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html:Ž‘€].‘¸Here“w¸ãe“willŽ¡‘6à¤study–è†the“skš¸ãew“proGduct“of“this“natural“extension“with“µG²(µm²),‘þIand“w˜e“will“sho˜w“thatŽ¡‘6à¤this–ªnew“skš¸ãew“proGduct“is“actually“the“natural“extension“of“the“sk˜ew“proGduct“usedŽ¡‘6à¤b¸ãy–ßÐJager“and“Liardet.‘9W‘ÿ*ªe“rst“recall“the“denition“of“natural“extension,‘osee“alsoŽ¡‘6à¤[ïhtml:RŽ‘\rï html:Ž‘\r]–UUor“[ïhtml:BrŽ‘ï html:Ž‘].Ž©F˜‘6à¤ÕDenition‘ÕTïhtml:ï html:1.ŽŽ‘z…äÔL›ÿ}'et–·µT‘wFÔand“µS‘§DÔb˜e“two“me˜asur˜e“pr˜eserving“tr˜ansformations“of“²(µX:©;–ª¨¸BMÛµ;“²)Ž¡‘6à¤Ôand›¿‰²(µY‘Ž9;–ª¨¸C‘•Wµ;“‘¢ö²)˜Ôr–ÿ}'esp“e“ctively.‘{The˜tr“ansformation˜µS‘SÔis˜said˜to˜b“e˜a˜factor˜map˜of˜µT‘#ÔifŽ¡‘6à¤ther›ÿ}'e–“çexists“a“me˜asur˜able“map“µ‘Í™²:‘qÀ(µX:©;–ª¨¸BMÛµ;“²)–Ç¸!“²(µY‘Ž9;–ª¨¸C‘•Wµ;“‘¢ö²)–“çÔsuch“thatŽŸÒ–’•íx²(i)Ž’£BËµ‘j²=›Çµ–8à¸“µ[ÙŸûÞÿ·±1Ž‘M²;‘Ç(ii)Ž‘'ã€µ‘”¹¸“µT‘*§²=˜µS‘Ìm¸“µ‘:ŽŽŽŽ¦‘6à¤ÕDenition‘ÕTïhtml:ï html:2.ŽŽ‘z…äÔA¾“n› 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(Ö¸FŸÿ´nŽ‘q~¸FŸÿ´n±+1ŽŽŽŽŽ‘-qµ;ŽŸqË‘6à¤²where–|(¸FŸÿ´nŽ‘q~²)Ÿÿ´n·±0Ž‘µn²is“the“FibGonacci-sequence“1µ;–ª¨²1µ;“²2µ;“²3µ;“²5µ;“²8µ;“¸““Ž‘s².‘WÔHence–|for“all“irrationalŽ¡‘6à¤µx–UU²and“all“µy‘"ñ¸2‘Ç²[0µ;‘ª¨²1]ŽŸ;’§ÌtlimŽŸ’¤L˜´n·!1ŽŽ’ºÚˆ¸jŽ’½¡¥N‘y=ŸûÞÿ´nŽ›ê»²(µx;‘ª¨y[Ù²)–8à¸“N‘y=ŸûÞÿ´nŽ˜²(µx;‘ª¨²0)¸jŽŽ’tµ²=‘qÀ0µ;ŽŸ&Ô‘6à¤²and–8ãthe“con•¸ãv“ergence–8ãis“uniform.‘÷But“then“for“all“µy‘"ñ¸2‘Ç²[0µ;›ª¨²1]“the“sequence“(¸N‘y=Ÿü^ÿ´nŽ‘ê»²(µx;˜y[Ù²))Ÿÿ´n·±0ŽŽ¡‘6à¤²has–sthe“same“distribution“as“¸N‘y=Ÿü^ÿ´nŽ‘ê»²(µx;›ª¨²0))Ÿÿ´n·±0Ž‘ò².