| {"id":"trans et nb rotation ratio","label":"trans et nb rotation ratio","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Si \\(\\rho (T) \\in \\mathbb {Q}/\\mathbb {Z}\\), alors aucune orbite de \\(T\\) n’est dense dans \\({\\mathbb {T}}\\). De façon équivalente: si \\(T\\) est transitive, alors nécessairement \\(\\rho (T) \\notin \\mathbb {Q}/\\mathbb {Z}\\).","doc_url":null,"latex_link":"sect0002.html#trans et nb rotation ratio","lean_link":null,"lean_decl_name":"trans et nb rotation ratio","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"delta orbite","label":"delta orbite","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soient \\(n,m \\in \\mathbb {Z}\\) avec \\(n\\neq m\\). Soit \\(x_0 \\in {\\mathbb {T}}\\). Notons \\(\\Delta := [T^n(x_0),T^m(x_0)]\\) et \\(\\Delta ':=[T^m(x_0),T^n(x_0)]\\) les deux arcs de cercle joignant \\(T^n(x_0)\\) et \\(T^m(x_0)\\).Alors, pour tout \\(k_0 \\in {\\mathbb {N}}\\), il existe des entiers \\(k_1\\geq k_0\\) et \\(k_1'\\geq k_0\\) tels que \\[ {\\mathbb {T}}=\\overset {k_1}{\\underset {j=k_0}{\\bigcup }}T^{j(m-n)}\\Delta =\\overset {k_1’}{\\underset {j=k_0}{\\bigcup }}T^{j(m-n)}\\Delta '. \\]","doc_url":null,"latex_link":"sect0002.html#delta orbite","lean_link":null,"lean_decl_name":"delta orbite","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"E+","label":"E+","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Propriétés autour de \\({\\mathcal{E}}^+\\) (\\({\\mathcal{E}}\\),\\(\\circ \\)) et (\\(\\mathcal{E}^+\\),\\(\\circ \\)) sont des sous-groupes de \\(\\mathrm{Hom}(\\mathbb {R})\\). Pour tout \\(f \\in \\mathcal{E}^+\\), \\(f - \\mathrm{Id_{\\mathbb {R}}}\\) est continue et \\(1-\\)périodique sur \\(\\mathbb {R}\\), donc bornée et uniformément continue. Pour tout \\(f \\in \\mathcal{E}^+\\), \\(f\\) est uniformément continue sur \\(\\mathbb {R}\\). Par récurrence, on démontre facilement que pour tous \\(f \\in \\mathcal{E}^+\\), \\(k \\in \\mathbb {Z}\\), et \\(x \\in \\mathbb {R}\\), \\(f(x+k)=f(x)+k\\) et \\(f^k(x+1)=f^k(x)+1\\) (\\(f^k\\) désigne la composée et non le produit).","doc_url":null,"latex_link":"sect0002.html#E+","lean_link":null,"lean_decl_name":"E+","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Tactic/Ring/Basic.lean","lean4_code":"instance {α : Q(Type u)} {E : Q($α) → Type} {e : Q($α)} [Inhabited (Σ e, E e)] :\n Inhabited (Result E e) :=\n let ⟨e', v⟩ : Σ e, E e := default; ⟨e', v, default⟩\n\nvariable {α : Q(Type u)} (sα : Q(CommSemiring $α)) {R : Type*} [CommSemiring R]\n\n/--\nConstructs the expression corresponding to `.const n`.\n(The `.const` constructor does not check that the expression is correct.)\n-/\n","line":245,"end_line":254,"proof_status":"statement_formalized"} |
| {"id":"a0000000047","label":"a0000000047","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Notons \\(d_q'=\\min (d_q,1/2)\\). L’application \\(\\iota \\) obtenue dans le corollaire 6 est une isométrie de (\\({\\mathcal{E}}^+/\\mathcal{T}_{\\mathrm{ent}}, d_q')\\) dans \\((\\mathrm{Hom}^+({\\mathbb {T}}),\\overline{d}_\\infty )\\).Autrement dit: \\[ \\forall \\ \\overline{f}, \\overline{g} \\in {\\mathcal{E}}^+/\\mathcal{T}_{\\mathrm{ent}},\\ \\overline{d}_\\infty (T_f,T_g)=\\min (d_q(\\overline{f},\\overline{g}), 1/2). \\]","doc_url":null,"latex_link":"sect0002.html#a0000000047","lean_link":null,"lean_decl_name":"a0000000047","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"conjugaison","label":"conjugaison","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Le nombre de rotation est invariant par conjugaison topologique: pour tous \\(T_1\\), \\(T_2\\) dans \\(\\mathrm{Hom}^+({\\mathbb {T}})\\), \\[ \\rho (T_{1}\\circ T_{2}\\circ T_{1}^{-1})=\\rho (T_{2}). \\]","doc_url":null,"latex_link":"sect0002.html#conjugaison","lean_link":null,"lean_decl_name":"conjugaison","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"a0000000141","label":"a0000000141","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"L’ensemble fermé \\(L_T\\) des valeurs d’adhérence de la suite \\((T^n(x))_{n\\geq 0}\\), qui ne dépend pas de \\(x \\in {\\mathbb {T}}\\), est appelé l’ensemble dérivé de \\(T\\).","doc_url":null,"latex_link":"sect0002.html#a0000000141","lean_link":null,"lean_decl_name":"a0000000141","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"a0000000095","label":"a0000000095","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Si \\(f:\\mathbb {R} \\to \\mathbb {R}\\) est une fonction uniformément continue. On définit l’application module de continuité \\(\\omega _f: \\mathbb {R} \\to \\mathbb {R}^+\\) par \\(\\omega _f(\\delta ) := \\sup E_{f}(\\delta )\\), où \\[ E_{f}(\\delta ):=\\lbrace |f(x_1) - f(x_2)|: (x_1;x_2) \\in \\mathbb {R}^2,\\ |x_1 - x_2| \\leq \\delta \\rbrace \\]","doc_url":null,"latex_link":"sect0002.html#a0000000095","lean_link":null,"lean_decl_name":"a0000000095","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"a0000000106","label":"a0000000106","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(f: X \\to X\\) un homéomorphisme. On dit que \\(f\\) est transitif si et seulement si il existe un point \\(x \\in X\\) tel que l’orbite \\(O_x\\) est dense dans \\(X\\). On dit que \\(f\\) est minimal si et seulement si pour tout \\(x \\in X\\), l’orbite \\(O_x\\) est dense dans \\(X\\). Autrement dit, \\(f\\) est minimal si toutes les orbites sont denses dans \\(X\\).","doc_url":null,"latex_link":"sect0002.html#a0000000106","lean_link":null,"lean_decl_name":"a0000000106","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"bornesnbrot","label":"bornesnbrot","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Lemme fondamentalSoit f un relèvement de T. Notons \\[ a=\\underset {x \\in \\mathbb {R}}{\\inf } f(x) - x \\text{ et } b=\\underset {x \\in \\mathbb {R}}{\\sup } f(x) - x. \\]","doc_url":null,"latex_link":"sect0002.html#bornesnbrot","lean_link":null,"lean_decl_name":"bornesnbrot","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"psi","label":"psi","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Notons \\(\\mathcal{C}_U(\\mathbb {R},\\mathbb {R})\\) l’ensemble des fonctions uniformément continues définies de \\(\\mathbb {R}\\) dans \\({\\mathbb {R}}\\). L’application \\(\\psi : \\mathcal{C}_U(\\mathbb {R},\\mathbb {R}) \\times \\mathcal{C}_U(\\mathbb {R},\\mathbb {R}) \\to \\mathcal{C}_U(\\mathbb {R},\\mathbb {R})\\) définie par \\(\\psi (f,g)=f\\circ g\\) est continue pour la topologie associée à la norme de la convergence uniforme.","doc_url":null,"latex_link":"sect0002.html#psi","lean_link":null,"lean_decl_name":"psi","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"equivrationnel","label":"equivrationnel","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"On a l’équivalence \\[ \\rho (T) \\in \\mathbb {Q}/\\mathbb {Z} \\Longleftrightarrow \\exists k \\in \\mathbb {N}^*,\\ T^k \\text{a un point fixe.} \\]","doc_url":null,"latex_link":"sect0002.html#equivrationnel","lean_link":null,"lean_decl_name":"equivrationnel","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"a0000000145","label":"a0000000145","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(T\\in \\mathrm{Hom}^-({\\mathbb {T}})\\) un homéomorphisme renversant l’orientation. Alors \\(T\\) possède exactement deux points fixes.","doc_url":null,"latex_link":"sect0002.html#a0000000145","lean_link":null,"lean_decl_name":"a0000000145","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"a0000000146","label":"a0000000146","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(T \\in \\mathrm{Hom}^-({\\mathbb {T}})\\). Alors: \\(T^2 \\in \\mathrm{Hom}^+({\\mathbb {T}})\\) et \\(\\rho (T^2)=\\overline{0}\\). Pour tout \\(x \\in {\\mathbb {T}}\\), les suites \\((T^{2n}(x))_{n\\geq }0\\), \\((T^{2n+1}(x))_{n\\geq }0\\), \\((T^{-2n}(x))_{n\\geq }0\\) et \\((T^{-2n-1}(x))_{n\\geq }0\\) convergent vers des points fixes de \\(T^2\\). Par conséquent, l’adhérence de l’orbite de \\(x\\) est