta","24",{"property":"article:modified_time","content":"2009-03-05T14:47:51.000Z"}],["$","meta","25",{"property":"article:author:name","content":"Philippe Mounaix"}],["$","meta","26",{"property":"article:author:id","content":"e942790f-ca38-45dc-8670-97981b70fb21"}],["$","meta","27",{"property":"article:author:alternatives","content":"Philippe Mounaix,P. Mounaix"}],["$","meta","28",{"property":"article:author:avatar","content":"default_1"}],["$","meta","29",{"property":"article:author:articles_count","content":"24"}],["$","meta","30",{"property":"article:author:name","content":"Pierre Collet"}],["$","meta","31",{"name":"twitter:card","content":"summary_large_image"}],["$","meta","32",{"name":"twitter:title","content":"Note on the Transition to Intermittency for the exponential of the Square of a Steinhaus Series"}],["$","meta","33",{"name":"twitter:description","content":"Intermittency of $\\mathcal{E}_N(x,g)=\\exp\\lbrack g| S_N(x)|^2\\rbrack$ as $N\\to +\\infty$ is investigated on a $d$-dimensional torus $\\Lambda$, when $S_N(x)$ is a finite Steinhaus series of $(2N+1)^d$ terms normalized to $<| S_N(x)|^2> =1$. Assuming ergodicity of $\\mathcal{E}_N(x,g)$ as $N\\to +\\infty$ in the domain $g<1$, where $\\lim_{N\\to +\\infty}<\\mathcal{E}_N(g)>$ exists, transition to intermittency is proved as $g$ increases past the threshold $g_{th}=1$. This transition goes together with a transition from (assumed) ergodicity at $g\\rbrack^{-1}\\int_{\\Lambda}\\mathcal{E}_N(x,g) d^dx=0$ at $g>g_{th}$. 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