‘&†Th¸ãus,‘ ·also“the“sequences“¸T‘Š«Ÿü^ÿ´nŽ‘ü)²(µx;˜²0µ;˜g[Ù²))Ÿÿ´n·±0ŽŽ¡‘6à¤²and‘UU¸T‘Š«Ÿü^ÿ´nŽ‘ü)²(µx;–ª¨yš[Ù;“g˜²))Ÿÿ´n·±0Ž‘G²ha•¸ãv“e–UUthe“same“distribution“for“all“µy›"ñ¸2‘Ç²[0µ;‘ª¨²1]“and“all“µg˜¸2‘ÇµG²(µm²).Ž¡‘Bà¤No¸ãw–Ëlet“µE‘<¸›¨…²[0µ;‘ª¨²1)–_Ø¸n“ÕQŽ‘.²bGe–Ëthe“set“of“those“irrational“µx˜¸2˜²[0µ;‘ª¨²1),‘~ifor“whic¸ãhŽ¦‘6à¤(¸N‘y=Ÿü^ÿ´nŽ‘ê»²(µx;›ª¨²0))Ÿÿ´n·±0Ž‘€ä²is–ÒòÕnot“²distributed“o•¸ãv“er–Òò “according“to“Ÿú~™‰fe†ŸgµŽŽŽ‘ Ùx²,‘2Ythen“Ÿ÷÷}‰feõTŸƒµEŽŽŽ‘¶ 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html:1.ŽŽ‘x$ÔL›ÿ}'et–Õ µa“Ôand“µm“Ôb˜e“two“inte˜gers,›åRµm–=¸“²2Ô,˜and–Õ let“²(µcŸÿ±1Ž–|sµ;›ª¨cŸÿ±2Ž“²)–=¸2“²[0µ;˜²1]Ÿü^ÿ±2Ž‘|sÔ.‘\üF‘ÿ;¼ur-Ž©‘6à¤thermor‘ÿ}'e,–“çlet“µA²(µx;–ª¨cŸÿ±1Ž›|sµ;“cŸÿ±2Ž˜µ;“N‘²)–“çÔb›ÿ}'e“the“c˜ar˜dinality“of“the“set“¸A²(µx;–ª¨cŸÿ±1Ž›|sµ;“cŸÿ±2Ž˜µ;“N‘²)Ô,–“çgiven“byŽ¤32‘Eô´¸fµn²;›UP1–Ç¸“µn“¸“µN‘ˆâ;˜²Ÿÿ´n·±1Ž‘u µ<“cŸÿ±1Ž‘|sµ;˜²Ÿÿ´nŽ‘8–µ<“cŸÿ±2Ž‘|sµ;˜pŸÿ´nŽ›8–¸“µa²(moGdŽ‘€µm²)‘“çÔandŽ‘ÿöµqŸÿ´nŽ˜¸“µb²(moGdŽ‘€µm²)¸g‘ª¨µ:Ž¡‘6à¤ÔThen–“çfor“almost“al‘‚Øl“µx“Ôone“hasŽŸhÕ’¡²limŽŸs3’œs´N‘À,·!1ŽŽŸù<$’¸hn²1Ž’¶XoŸw‰fe þŸ (ÖµNŽŽŽŽ’À« A²(µx;–ª¨cŸÿ±1Ž›|sµ;“cŸÿ±2Ž˜µ;“N‘²)‘qÀ=Ÿù<$‘¤óµG²(µcŸÿ±1Ž˜µ;“cŸÿ±2Ž˜²)Ž‘¤óŸw‰fe%¶KŸ (Ö‘RØµJ‘ö9²(µm²)ŽŽŽŽ‘,Žqµ;ŽŸ;‘6à¤Ôwher‘ÿ}'e›“çµG²(µcŸÿ±1Ž–|sµ;‘ª¨cŸÿ±2Ž“²)˜Ôis˜a˜distribution˜function˜with˜density˜µg[ÙÔ,˜given˜byŽŸ$è‘oß˜µg•[Ù²(µuÇ;‘ª¨“²)–qÀ=Ÿëô“«8ŽŸ “<ŽŸ“:ŽŽŸóŸü‘cT±1Ž‘ˆ‚Ÿ£&‰fe²Ÿ¿˜logŽ‘µ£2ŽŽŽŽŸü‘2'ú1Ž‘% þŸ£&‰fe kŸ_XŸúµ·pŽ‘‘ÉŸúµ‰W x¢ŸJá±1·±4´=ŽŽŽŽŽŽ‘CÞœµ;Ž‘P¥¹‘<ß>–Ç²0µ;‘UP›"ñ>“²0‘“çÔandŽ‘ÿöµ‘®§²+‘8àµ˜<“²1µ;ŽŽŸ 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html:Ž‘Ç],“Section“2,“it“wš¸ãas“sho˜wn“thatŽŸá8Ÿýxä‘eûd^ŽŸ‡‘b@¸µMŸÿ´nŽŽŽ‘t,Ý²=Ÿýxä‘Ä~ŽŸ‡‘ÇµMŸÿ´nŽŽŽ‘Az²forŽ‘$µµn–Ç²=“1µ;–ª¨:“:“:Ž‘ªŸ;‘ª¨k›‰w¸‘8à²1;Ÿýxä‘º¤^ŽŸ‡‘ÿøµMŸÿ´nŽŽŽ‘ì²=Ÿýxä‘Ä~ŽŸ‡“µMŸÿ´n±+1ŽŽŽ‘a}²forŽ‘.