dénombrable.","doc_url":null,"latex_link":"sect0002.html#a0000000146","lean_link":null,"lean_decl_name":"a0000000146","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"Denjoy","label":"Denjoy","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Exemple de DenjoySoit \\(\\alpha \\) un irrationnel fixé. Tout ensemble \\(L\\) fermé d’intérieur non vide sans point isolé peut être vu comme ensemble dérivé d’un homéomorphisme de nombre de rotation \\(\\alpha \\).","doc_url":null,"latex_link":"sect0002.html#Denjoy","lean_link":null,"lean_decl_name":"Denjoy","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"minimal équivaut transitif","label":"minimal équivaut transitif","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Il y a équivalence entre les 3 propriétés suivantes: Il existe \\(x\\in {\\mathbb {T}}\\) tel que \\(O_x\\) soit dense dans \\({\\mathbb {T}}\\) (autrement dit, \\(T\\) est transitif). \\(T\\) est conjugué à une rotation d’angle irrationnel. Pour tout \\(x\\in {\\mathbb {T}}\\), l’orbite \\(O_x\\) de \\(x\\) est dense dans \\({\\mathbb {T}}\\) (autrement dit T est minimal).","doc_url":null,"latex_link":"sect0002.html#minimal équivaut transitif","lean_link":null,"lean_decl_name":"minimal équivaut transitif","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"a0000000101","label":"a0000000101","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Le nombre de rotation \\(\\rho (T)\\) dépend continûment de T.","doc_url":null,"latex_link":"sect0002.html#a0000000101","lean_link":null,"lean_decl_name":"a0000000101","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"poincaré","label":"poincaré","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Supposons que \\(\\rho (T) \\notin \\mathbb {Q}/\\mathbb {Z}\\), ie supposons que pour tout relèvement \\( f\\) de \\(T\\), \\(\\alpha (f) \\in {\\mathbb {R}}\\setminus {\\mathbb {Q}}\\). Notons \\(\\alpha _T\\) l’unique représentant de \\(\\rho (T)\\) dans [0,1[. La classification de Poincaré distingue deux cas: Si \\(T\\) est transitive, alors \\(T\\) est conjuguée à la rotation \\(R_{\\alpha _T}\\). Si \\(T\\) n’est pas transitive, \\(T\\) est seulement semi-conjuguée à la rotation \\(R_{\\alpha _T}\\). Plus particulièrement, il existe une application continue surjective \\(h\\) telle que \\[ h\\circ T = R_{\\alpha _T} \\circ h. \\]","doc_url":null,"latex_link":"sect0002.html#poincaré","lean_link":null,"lean_decl_name":"poincaré","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"ensemble dérivé","label":"ensemble dérivé","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(x \\in {\\mathbb {T}}\\). Notons \\(L_x\\) l’ensemble des valeurs d’adhérence de la suite \\((T^n(x))_{n\\geq 0}\\). L’ensemble \\(L_x\\) possède les trois propriétés suivantes: Il est invariant par \\(T\\) et \\(T^{-1}\\). Il ne dépend pas de \\(x\\). Il vaut soit \\({\\mathbb {T}}\\) tout entier, soit un fermé de \\({\\mathbb {T}}\\) d’intérieur vide sans point isolé.","doc_url":null,"latex_link":"sect0002.html#ensemble dérivé","lean_link":null,"lean_decl_name":"ensemble dérivé","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"distance sur HomT","label":"distance sur HomT","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"La métrique \\(d_q\\) définit la topologie quotient sur \\({\\mathcal{E}}^+/\\mathcal{T}_{\\mathrm{ent}}\\).","doc_url":null,"latex_link":"sect0002.html#distance sur HomT","lean_link":null,"lean_decl_name":"distance sur HomT","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"isom relation d'équiv sur les homéo","label":"isom relation d'équiv sur les homéo","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Isomorphisme entre \\(\\mathcal{E}^+/\\mathcal{T}_{\\mathrm{ent}}\\) et \\(\\mathrm{Hom}^+({\\mathbb {T}})\\)L’application \\(f\\mapsto T_f\\) de \\({\\mathcal{E}}^+\\) dans \\(\\mathrm{Hom}^+({\\mathbb {T}})\\) passe au quotient et fournit un isomorphisme \\(\\iota \\) de groupes de \\({\\mathcal{E}}^+/\\thicksim ={\\mathcal{E}}^+/\\mathcal{T}_{\\mathrm{ent}}\\) vers \\(\\mathrm{Hom}^+({\\mathbb {T}})\\).L’application \\(\\iota \\) est définie par: \\[ \\forall \\ \\overline{f} \\in {\\mathcal{E}}^+/\\mathcal{T}_{\\mathrm{ent}}, \\ \\iota (\\overline{f})= T_f. \\]","doc_url":null,"latex_link":"sect0002.html#isom relation d'équiv sur les homéo","lean_link":null,"lean_decl_name":"isom relation d'équiv sur les homéo","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"T[x;y]","label":"T[x;y]","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soient \\(x,y \\in {\\mathbb {R}}/{\\mathbb {Z}}\\), on a l’égalité \\(T([x,y])=[T(x),T(y)].