Thuật toán kiểm tra điều kiện tập mở đối với một lớp hệ hàm lặp

❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ▲➟ ❍♦➭✐ ❚❤➢➡♥❣ ❚❤✉❐t t♦➳♥ ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ➤è✐ ✈í✐ ♠ét ❧í♣ ❤Ư ❤➭♠ ❧➷♣ ▲✉❐♥ ✈➝♥ ❚❤➵❝ sü ❚♦➳♥ ❤ä❝ ◆❣❤Ö ❆♥ ✲ ✷✵✶✺ ❇é ●✐➳♦ ❞ô❝ ✈➭ ➜➭♦ t➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ▲➟ ❍♦➭✐ ❚❤➢➡♥❣ ❚❤✉❐t t♦➳♥ ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ➤è✐ ✈í✐ ♠ét ❧í♣ ❤Ư ❤➭♠ ❧➷♣ ▲✉❐♥ ❱➝♥ ❚❤➵❝ ❙ü ❚♦➳♥ ❍ä❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❚♦➳♥ ●✐➯✐ tÝ❝❤ ▼➲ sè✿ ✻✵✳✹✻✳✵✶✳✵✷ ❈➳♥ ❜é ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝ ❚❙✳ ❱ị ❚❤Þ ❍å♥❣ ❚❤❛♥❤ ◆❣❤Ư ❆♥ ✲ ✷✵✶✺ ▼ơ❝ ▲ơ❝ ❚r❛♥❣ ▼ơ❝ ❧ơ❝ ✶ ▲ê✐ ♥ã✐ ➤➬✉ ✸ ❈❤➢➡♥❣ ✶✳ ❍Ö ❤➭♠ ❧➷♣ ✈➭ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶ ❍Ö ❤➭♠ ❧➷♣ ✈➭ t❐♣ tù ➤å♥❣ ❞➵♥❣ ✶✳✷ ➜é ➤♦ ❍❛✉s❞♦r❢❢ ✈➭ ❝❤✐Ị✉ ❍❛✉s❞♦r❢❢ ✶✳✸ ➜✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ✶✳✹ ▼ét sè ➤✐Ị✉ ❦✐Ư♥ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ✶✳✺ ❙è P✐s♦t ✶✳✻ ➜å t❤Þ ✈➭ ➤➢ê♥❣ ➤✐ ❈❤➢➡♥❣ ✷✳ ❚❤✉❐t t♦➳♥ ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ➤è✐ ✈í✐ ♠ét ❧í♣ ❤Ư ❤➭♠ ❧➷♣ ✶✹ ✷✳✶ ❈➡ së ❝ñ❛ t❤✉❐t t♦➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✷ ➜å t❤Þ ✈➭ t❤✉❐t t♦➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✸ ▼ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ❑Õt ❧✉❐♥ ✸✷ ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ✸✸ ✶ ✷ ❧ê✐ ♥ã✐ ➤➬✉ ❍×♥❤ ❤ä❝ ❋r❛❝t❛❧ ❧➭ ♠ét ❧Ü♥❤ ✈ù❝ ♠í✐ ♠❰ ✈➭ ❝ã ♥❤✐Ị✉ ø♥❣ ❞ơ♥❣✳ ❝❤Ý♥❤ ➤Ĩ ♥❣❤✐➟♥ ❝ø✉ ❤×♥❤ ❤ä❝ ❋r❛❝t❛❧ ❧➭ ➤é ➤♦ ✈➭ ❝❤✐Ị✉ ❍❛✉s❞♦r❢❢✳ ❈➠♥❣ ❝ơ ❱✐Ư❝ tÝ♥❤ ❝❤✐Ị✉ ❍❛✉s❞♦r❢❢ ❧➭ r✃t ❝➬♥ t❤✐Õt ♥❤➢♥❣ ❧➵✐ r✃t ❦❤ã✳ ❚❤❡♦ ❏✳ ❊✳ ❍✉t❝❤✐♥s♦♥ ✭✶✾✽✶✮✱ ❝ø ❝ã ♠ét ❤ä ❤÷✉ ❤➵♥ ❝➳❝ ➳♥❤ ①➵ ❝♦ tr➟♥ ❣ä✐ ➤ã ❧➭ ❤Ö ❤➭♠ ❧➷♣✳ Rn t❤× ❧✉➠♥ s✐♥❤ r❛ ♠ét t❐♣ ❋r❛❝t❛❧✳ ❚❛ ➜è✐ ✈í✐ ♥❤÷♥❣ t❐♣ ❋r❛❝t❛❧ s✐♥❤ ❜ë✐ ❤Ư ❤➭♠ ❧➷♣ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ✭❖♣❡♥ ❙❡t ❈♦♥❞✐t✐♦♥ ✲ ❖❙❈✮ ♥❣➢ê✐ t❛ ➤➲ t❤✐Õt ❧❐♣ ➤➢ỵ❝ ❝➠♥❣ t❤ø❝ r✃t ➤➡♥ ❣✐➯♥ ➤Ĩ tÝ♥❤ ❝❤✐Ị✉ ❍❛✉s❞♦r❢❢✳ ❉♦ ✈❐②✱ ♠ét ❝➞✉ ❤á✐ tù ♥❤✐➟♥ ➤➢ỵ❝ ➤➷t r❛ ❧➭✿ ❧✐Ư✉ ❝ã t❤✉❐t t♦➳♥ ♥➭♦ ➤Ĩ ❦✐Ĩ♠ tr❛ ♠ét ❤Ư ❤➭♠ ❧➷♣ ➤➲ ❝❤♦ ❝ã