¯¸µn“²=“µk˜²+‘8à1µ;–ª¨:“:“:ŽŽŸèU‘6à¤²andŽŸ5*Ÿýxä’¦^ŽŸ‡’©]úµMŸÿ´kŽŽŽ’¼nÙ²=Ÿýxä‘,l~ŽŸ‡‘qÀµMŸÿ´k+B±+1ŽŽŽ‘ÛŠŸñæ^«ŽŸùæd‘+7þ¸µ"Ÿÿ´k+B±+1ŽŽ‘P´u²0ŽŽ¡‘/µ"Ÿÿ´k+B±+1ŽŽ‘P´u²1ŽŽŽ‘Z´vŸñæ^«ŽŽ‘ef:µ:ŽŸÎÄ‘6à¤²F‘ÿ*ªrom–pthis“it“follo¸ãws“that“(µcŸÿ´nŽ–q~µ=dŸÿ´nŽ“²)Ÿÿ´n·±1Ž›ýc²is–pobtained“from“(µrŸÿ´nŽ–q~µ=sŸÿ´nŽ“²)Ÿÿ´n·±1Ž˜²b¸ãy‘pskippingŽ¡‘6à¤the–UUterm“µrŸÿ´kŽ–ëµ=sŸÿ´kŽ“µ:–UU²See“also“[ïhtml:KŽ‘Çï html:Ž‘Ç],“Sections“2“and“4.Ž¡‘Bà¤A‘·Rsimple›·zw•¸ãa“y˜to˜deriv“e˜a˜strategy˜for˜singularization˜is˜giv“en˜b“y˜a˜ÔsingularizationŽ¡‘6à¤ar–ÿ}'e“a‘“çµS‘“Ô.Ž©Œ‘6à¤ÕDenition‘ÕTïhtml:ï html:3.ŽŽ‘z…äÔA–“çsubset“µS‘'tÔfr›ÿ}'om“² “Ôis“c˜al‘‚Øle˜d“a“singularisation“ar˜e˜a“if“it“satisesŽŸ¿‘Eü²(i)Ž‘RlOµS›Z¥¸2–ÇB‘ŒjÔandŽ‘¿‘µ²(µ@‘Ž8S‘“²)“=“0;‘ÿø(ii)Ž‘hµS˜¸“²([Ÿù<$‘331Ž‘33Ÿw‰feŸ (Ö2ŽŽŽŽ‘fgµ;›ª¨²1)–8à¸n“ÕQŽ‘ 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html:“that“for“an˜y“µS‘“²-expansion“and“forŽ¡‘6à¤almost–Õ~evš¸ãery“µx“²the“sequence“of“n˜umerators“(µrŸÿ´nŽ›q~²)Ÿÿ´n·±1Ž‘Ãq²and“denominators“(µsŸÿ´nŽ˜²)Ÿÿ´n·±1ŽŽ¡‘6à¤²of–j¹the“µS‘“²-con•¸ãv“ergen“ts›j¹(µrŸÿ´nŽ–q~µ=sŸÿ´nŽ“²)Ÿÿ´n·±1Ž‘X¬²of˜µx˜²ha•¸ãv“e˜|˜moGd˜µm˜²|˜the˜same˜asymptoticŽ¡‘6à¤bGehaš¸ãviour–‘þas“the“sequence“of“n˜umerators“(µpŸÿ´nŽ›q~²)Ÿÿ´n·±1Ž‘ñ²and“denominators“(µqŸÿ´nŽ˜²)Ÿÿ´n·±1Ž‘ñ²ofŽ¡‘6à¤the›UUR•¸ãCF-con“v“ergen“ts˜of˜µx².Ž¡‘Bà¤T‘ÿ*ªo–UUbGe“more“precise,“wš¸ãe“ha˜v˜e“the“follo˜wing“corollary‘ÿ*ª.ŽŸ_‘6à¤ÕCorollary‘þ8ïhtml:ï html:4.ŽŽ‘x$ÔL›ÿ}'et–£µr¸ä;‘UPs“Ôand“µm“Ôb˜e“thr˜e˜e“inte˜gers,‘¦Õsuch“that“µm–â‡¸“²2–£Ôand“²(µr¸ä;–ª¨s;“m²)–â‡=“1Ô.