\\)","doc_url":null,"latex_link":"sect0002.html#T[x;y]","lean_link":null,"lean_decl_name":"T[x;y]","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"k rho","label":"k rho","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Pour tout \\(k \\in \\mathbb {Z}\\), \\(\\rho (T^k)= k \\times \\rho (T)\\). En particulier, on a \\(\\rho (\\mathrm{Id_{{\\mathbb {T}}}})=0\\) et \\(\\rho (T^{-1})=-\\rho (T)\\).","doc_url":null,"latex_link":"sect0002.html#k rho","lean_link":null,"lean_decl_name":"k rho","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"a0000000051","label":"a0000000051","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\((u_n)_{n \\in \\mathbb {N}}\\) une suite réelle.On dit que \\((u_n)_{n \\in \\mathbb {N}}\\) est sous-additive si pour tout couple d’entier naturels \\((p,q)\\), on a: \\(u_{p+q} \\leq u_p + u_q\\)On dit que \\((u_n)_{n \\in \\mathbb {N}}\\) est sur-additive si pour tout couple d’entiers naturels \\((p, q)\\), on a: \\(u_{p+q} \\geq u_p + u_q\\)","doc_url":null,"latex_link":"sect0002.html#a0000000051","lean_link":null,"lean_decl_name":"a0000000051","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"a0000000110","label":"a0000000110","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soient \\(X\\) et \\(Y\\) deux espaces topologiques et \\(f:X\\to X\\), \\(g:Y \\to Y\\) deux homéomorphismes. On dit que \\(f\\) et \\(g\\) sont semi-conjuguées (resp. conjuguées) s’il existe une application continue surjective (resp. bijective) \\(h : Y \\to X\\) telle que \\(f\\circ h = h \\circ g\\).","doc_url":null,"latex_link":"sect0002.html#a0000000110","lean_link":null,"lean_decl_name":"a0000000110","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"préli relèvement","label":"préli relèvement","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(y_0 \\in {\\mathbb {R}}\\), et \\(\\varepsilon \\in \\ ]0,1/2[\\). La restriction de \\(\\pi \\) à \\([y_0-\\varepsilon , y_0 +\\varepsilon ]\\) est un homéomorphisme de \\([y_0-\\varepsilon , y_0 +\\varepsilon ]\\) vers son image.","doc_url":null,"latex_link":"sect0002.html#préli relèvement","lean_link":null,"lean_decl_name":"préli relèvement","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"ouvertUr","label":"ouvertUr","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(r \\in \\mathbb {Z}\\).Posons \\(U_r:=\\lbrace f \\in \\mathcal{E}^+: \\mathrm{Im}(f-Id_{\\mathbb {R}}) \\subset \\ ]r,r+1[\\rbrace \\). \\(U_r\\) est un ouvert de \\(\\mathcal{E}^+\\) muni de la topologie issue de la norme de la convergence uniforme.","doc_url":null,"latex_link":"sect0002.html#ouvertUr","lean_link":null,"lean_decl_name":"ouvertUr","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"a0000000111","label":"a0000000111","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit f un relèvement de T. Supposons \\(\\alpha (f) \\notin \\mathbb {Q}\\).Soient \\(n_1,n_2, m_1, m_2 \\in \\mathbb {Z}\\), il y a équivalence entre: \\(\\forall \\ x \\in \\mathbb {R}, \\ f^{n_1}(x)+m_1 {\\lt} f^{n_2}(x)+m_2\\). \\(n_1 \\alpha (f) + m_1 {\\lt} n_2 \\alpha (f) + m_2\\). \\(\\exists \\ x \\in \\mathbb {R}, \\ f^{n_1}(x)+m_1 {\\lt} f^{n_2}(x)+m_2\\).","doc_url":null,"latex_link":"sect0002.html#a0000000111","lean_link":null,"lean_decl_name":"a0000000111","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"a0000000033","label":"a0000000033","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"On définit sur \\({\\mathcal{E}}^+\\) la relation d’équivalence \\(\\thicksim \\) par: \\[ f\\ \\thicksim \\ g \\Longleftrightarrow f-g \\text{ est constante et à valeurs dans\\ } \\mathbb {Z}. \\]","doc_url":null,"latex_link":"sect0002.html#a0000000033","lean_link":null,"lean_decl_name":"a0000000033","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"passage au quotient","label":"passage au quotient","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Passage au quotientSoit \\(f\\in {\\mathcal{E}}\\), i.e. \\(f: \\mathbb {R} \\to \\mathbb {R}\\) une fonction continue et strictement monotone telle que \\(x\\mapsto f(x+1) -f(x)\\) est constante égale à \\(\\pm 1\\).Alors, il existe une unique application \\(T_f \\in \\mathrm{Hom}({\\mathbb {T}})\\), telle que \\(T_f\\circ \\pi = \\pi \\circ f\\).","doc_url":null,"latex_link":"sect0002.html#passage au quotient","lean_link":null,"lean_decl_name":"passage au quotient","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"rho commute","label":"rho commute","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soient \\(T_1\\), \\(T_2 \\in \\mathrm{Hom}^+({\\mathbb {T}})\\) tels que \\(T_1\\circ T_2=T_1 \\circ T_2\\). Alors, \\[ \\rho (T_1\\circ T_2)=\\rho (T_1)+\\rho (T_2). \\]","doc_url":null,"latex_link":"sect0002.html#rho commute","lean_link":null,"lean_decl_name":"rho commute","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"phin","label":"phin","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\(n \\in \\mathbb {N}\\), l’application \\(\\varphi _n : \\mathcal{C}_U(\\mathbb {R},\\mathbb {R}) \\to \\mathcal{C}_U(\\mathbb {R},\\mathbb {R})\\) définie par \\(\\varphi _n(f)=f^n\\) est continue pour la topologie associée à la norme de la convergence uniforme.","doc_url":null,"latex_link":"sect0002.html#phin","lean_link":null,"lean_decl_name":"phin","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"suiteadd","label":"suiteadd","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soit \\((u_n)_{n \\in \\mathbb {N}}\\) une suite réelle. Si \\((u_n)_{n \\in \\mathbb {N}}\\) est sous-additive, alors \\[ \\frac{u_n}{n} \\underset {n \\to + \\infty }{\\longrightarrow } \\underset {p \\geq 1}{\\inf }\\ \\frac{u_p}{p} \\in \\mathbb {R} \\cup \\lbrace - \\infty \\rbrace . \\]","doc_url":null,"latex_link":"sect0002.html#suiteadd","lean_link":null,"lean_decl_name":"suiteadd","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"a0000000142","label":"a0000000142","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"On a les équivalences suivantes: \\(L_T= {\\mathbb {T}}\\). \\(T\\) est minimal. \\(T\\) est transitif.","doc_url":null,"latex_link":"sect0002.html#a0000000142","lean_link":null,"lean_decl_name":"a0000000142","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"equiv amel","label":"equiv amel","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Soient f \\(\\in \\mathcal{E}^+\\) et \\(k,r \\in \\mathbb {Z}\\). Alors \\[ \\mathrm{Im}(f^k - \\mathrm{Id})\\subset \\ ]r,r+1[ \\Longleftrightarrow \\alpha (f^k) \\in ]r,r+1[. \\]","doc_url":null,"latex_link":"sect0002.html#equiv amel","lean_link":null,"lean_decl_name":"equiv amel","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
| {"id":"modulecont","label":"modulecont","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://vbeffara.github.io/Cours_ED/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Pour toute fonction \\(f\\) uniformément continue, le module de continuité \\(w_f\\) possède les 3 propriétés suivantes: \\(\\omega _f\\) est croissant. \\(\\omega _f\\) est sous-additif: \\(\\forall \\delta _1, \\delta _2 \\in \\mathbb {R}^+\\), \\(\\omega _f(\\delta _1 + \\delta _2) \\leq \\omega _f(\\delta _1) +\\omega _f(\\delta _2)\\). \\(\\omega _f(\\delta ) \\underset {\\delta \\to 0}{\\longrightarrow }0\\).","doc_url":null,"latex_link":"sect0002.html#modulecont","lean_link":null,"lean_decl_name":"modulecont","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"} |
|
|