Ž¡‘6à¤Then–“çfor“almost“al‘‚Øl“µx“Ôone“hasŽŸ‡f‘Xüd²limŽŸs3‘Tjo´N‘À,·!1ŽŽŸù<$‘p_Ä²1Ž‘nOÅŸw‰fe þŸ (ÖµNŽŽŽŽ‘x¢ö²#‘ª¨Ÿñæ^«Ž‘ *©µn‘ª¨²;›UP1–Ç¸“µn“¸“µN‘ˆâ;˜Ÿñæ^«ŽŸùæd‘ÈIµrŸÿ´nŽŽŽ¡‘±ÄµsŸÿ´nŽŽŽŽ‘ÓBŸñæ^«ŽŽ‘*¡v¸‘qÀŸñæ^«ŽŸùæd‘Î4µpŽŽ¡‘èßqŽŽŽ‘Ö4Ÿñæ^«ŽŽ‘%‡ø²(moGdŽ‘€µm²)Ÿñæ^«ŽŽ’Ô’?²=Ÿù<$‘Wé1Ž‘O›Ÿw‰feœŸ (ÖµJ‘ö9²(µm²)ŽŽŽŽ‘!>µ:ŽŸó.‘6à¤ÕAc•®9kno“wledgemen“ts–ƒÃ²W‘ÿ*ªe“w•¸ãan“t–ƒÃto“thank“the“referee,‘^Henk“Jager“and“Mark“Stein-Ž¡‘6à¤bGerger–UUfor“sevš¸ãeral“remarks“whic˜h“impro˜v˜ed“the“presen˜tation“of“this“papGer.Ž‘6à¤Ÿßïhtml:ï html:Ÿ9ÍReferencesŽŸ¹[ïhtml:ï html:Ba]ŽŽ‘ŠD.‘B}Barbï html:Br]ŽŽ‘ŠJ.–R.“BroÃŽwn,‘,ÉEr•¥go“dic›?¸The“ory˜and˜T‘ÿWwop“olo“gic“al˜Dynamics¹,–,Academic›Press,“New˜Y‘ÿJªork,“SanŽ¡‘ŠF‘ÿJªrancisco,–ÕXLondon,“1976.Ž¡[ïhtml:ï html:BJW]ŽŽ‘ºW.–)ãBosma,›?H.“Jager,˜and“F.“Wiedijk,˜ÉSome–ametricš¥al“observations“on“the“appr˜oximationŽ¡‘Šby›~c•¥ontinue“d˜fr“actions¹,–ÕXIndag.“Math.,“Ø45“¹(1983),“281{299.ŽŽŽŒ‹PÅh |4ù ýš‘6à¤ïhtml:ï html:²80Ž’‘r§ÙKarma–$øDajani“and“Cor“Kr‘ÿi>aaikampŽŽ N ýÌ‘6à¤¹[ïhtml:ï html:DK]ŽŽ‘Oj°K.–«Daxäjani“and“C.“Kraaik‡amp“ÉGenerš¥alization–Ù"of“a“the˜or˜em“of“Kusmin¹,‘à‚Monatsh.‘«Math.,Ž¤ ‘Oj°Ø118–ÕX¹(1994),“55{73.Ž¡‘6à¤[ïhtml:ï html:DP]ŽŽ‘Oj°R.–·ƒDescomšÃŽbï html:E]ŽŽ‘Oj°P‘ÿJª.–ÕXErd‘û¿üos,“ÉSome–~rš¥esults“on“Diophantine“appr˜oximation¹,–ÕXActa“Arith.,“ØV“¹(1959),“359{369.Ž¡‘6à¤[ïhtml:ï html:H]ŽŽ‘Oj°S.‘.žHartman,‘„ïÉSur–Scune“cš¥ondition“supplÇÒ‘ûÔŸementair˜e“dans“les“appr˜oximations“diophantiques¹,Ž¡‘Oj°Coll.–ÕXMath.,“Ø2“¹(1949),“48{51.Ž¡‘6à¤[ïhtml:ï html:IN]ŽŽ‘Oj°Sh.–…Ito“and“H.“Nak‡ada,‘.JÉOn–Pjnaturš¥al“extensions“of“tr˜ansformations“r˜elate˜d“to“Diophantine“ap-Ž¡‘Oj°pr¥oximations¹,‘—‡Prošï html:I]ŽŽ‘Oj°Sh.‘NcIto,‘l¦ÉAÇÒlgorithms–‚ýwith“meš¥diant“c˜onver˜gents“and“their“metric˜al“the˜ory¹,‘l¦Osak‡a–NcJ.“Math.,Ž¡‘Oj°Ø26–ÕX¹(1989),“557{578.Ž¡‘6à¤[ïhtml:ï html:JL]ŽŽ‘Oj°H.–ˆJager“and“P‘ÿJª.“Liardet,‘µÉDistributions–í~arithmÇÒ‘ûÔŸetiques“des“demoninateurs“de“c•¥onver“gents‘í~deŽ¡‘Oj°fr•¥actions‘~c“ontinues¹,–ÕXIndag.“Math.,“Ø50“¹(1988),“181{197.Ž¡‘6à¤[ïhtml:ï html:Kok]ŽŽ‘Oj°J.‘ó7Koksma,‘ qÉSur–@Tl'apprš¥oximation“des“nombr˜es“irr˜ationels“sous“une“c˜ondition“supplÇÒ‘ûÔŸementair˜e¹,Ž¡‘Oj°Simon–ÕXStevin“Ø28“¹(1951),“199{202.Ž¡‘6à¤[ïhtml:ï html:K]ŽŽ‘Oj°C.‘æÊKraaik‡amp,‘ë'ÉA‘"±new–"µclass“of“c•¥ontinue“d›"µfr“action˜exp“ansions¹,–ë'Acta‘æÊArithm.,“ØL‘þé>VIE±I‘æÆ¹(1991),Ž¡‘Oj°1{39.Ž¡‘6à¤[ïhtml:ï html:L]ŽŽ‘Oj°P‘ÿJª.‘h^Liardet,‘ ÉDistributions››#A•ÇÒrithm“‘ûÔŸetiques˜des˜Demoninateurs˜de˜Conver•¥gents˜de˜F‘ÿWwr“actionsŽ¡‘Oj°Continues¹,›Ä1Sem.–¿èTh.“des“NomÃŽbres“Bordeaux“1986{1987,˜Expï html:M]ŽŽ‘Oj°R.‘ªêMoï html:NIT]ŽŽ‘Oj°H.–ztNak‡ada,›£»Sh.“Ito,˜and“S.“T‘ÿJªanak‡a,˜S.“ÉOn–«òthe“invariant“me•¥asur“e–«òfor“the“tr¥ansformationŽ¡‘Oj°asso•¥ciate“d–~with“some“r•¥e“al›~c“ontinue“d˜fr“action¹,–ÕXKeio“Eng.“Rep.,“Ø30“¹(1977),“159{175.Ž¡‘6à¤[ïhtml:ï html:Na]ŽŽ‘Oj°H.‘å+Nak‡ada,‘) ÉMetric•¥al› the“ory˜for˜a˜class˜of˜c“ontinue“d˜fr“action˜tr“ansformations˜and˜theirŽ¡‘Oj°natur¥al‘~extensions¹,–ÕXT‘ÿJªokyÃŽo“J.“Math.,“Ø4“¹(1981),“399{426.Ž¡‘6à¤[ïhtml:ï html:No]ŽŽ‘Oj°V.–ü)N.“Nolte,‘ÅÞÉSome›Ypr•¥ob“abilistic˜r“esults˜on˜the˜c“onver“gents˜of˜c“ontinue“d˜fr“actions¹,Ž¡‘Oj°Indag.–ÕXMath.“(N.S.)“Ø1“¹(1990),“381{389.Ž¡‘6à¤[ïhtml:ï 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