t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ❤❛② ❦❤➠♥❣❄ ❇➭✐ t♦➳♥ ♥➭② ➤➲ ✈➭ ➤❛♥❣ ➤➢ỵ❝ ♥❤✐Ị✉ ♥❤➭ t♦➳♥ ❤ä❝ q✉❛♥ t➞♠ ♥❣❤✐➟♥ ❝ø✉✳ ●➬♥ ➤➞②✱ tr♦♥❣ ✭❬✼❪✮✱ ♥❣➢ê✐ t❛ ➤➲ t×♠ r❛ ❝➞✉ tr➯ ❧ê✐ ❝❤♦ ❝➞✉ ❤á✐ ♥➭② tr♦♥❣ ♠ét tr➢ê♥❣ ❤ỵ♣ ❝ơ t❤Ĩ ❝đ❛ ❤Ư ❤➭♠ ❧➷♣✳ ➜✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ➤➢ỵ❝ ➤➢❛ r❛ ➤➬✉ t✐➟♥ ❜ë✐ P✳ ❆✳ P✳ ▼♦r❛♥ ✈➭♦ ♥➝♠ ✶✾✹✻ ✭❬✶✶❪✮✱ tr♦♥❣ ➤ã ➤✐Ò✉ ệ ợ tỏ tì ộ sr ủ t❐♣ tù ➤å♥❣ ❞➵♥❣ s✐♥❤ ❜ë✐ ❤Ö ❤➭♠ ❧➷♣ ♥➭② ❝ã ❣✐➳ trÞ ❞➢➡♥❣✳ ❙ư ❞ơ♥❣ ♥❤ã♠ t➠♣➠ ❝➳❝ ➳♥❤ ①➵ ➤å♥❣ ❞➵♥❣ ✭❬✷❪✮✱ ♥➝♠ ✶✾✾✷✱ ❈✳ ❇❛♥❞t ✈➭ ❙✳ ●r❛❢ ➤➲ ❝❤Ø r❛ ♠ét ➤✐Ị✉ ❦✐Ư♥ ➤Ĩ ❤Ư ➳♥❤ ①➵ ❝♦ t❤á❛ ♠➲♥ ❖❙❈✳ ➜Õ♥ ♥➝♠ ✶✾✾✹✱ ❆✳ ❈❤✐❡❢ ✭❬✶❪✮ ➤➲ ❝ã ❦Õt q✉➯ ✈Ị ♠Ư♥❤ ➤Ị ♥❣➢ỵ❝ ❝đ❛ P✳ ❆✳ P✳ ▼♦r❛♥✳ ◆➝♠ ✷✵✵✻ ✭❬✸❪✮✱ ❈✳ ❇❛♥❞t✱ ◆✳ ❱✳ ❍✉♥❣ ✈➭ ❘✳ ❍✉✐ ➤➲ ➤➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ t❐♣ ♠ë tr✉♥❣ t➞♠ ✈➭ ❝❤Ø r❛ ♠è✐ q✉❛♥ ❤Ö ❣✐÷❛ ♥ã ✈í✐ ➤✐Ị✉ ❦✐Ư♥ ❖❙❈ ✈➭ ♥➝♠ ✷✵✶✹ ✭❬✼❪✮✱ ❍✳ ◗✳ ●✉♦✱ ◗✳ ❲❛♥❣ ✈➭ ▲✳ ❋✳ ❳✐ ➤➲ ➤➢❛ r❛ t❤✉❐t t♦➳♥ ❞ù❛ ✈➭♦ ➤å t❤Þ ✈➭ ➤➢ê♥❣ ➤✐ t❤✐Õt ❧❐♣ tõ ♠ét ❤Ư ❤➭♠ ❧➷♣ ➤Ĩ ❦✐Ĩ♠ tr❛ ♠ét ❧í♣ ❤Ư ❤➭♠ ❧➷♣ ❝ã t❤á❛ ♠➲♥ ❖❙❈ ❤❛② ❦❤➠♥❣✳ ❱× ✈❐②✱ ➤Ĩ t❐♣ ❞✉②Ưt ✈í✐ ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝ ✈➭ t×♠ ❤✐Ĩ✉ ✈Ị ✈✃♥ ➤Ị ♥➭② ❝❤ó♥❣ t➠✐ ❝❤ä♥ ➤Ị t➭✐ ♥❣❤✐➟♥ ❝ø✉ ❝❤♦ ❧✉❐♥ ✈➝♥ ❝đ❛ ♠×♥❤ ❧➭✿ ✧ ❚❤✉❐t t♦➳♥ ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ➤è✐ ✈í✐ ♠ét ❧í♣ ❤Ư ❤➭♠ ❧➷♣✧✳ ▼ơ❝ ➤Ý❝❤ ❝đ❛ ❧✉❐♥ ✈➝♥ ❧➭ t❤➠♥❣ q✉❛ ✈✐Ư❝ t×♠ ❤✐Ĩ✉ t➭✐ ❧✐Ư✉ ♥❣❤✐➟♥ ❝ø✉ ✈Ị ❤Ư ❤➭♠ ❧➷♣✱ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ✈➭ t❤✉❐t t♦➳♥ ➤Ĩ ❦✐Ĩ♠ tr❛ ♠ét ❤Ư ❤➭♠ ❧➷♣ ❝ã t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ❤❛② ❦❤➠♥❣❄ ✸ ◆❣♦➭✐ ♣❤➬♥ ▼ë ➤➬✉✱ ❑Õt ❧✉❐♥ ✈➭ ❚➭✐ ❧✐Ö✉ t ộ ợ trì tr ❤❛✐ ❝❤➢➡♥❣✳ ❈❤➢➡♥❣ ✶✳ ❍Ư ❤➭♠ ❧➷♣ ✈➭ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ❈❤➢➡♥❣ ♥➭② tr×♥❤ ❜➭② ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ së ❞ï♥❣ ❝❤♦ t♦➭♥ ❧✉❐♥ ✈➝♥ ♥❤➢✿ ❈➳❝ ❧♦➵✐ ➳♥❤ ①➵✱ ❤Ö ❤➭♠ ❧➷♣✱ t❐♣ tù ➤å♥❣ ❞➵♥❣✱ ➤é ➤♦ ✈➭ ❝❤✐Ị✉ ❍❛✉s❞♦r❢❢✱ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ✈➭ t❐♣ ♠ë ♠➵♥❤✳ ❈❤➢➡♥❣ ✷✳ ❚❤✉❐t t♦➳♥ ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ➤è✐ ✈í✐ ♠ét ❧í♣ ❤Ư ❤➭♠ ❧➷♣ ❈❤➢➡♥❣ ♥➭② trì tt t ự tị ➤✐ ➤Ĩ ❦✐Ĩ♠ tr❛ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ❝đ❛ ♠ét ❤Ö ❤➭♠ ❧➷♣ ❧✐➟♥ q✉❛♥ ➤Õ♥ sè P✐s♦t✳ ▲✉❐♥ ✈➝♥ ợ t t tì ➤➳♦ ❝đ❛ ❝➠ ❣✐➳♦ ❚❙✳ ❱ị ❚❤Þ ❍å♥❣ ❚❤❛♥❤✳ ❚➳❝ ❣✐➯ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ s➞✉ s➽❝ ♥❤✃t ➤Õ♥ ❝➠✱ ♥❣➢ê✐ ➤➲ ❤➢í♥❣ ❞➱♥ t➳❝ ❣✐➯ ♥❤÷♥❣ ❦✐Õ♥ t❤ø❝✱ ❦✐♥❤ ♥❣❤✐Ö♠ tr♦♥❣ ✈✐Ö❝ ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝✳ ❚➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❇❛♥ ❝❤đ ♥❤✐Ư♠ P❤ß♥❣ ❙❛✉ ➤➵✐ ❤ä❝✱ ❇❛♥ ❝❤đ ♥❤✐Ư♠ ❑❤♦❛ s➢ ♣❤➵♠ ❚♦➳♥ ❤ä❝ ✲ q✉ý ❚❤➬② ❣✐➳♦✱ ❈➠ ❣✐➳♦ tr♦♥❣ tæ ●✐➯✐ tÝ❝❤ ❝ñ❛ ❦❤♦❛ ❙➢ ♣❤➵♠ ❚♦➳♥ ❤ä❝ ✲ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❱✐♥❤ ➤➲ ♥❤✐Ưt t×♥❤ ❣✐➯♥❣ ❞➵② ✈➭ ❣✐ó♣ ➤ì t➳❝ ❣✐➯ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣✳ ❈✉è✐ ❝ï♥❣ t➳❝ ❣✐➯ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❝➳❝ ❜➵♥ ❤ä❝ ✈✐➟♥ tr♦♥❣ ❧í♣ ❈❛♦ ❤ä❝ ✷✶✲ ❝❤✉②➟♥ ♥❣➭♥❤ ●✐➯✐ tÝ❝❤ ✈➭ ❣✐❛ ➤×♥❤ ➤➲ ❝é♥❣ t➳❝✱ ❣✐ó♣ ➤ì ✈➭ ➤é♥❣ ✈✐➟♥ t➳❝ ❣✐➯ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ♥❣❤✐➟♥ ❝ø✉✳ ▼➷❝ ❞ï ➤➲ r✃t ❝è ❣➽♥❣✱ ♥❤➢♥❣ ❧✉❐♥ ✈➝♥ ❦❤➠♥❣ t❤Ĩ tr➳♥❤ ❦❤á✐ ♥❤÷♥❣ ❤➵♥ ❝❤Õ✱ t❤✐Õ✉ sãt✳ ❑Ý♥❤ ♠♦♥❣ q✉ý ❚❤➬② ❈➠ ✈➭ ❜➵♥ ❜❒ ❣ã♣ ý ➤Ó ❧✉❐♥ ✈➝♥ ➤➢ỵ❝ ❤♦➭♥ t❤✐Ư♥ ❤➡♥✳ ❚P✳ ❍å ❈❤Ý ▼✐♥❤✱ ♥❣➭② ✸✵ t❤➳♥❣ ✵✾ ♥➝♠ ✷✵✶✺ ✹ ❝❤➢➡♥❣ ✶ ❍Ö ❤➭♠ ❧➷♣ ✈➭ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ❈❤➢➡♥❣ ♥➭② tr×♥❤ ❜➭② ❝➳❝ ❦✐Õ♥ t❤ø❝ ❝➡ së ❞ï♥❣ ❝❤♦ t♦➭♥ ❧✉❐♥ ✈➝♥ ♥❤➢✿ ❈➳❝ ❧♦➵✐ ➳♥❤ ①➵✱ ❤Ö ❤➭♠ ❧➷♣✱ t❐♣ tù ➤å♥❣ ❞➵♥❣✱ ➤é ➤♦ ✈➭ ❝❤✐Ị✉ ❍❛✉s❞♦r❢❢✱ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ✈➭ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë ♠➵♥❤✳ ✶✳✶ ❍Ư ❤➭♠ ❧➷♣ ✈➭ t❐♣ tù ➤å♥❣ ❞➵♥❣ ✶✳✶✳✶ ➜Þ♥❤ ♥❣❤Ü❛ ✭❬✶✶❪✮✳ ●✐➯ sö D ⊂ Rn , D = ∅ ✭t❤➢ê♥❣ ❧✃② D = Rn ✮ ✈➭ ❦ý ❤✐Ö✉ |a − b| ❧➭ ❦❤♦➯♥❣ ❝➳❝❤ ❣✐÷❛ ❤❛✐ ➤✐Ĩ♠ a, b tr♦♥❣ Rn ✳ ✶✮ ➳ ♥❤ ①➵ f : D → D ➤➢ỵ❝ ❣ä✐ ❧➭ ❝♦ tr➟♥ D ♥Õ✉ tå♥ t➵✐ c ∈ [0; 1) s❛♦ ❝❤♦ |f (x) − f (y)| ≤ c |x − y| ∀x, y ∈ D✱ c ➤➢ỵ❝ ❣ä✐ ❧➭ tû sè ❝♦✳ ✷✮ ➳ ♥❤ ①➵ f : D → D ➤➢ỵ❝ ❣ä✐ ❧➭ ➤å♥❣ ❞➵♥❣ tr➟♥ D ♥Õ✉ tå♥ t➵✐ c > s❛♦ ❝❤♦ |f (x) − f (y)| = c |x − y| ∀x, y ∈ D✱ c ➤➢ỵ❝ ❣ä✐ ❧➭ tû sè ➤å♥❣ ❞➵♥❣✳ ✶✳✶✳✷ ➜Þ♥❤ ♥❣❤Ü❛ ✭❬✶✶❪✮✳ ❈❤♦ D ❧➭ ♠ét t❐♣ ❝♦♥ ➤ã♥❣ tr♦♥❣ (Rn , d) ✈í✐ d ợ n ị |xi yi |2 ✱ d (x, y) = |x − y| = i=1 tr ó ỗ x = (x1 , , xn ) , y = (y1 , , yn ) ∈ Rn x Rn ỗ t A ⊂ Rn t❛ ①➳❝ ➤Þ♥❤ ♥❤➢ s❛✉ d (x, A) = inf {d (x, a) : a ∈ A} ✺ từ x ế A ỗ số tự ❞➢➡♥❣ δ ✱ t❛ ➤➷t Aδ = {x ∈ Rn : d (x, A) ≤ δ} ❧➭ t❐♣ ❣å♠ ♥❤÷♥❣ ➤✐Ó♠ ❝➳❝❤ A ♠ét ❦❤♦➯♥❣ ❦❤➠♥❣ q✉➳ δ ✳ ❚❛ ❣ä✐ Aδ ❧➭ δ − bao ❝ñ❛ A✳ ●ä✐ t❤✉é❝ K ❧➭ ❧í♣ t✃t ❝➯ ❝➳❝ t❐♣ ❝♦♥ ❝♦♠♣❛❝t✱ ❦❤➳❝ rỗ ủ D t A, B K t ❦ý ❤✐Ö✉ dH (A, B) = inf {δ > : A ⊂ Bδ , B ⊂ Aδ } (1.1) ị ý ữ dH ị tì dH ột tr tr K✳ (K, dH ) ❧➭ ❦❤➠♥❣ ❣✐❛♥ ♠➟tr✐❝ ➤➬② ➤ñ✳ ị ĩ ợ ọ ột ột ọ ❤÷✉ ❤➵♥ ➳♥❤ ①➵ ❝♦ ❤Ư ❤➭♠ ❧➷♣ ✶✳✶✳✺ ▼Ư♥❤ ➤Ò ✭❬✽❪✮✳ {Si }N i=1 ✭■❋❙✲■t❡r❛t❡❞ ❋✉♥❝t✐♦♥ ❙②st❡♠✮ tr➟♥ ❈❤♦ N ➳♥❤ ①➵ ❝♦ {Si }N i=1 tr➟♥ ✈í✐ Si : D → D D✳ D✱ ➳♥❤ ①➵ S : K K ợ ị N E S (E) = Si (E)✳ ✭✶✳✷✮ i=1 ❑❤✐ ➤ã✱ S ❧➭ ➳♥❤ ①➵ ❝♦✳ ❚õ ➜Þ♥❤ ❧ý ✶✳✶✳✸✱ ▼Ư♥❤ ➤Ị ✶✳✶✳✺ ✈➭ t❤❡♦ ♥❣✉②➟♥ ❧ý ➳♥❤ ①➵ ❝♦ ❇❛♥❛❝❤ t❛ ❝ã ❦Õt q✉➯ s❛✉✳ ✶✳✶✳✻ ➜Þ♥❤ ❧ý ✭❬✼❪✮✳ ❈❤♦ ❤Ư ❤➭♠ ❧➷♣ {Si }m i=1 ✭✶✳✷✮✳ ❑❤✐ ➤ã✱ ❧✉➠♥ tå♥ t➵✐ ❞✉② ♥❤✃t ♠ét t❐♣ ❝ã t❐♣ E∈K S ✱ t❛ ❝ã F = s❛♦ ❝❤♦ ∞ k ✈➭ S ợ ị F K s❛♦ ❝❤♦ S (F ) = F ✳ ❍➡♥ ♥÷❛✱ ♥Õ✉ Si (E) ⊂ E (1 ≤ i ≤ m) t❤× ✈í✐ Sk ❧➭ sù ❧➷♣ ❧➵✐ ❦ ❧➬♥ ➳♥❤ ①➵ S (E)✳ k=1 ✶✳✶✳✼ ➜Þ♥❤ ♥❣❤Ü❛ ✶✮ ❚❐♣ t❐♣ ❤ót F ✭❬✽❪✮✳ ❈❤♦ ❤Ư ❤➭♠ ❧➷♣ {Si }m i=1 tr D ó ợ ị tr ị ý ợ ọ (attractor) t rt ❝đ❛ ❤Ư ❤➭♠ ❧➷♣ {Si }m i=1 ✳ ✻ t❐♣ ❜✃t ❜✐Õ♥ ❤❛② ✷✮ ◆Õ✉ Si (1 ≤ i ≤ m) ❧➭ ➳♥❤ ①➵ ➤å♥❣ ❞➵♥❣ t❤× t❐♣ ❜✃t ❜✐Õ♥ F tù ➤å♥❣ ❞➵♥❣ ✶✳✷ ➤➢ỵ❝ ❣ä✐ ❧➭ t❐♣ ✭s❡❧❢ ✲ s✐♠✐❧❛r s❡t✮✳ ➜é ➤♦ ❍❛✉s❞♦r❢❢ ✈➭ ❝❤✐Ò✉ ❍❛✉s❞♦r❢❢ P ❤➬♥ ♥➭② ❝❤ó♥❣ t➠✐ tr×♥❤ ❜➭② ❦❤➳✐ ♥✐Ư♠ ✈➭ ❝➳❝ tÝ♥❤ ❝❤✃t ❝➡ ❜➯♥ ✈Ò ➤é ➤♦✱ ➤é ➤♦ ❍❛✉s❞♦r❢❢ ✈➭ ❝❤✐Ị✉ ❍❛✉s❞♦r❢❢✳ ✶✳✷✳✶ ➜Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ✭❬✶✶❪✮✳ ❈❤♦ X ❧➭ ♠ét t❐♣ ❤ỵ♣ tï② ý ✈➭ C ❧➭ ➤➵✐ sè ❝➳❝ t❐♣ ❝♦♥ X✳ ✶✮ ❍➭♠ t❐♣ µ : C → R ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét ➤é ➤♦ tr➟♥ C ♥Õ✉ t❤á❛ ♠➲♥ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ s❛✉ ✐✮ µ (A) ≥ ✈í✐ ♠ä✐ A ∈ C ❀ ✐✐✮ µ (φ) = 0❀ ∞ ✐✐✐✮ µ ❧➭ σ ✲ ❝é♥❣ tÝ♥❤✱ tø❝ ❧➭ ♥Õ✉ Ai ∈ C (i = 1, 2, ) , Ai ∩Aj = φ (i = j) , Ai C i=1 tì Ai ợ ọ ột ộ (Ai )✳ = i=1 i=1 ➤♦ ♥❣♦➭✐ tr➟♥ C ♥Õ✉ µ t❤á❛ ♠➲♥ ✐✮✱ ✐✐✮ ✈➭ t❤❛② ✐✐✐✮ ❜ë✐ ✐✐✐✬✮ ❧➭ ∞ ✐✐✐✬✮ ◆Õ✉ Ai ∈ C (i = 1, 2, ) , Ai C tì i=1 Ai i=1 ị ĩ ợ ị ❜ë✐ ✭❬✶✶❪✮✳ ✶✮ ❈❤♦ µ (Ai )✳ i=1 F ⊂ Rn , F = ∅✳ ❑❤✐ ➤ã✱ ➤➢ê♥❣ ❦Ý♥❤ ❝ñ❛ t❐♣ F |F | = sup {d (x, y) : x, y ∈ F }✳ ✼ ✷✮ ❈❤♦ {Ui } ❧➭ ♠ét ❤ä ❝➳❝ t❐♣ ❝♦♥ tr♦♥❣ R n ∞ ế F Ui tì {Ui } ợ i=1 ❣ä✐ ❧➭ ♠ét ♣❤đ ❝đ❛ {Ui } ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ét δ ✲ ♣❤đ F ✳ ❧➭ sè ❝❤♦ tr➢í❝ t❤× ❱í✐ F ✳ ◆Õ✉ t❤➟♠ ➤✐Ị✉ ❦✐Ư♥ < |Ui | ≤ δ ✱ ✈í✐ ♠ä✐ ✐✱ tr♦♥❣ ➤ã δ > F ⊂ Rn , s ≥ ✈➭ δ > 0✱ t❛ ➤➷t Hδs (F ) ❑❤✐ ➤ã✱ ∞ = inf { |Ui |s : {Ui } ❧➭ δ− ♣❤ñ F }✳ i=1 Hδs (F ) ❧➭ ❤➭♠ ♥❣❤Þ❝❤ ❜✐Õ♥ t❤❡♦ δ ✱ ♥❣❤Ü❛ ❧➭ ♥Õ✉ δ1 ≤ δ2 t❤× Hδs2 (F ) ≤ Hδs1 (F )✳ ✶✳✷✳✸ ➜Þ♥❤ ❧ý ✭❬✶✶❪✮✳ ❈❤♦ C ❧➭ ❧í♣ ❝➳❝ t❐♣ ủ Rn ỗ s0 t Hs : C → Rn ①➳❝ ➤Þ♥❤ ❜ë✐ Hs (F ) = lim Hδs (F ) ✈í✐ ♠ä✐ F ∈ C δ→0 ❧➭ ♠ét ➤é ➤♦ ♥❣♦➭✐ tr➟♥ C✳ ❚õ ➜Þ♥❤ ❧ý ✶✳✷✳✸ t❛ ➤✐ ➤Õ♥ ➤Þ♥❤ ♥❣❤Ü❛ s❛✉✳ ✶✳✷✳✹ ➜Þ♥❤ ♥❣❤Ü❛ ➤➢ỵ❝ ❣ä✐ ❧➭ ✭❬✶✶❪✮✳ ➤é ➤♦ ❍❛✉s❞♦r❢❢ ➜é ➤♦ s✐♥❤ ❜ë✐ ➤é ➤♦ ♥❣♦➭✐ tr➟♥ σ ✲ ➤➵✐ sè τ ❝➳❝ t❐♣ ❝♦♥ Hs tr♦♥❣ ➜Þ♥❤ ❧ý ✶✳✷✳✸ Hs ✲ ➤♦ ➤➢ỵ❝ ❝đ❛ Rn ✳ ❚❐♣ F ⊂ Rn t❤á❛ ♠➲♥ < Hs (F ) < ∞ ➤➢ỵ❝ ❣ä✐ ❧➭ s ✲ t❐♣✳ ✶✳✷✳✺ ▼Ư♥❤ ➤Ị ♥❤✃t ♠ét ❣✐➳ trÞ ✭❬✶✶❪✮✳ ❈❤♦ ❧➭ t❐♣ ❇♦r❡❧✳ ❑❤✐ ➤ã✱ ❧✉➠♥ tå♥ t➵✐ ❞✉② sF ∈ [0; +∞) ➤Ó ✶✮ Hs (F ) = 0, ∀s > sF ✱ ✷✮ Hs (F ) = +∞, ∀s < sF ✳ ✶✳✷✳✻ ➜Þ♥❤ ♥❣❤Ü❛ ✭❬✶✶❪✮✳ ▼Ư♥❤ ➤Ị ✶✳✷✳✺ ➤➢ỵ❝ ❣ä✐ ❧➭ ✶✳✷✳✼ ◆❤❐♥ ①Ðt ✶✮ ∅ = F ⊂ Rn ✳ ◆Õ✉ ❈❤♦ F ⊂ Rn ✳ ❝❤✐Ị✉ ❍❛✉s❞♦r❢❢ ❙è ❝đ❛ sF ∈ [0; +∞] ➤➢ỵ❝ ♥ã✐ tí✐ tr♦♥❣ F ✱ ❦ý ❤✐Ư✉ dimH F ✳ F ⊂ Rn t❤× dimH F = inf {s : Hs (F ) = 0} = sup {s : Hs (F ) = ∞}✱ ✷✮ ◆Õ✉ tå♥ t➵✐ s ∈ [0; +∞] ➤Ó < Hs (F ) < ∞ t❤× dimH F = s✳ ✽ ❚❤❡♦ ◆❤❐♥ ①Ðt ✷✳✶✳✻ ✭✐✮ t❛ ❝ã E ⊂ x : |x| ≤ b0 1−λ ✱ ♥➟♥ πR (pτk , bτk )−1 (pσk , bσk ) ≤ |x1 | + λpσk −pτk |x2 | b0 ≤ (1 + λpσk −pτk ) 1−λ ≤ + λT −1 ❚❤❡♦ ❇ỉ ➤Ị ✷✳✶✳✷✱ t❛ ❝ã ✷✳✶✳✶✵ ➜Þ♥❤ ♥❣❤Ü❛ ❚❛ ♥ã✐ r➺♥❣ b0 1−λ πR (pτk , bτk )−1 (pσk , bσk ) ∈ λT −2 R λ−1 ✭❬✺❪✮✳ ●✐➯ sö ✳ {αk }tk=0 ✈➭ {βk }tk=0 ⊂ Σ∗ ✈í✐ t ∈ N ❤❛② t = ∞✳ t t {(qk , ck )}t+1 k=1 ❝ã ❝✃✉ tró❝ ➤Ư q✉② ➤è✐ ✈í✐ {αk }k=0 , {βk }k=0 ♥Õ✉ (q1 , c1 ) = (pβ0 , bβ0 )−1 (pα0 , bα0 ) ✳ ✳ ✳ (qk+1 , ck+1 ) = (pβk , bβk )−1 (qk , ck ) (pαk , bαk ) ✈í✐ ≤ k ≤ t✳ ✷✳✶✳✶✶ ➜Þ♥❤ ❧ý ✭❬✺❪✮✳ ❍Ư ❤➭♠ ❧➷♣ {λpi x + bi }m i=1 ❦❤➠♥❣ t❤á❛ ♠➲♥ ❖❙❈ ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ tå♥ t➵✐ sè ♥❣✉②➟♥ ❞➢➡♥❣ ▼ ✈➭ ❝➳❝ ❞➲② M ∗ {αk }M k=0 , {βk }k=0 ⊂ Σ ✈í✐ ≤ pαk , pβk ≤ 2T − ✈➭ |α0 | = |β0 | = 1✱ α0 = β0 s❛♦ ❝❤♦ ✈í✐ ♠ä✐ k ≥ t❛ ❝ã (qM +1 , cM +1 ) = (0, 0) (qk , ck ) ∈ Ξ ♠➭ +1 M M {(qk , ck )}M k=1 ❝ã ❝✃✉ tró❝ ➤Ư q✉② ➤è✐ ✈í✐ {αk }k=0 , {βk }k=0 ◆❤➢ ❝❤ó♥❣ t❛ ➤➲ ❜✐Õt✱ ✳ ♥Õ✉ ❤Ư ❤➭♠ ❧➷♣ t❤á❛ ♠➲♥ ❙❙❈ t❤× t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❖❙❈✳ ❱× t❤Õ ❝➳❝ ❦Õt q✉➯ s❛✉ ♥❣❤✐➟♥ ❝ø✉ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ➤Ĩ ❤Ư ❤➭♠ ❧➷♣ {λpi x + bi }m i=1 t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❙❙❈✳ ✷✳✶✳✶✷ ▼Ư♥❤ ➤Ị t❤× tå♥ t➵✐ ❝➳❝ ❞➲② ✭❬✺❪✮✳ {σk }k ●✐➯ sö i1 , i2 , ; j1 , j2 , ✈➭ {τk }k ⊂ Σ∗ ∈ Σ✳ ◆Õ✉ Si1 ,i2 , (E) = Sj1 ,j2 , (E) s❛♦ ❝❤♦ ✈í✐ ♠ä✐ τk ≺ j1 j2 t❤á❛ ♠➲♥ pσk , pτk ∈ ((k − 1) T, kT ] (pτk , bτk )−1 (pσk , bσk ) ∈ Ξ ✷✵ k ≥ t❛ ❝ã σk ≺ i1 i2 ✈➭ ❈❤ø♥❣ ♠✐♥❤✳ ❱í✐ ❜✃t ❦ú k ≥ 1✱ pσk ∈ (kT, (k + 1) T ] ✈➭ ❧✃② k ì t ố ị t ủ Z (pk , bτk )−1 (pσk , bσk ) = pσk − pτk ✈➭ σk ♥❣➽♥ ♥❤✃t ❝ñ❛ i1 i2 s❛♦ ❝❤♦ j1 j2 s❛♦ ❝❤♦ pτk ∈ (kT, (k + 1) T ]✳ pσk , pτk ∈ ((k − 1) T, kT ) t❛ s✉② r❛ πZ (pτk , bτk )−1 (pσk , bσk ) ≤ T − 1✳ ❚õ ❣✐➯ t❤✐Õt Si1 ,i2 , (E) = Sj1 ,j2 , (E) t❛ s✉② r❛ Sσk (E) ∩ Sτk (E) = ∅ ❤❛② E ∩ Sσ−1 Sτk (E) = ∅ ✈í✐ Sσ−1 Sτk (x) = x1 = λpσk −pτk x + πR (pτk , bτk )−1 (pσk , bσk ) k k ●✐➯ sö r➺♥❣ x1 = λpσk −pτk (x2 ) + πR (pτk , bτk )−1 (pσk , bσk ) ✈í✐ ✳ x1 , x2 ∈ E ✳ max|bi | ❚õ ◆❤❐♥ ①Ðt ✷✳✶✳✼✭✐✮ ❧➭ E⊂ x : |x| ≤ i 1−λ ✱ t❛ s✉② r❛ πR (pτk , bτk )−1 (pσk , bσk ) ≤ |x1 | + λpσk −pτk |x2 | pσk −pτk ≤ (1 + λ ≤ + λT −1 ❚❤❡♦ ❇ỉ ➤Ị ✷✳✶✳✷✱ t❛ ❝ã ✷✳✶✳✶✸ ▼Ư♥❤ ➤Ị ✭❬✺❪✮✳ ) b0 1−λ b0 1−λ πR (pτk , bτk )−1 (pσk , bσk ) = λT −2 R λ−1 ●✐➯ sö i1 , i2 , ; j1 , j2 , ế ỗ k tồ σk ≺ i1 i2 ✈➭ τk ≺ j1 j2 s❛♦ ❝❤♦ pσk , pτk ∈ ((k − 1) T, kT ] ✈➭ πR (pτk , bτk )−1 (pσk , bσk ) t❤× ≤ + λT −1 b0 1−λ Si1 i2 (E) = Sj1 j2 (E)✳ ❈❤ø♥❣ ♠✐♥❤✳ ❳Ðt ❦❤♦➯♥❣ ❝➳❝❤ ❣✐÷❛ Sσk (0) ✈➭ Sτk (0)✳ ❚❛ ❝ã✿ Sτ−1 Sσk (x) = λpσk −pτk x + πR (pτk , bτk )−1 (pσk , bσk ) k ♥➟♥ Sτ−1 Sσk (0) = πR (pτk , bτk )−1 (pσk , bσk ) k ✳ ➜✐Ò✉ ♥➭② ❦Ð♦ t❤❡♦ b0 →0 |Sσk (0) − Sτk (0)| = λpτk Sτk −1 Sσk (0) − ≤ λ(k−1)T + λ−T +1 1−λ ❦❤✐ k → ∞ ❞♦ λ < 1✳ ❉♦ ➤ã✱ ❦❤✐ ❝❤♦ k → ∞ t❛ ❝ã Si1 i2 (0) = Sj1 j2 (0)✳ ◆❤➢ ✈❐②✱ t❛ ❝ã ✷✶ t➵✐ Si1 i2 (E) = Sj1 j2 (E)✳ ❚õ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✷ ✈➭ ✷✳✶✳✶✸ t❛ t❤✉ ➤➢ỵ❝ ❦Õt q✉➯ s❛✉✳ ✷✳✶✳✶✹ ➜Þ♥❤ ❧ý ✭❬✺❪✮✳ ❍Ư ❤➭♠ ❧➷♣ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ tå♥ t➵✐ ❝➳❝ ❞➲② ♠ä✐ k ❝ã ❝✃✉ tró❝ ➤Ư q✉② ➤è✐ ✈í✐ ❈❤ø♥❣ ♠✐♥❤✳ t➵✐ ❦❤➠♥❣ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ö♥ ❙❙❈ ∞ ∗ {αk }∞ k=0 ✱ {βk }k=0 tr♦♥❣ Σ ✈í✐ ≤ pαk , pβk ≤ 2T − ✈í✐ |α0 | = |β0 | = 1, α0 = β0 ✱ s❛♦ ✈➭ {λpi x + bi }m i=1 ❝❤♦ (qk , ck ) ∈ Ξ ✈í✐ k ≥ 1✳ tr♦♥❣ ➤ã (qk , ck )k ✱ ∞ ({αk }∞ k=0 , {βk }k=0 )✳ ◆Õ✉ ❤Ö ❤➭♠ ❧➷♣ ➤➲ ❝❤♦ ❦❤➠♥❣ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❙❙❈ t❤× tå♥ i1 i2 ✈➭ j1 j2 ∈ Σ ✈í✐ i1 = j1 s❛♦ ❝❤♦ Si1 ,i2 , (E) = Sj1 ,j2 , (E)✳ ❙ư ❞ơ♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ♥❤➢ tr♦♥❣ ❝❤ø♥❣ ♠✐♥❤ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✷✱ {σk }k ; {τk }k ✈í✐ ❑❤✐ ➤ã✱ ❧✃② t❛ ❝ã ❞➲② σ1 = i1 = α0 ✈➭ τ1 = j1 = β0 ✳ (qk , ck ) = (pτk , bτk )−1 (pσk , bσk )❀ αk = σk+1 \σk ✈➭ βk = τk+1 \τk ✈í✐ ≤ pαk , pβk ≤ 2T − 1✳ ❱× Sτ−1 Sσk+1 = Sβ−1 Sτ−1 Sσk Sαk k+1 k k ♥➟♥ t❛ ❝ã (qk+1 , ck+1 ) = pτk+1 , bτk+1 −1 pσk+1 , bσk+1 = (pβk , bβk )−1 (pτk , bτk )−1 (pσk , bσk ) (pαk , bαk ) = (pβk , bβk )−1 (qk , ck ) (pαk , bαk ) ❚❤❡♦ ▼Ö♥❤ ➤Ò ✷✳✶✳✶✷✱ t❛ ❝ã (qk , ck ) ∈ Ξ ✈í✐ ♠ä✐ k ≥ 1✳ ◆❣➢ỵ❝ ❧➵✐✱ ♥Õ✉ tå♥ t➵✐ ❝➳❝ ❞➲② {αk }k≥0 ✈➭ {βk }k≥0 tr♦♥❣ Σ∗ t❤á❛ ♠➲♥ ❝➳❝ ②➟✉ ❝➬✉ tr♦♥❣ ❣✐➯ t❤✐Õt ❝đ❛ ➤Þ♥❤ ❧ý✳ ❚❛ ❧✃② σk = α0 ∗ α1 ∗ αk−1 ✈➭ τk = β0 ∗ β1 ∗ βk−1 ✳ ❈❤♦ ✈í✐ ♠ä✐ k → ∞✱ t❛ ❝ã i1 , i2 , ; j1 , j2 , ∈ Σ s❛♦ ❝❤♦ σk ≺ i1 , i2 , ✈➭ τk ≺ j1 , j2 , k ✈➭ ✷✷ πR (pτk , bτk )−1 (pσk , bσk ) b0 1−λ ≤ + λT −1 ❚❤❡♦ ▼Ư♥❤ ➤Ị ✷✳✶✳✶✸ ✈➭ ➜Þ♥❤ ♥❣❤Ü❛ ✶✳✸✳✷✱ t❛ s✉② r❛ ❤Ư ❤➭♠ ❧➷♣ ❦❤➠♥❣ t❤á❛ ♠➲♥ ❙❙❈✳ ❱❐②✱ t❛ ❝ã ➤✐Ò✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ✷✳✷ ➜å t❤Þ ✈➭ t❤✉❐t t♦➳♥ P❤➬♥ ♥➭② tr×♥❤ ❜➭② t❤✉❐t t♦➳♥ ❦✐Ĩ♠ tr❛ ✈í✐ ❤❛✐ ❜é ❣✐➳ trÞ (b1 , , bm ) ∈ Zm ♥❤➢ t❤Õ ♥➭♦ t❤× ❤Ư ❤➭♠ ❧➷♣ (p1 , , pm ) ∈ Nm −1 S = {λpi x + bi }m i=1 ✭λ ✈➭ ❧➭ P✳❱ sè✮ t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ t❐♣ ♠ë✳ ✷✳✷✳✶ ❈➳❝ ❦ý ❤✐Ư✉ ✳ ✶✮ ❱í✐ i=1,m C ={x ∈ R : |x| ≤ t❛ ❦ý ❤✐Ö✉ (1+λ−T +1 ) b0 1−λ i=1,m ; x = λT −2 R λ−1 ✈í✐ R ∈ Z − Z} Ξ = {n ∈ Z : |n| ≤ T − 1} × C = {(n, x) : n ∈ Z : |n| ≤ T − 1, x ∈ C} ✈➭ ➤➢ỵ❝ ❣ä✐ ❧➭ ✷✮ ●✐➯ sö T = max pi ✱ b0 = max |bi | ✈➭ (p, b) t❐♣ ➤Ø♥❤✱ ỗ (p , b ) ỉ v = (p, b) ∈ Ξ✳ ❧➭ ❤❛✐ ➤Ø♥❤✳ ❚❛ ♥ã✐ ❝ã ♠ét ❝➵♥❤ trù❝ t✐Õ♣ e tõ ➤Ø♥❤ (p, b) ➤Õ♥ (p , b ) ♥Õ✉ ✈➭ ❝❤Ø ♥Õ✉ tå♥ t➵✐ α, β ∈ Σ∗ s❛♦ ❝❤♦ ≤ pα , pβ < 2T − ✭❝❤➻♥❣ ❤➵♥ α = α1 αk ∈ Σk t❤× pα = pα1 + + pαm < 2T − 1✮ s❛♦ ❝❤♦ (p , b ) = (pβ , bβ )−1 (p, b) (pα , bα )✳ ✭✶✮ ❑ý ❤✐Ö✉ t❐♣ ❝➳❝ ❝➵♥❤ ❧➭ ✸✮ ➜➷t L ={(α, β) : α, β ∈ Σ∗ ; α, β Λ ={(pi , bi )−1 (pj , bj ) : i = j ✈➭ ✷✸ t❤á❛ ♠➲♥ ✭✶✮ λ−pi |bj − bi | ≤ + λ−T +1 }✳ b0 1−λ }✳ ✷✳✷✳✷ ➜Þ♥❤ ♥❣❤Ü❛ ✭❬✼❪✮✳ ❚❐♣ s➽♣ t❤ø tù ❣å♠ ❝➳❝ ❜é ➤Ø♥❤ ✈➭ ❝➵♥❤ G = (, L) L ợ ị ♣❤➬♥ ❈➳❝ ❦ý ❤✐Ư✉ ✷✳✷✳✶ ➤➢ỵ❝ ❣ä✐ ❧➭ ➤å tr♦♥❣ ➤ã t➢➡♥❣ ø♥❣ ✈í✐ ❤Ư ❤➭♠ ❧➷♣ ✷✳✷✳✸ ◆❤❐♥ ①Ðt t❤Þ m pi {Si }m i=1 = {λ x + bi }i=1 ✳ ✭❬✼❪✮✳ ✶✮ (pi , bi )−1 (pj , bj ) = (pj − pi , λ−pi bj − λ−pi bi ) ✈× (pi , bi )−1 (pj , bj ) (x) = (pi , bi )−1 (λpi x + bj ) = λ−pi (λpj x + bj ) − bi λ−pi = λpj −pi x + λ−pi bj − bi λ−pi = (pj − pi , λ−pi bj − λ−pi bi ) ✷✮ ❱× λpi (bj bi ) Z Z ỗ tö tr♦♥❣ Λ ❧➭ (pi , bi )−1 (pj , bj ) = (pj − pi ; λ−pi (bj − bi )) t❤á❛ ♠➲♥ pj − pi ∈ Z✱ |pj − pi | ≤ max {pj , pi } − = T − ✈➭ λ−pi (bj − bi ) ∈ Z − Z ♥➟♥ λ−pi (bj − bi ) ∈ C ✳ ❉♦ ➤ã✱ t❛ ❝ã Λ ⊂ Ξ✳ ✸✮ ➜➷t y = x.λ−T +2 = R λ−1 ❱× λ−T +2 λ d s✉② r❛ λ−1 y + b0 > d λ−1 y + b0 < d✳ ▲➵✐ ➤➷t ✷✹ ✈í✐ ✈í✐ ♠ä✐ R ∈ Z − Z }✳ b0 ∈ B − B ✱ ♥➟♥ y ∈ D t❤× Dk ={y ∈ D : x = R λ−1 ❱× ✈í✐ ❜✃t ❦ú ✈í✐ ✈í✐ deg (R) ≤ k }✳ Rk ∈ Z −Z ✈í✐ deg (Rk ) = k ✱ ❧✉➠♥ tå♥ t➵✐ ➤❛ t❤ø❝ Rk−1 ∈ Z −Z deg (Rk−1 ) = k − ✈➭ b ∈ B − B s❛♦ ❝❤♦ Rk λ−1 = λ−1 Rk−1 λ−1 + b✳ ❉♦ ➤ã✱ Rk−1 λ−1 ∈ Dk−1 ♥Õ✉ Rk λ−1 ∈ Dk ✳ ❱× t❤Õ✱ t❛ ❝ã t❤Ĩ ❜✐Ĩ✉ ❞✐Ơ♥ Dk = {y : |y| ≤ d} ∩ λ−1 Dk−1 + (B − B) ✷✳✷✳✹ ➜Þ♥❤ ❧ý ✭❬✼❪✮✳ ❈❤♦ ❤Ư ❤➭♠ ❧➷♣ {λpi x + bi }m i=1 ✳ ✳ ❑❤✐ ➤ã✱ ❤Ö ❤➭♠ ❧➷♣ ❦❤➠♥❣ t❤á❛ ♠➲♥ ❖❙❈ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ tå♥ t➵✐ ♠ét ➤➢ê♥❣ ➤✐ tr♦♥❣ ● ①✉✃t ♣❤➳t tõ ♠ét ➤Ø♥❤ tr♦♥❣ Λ ✈➭ ❦Õt t❤ó❝ t➵✐ ➤Ø♥❤ ✭✵✱✵✮✳ ❈❤ø♥❣ ♠✐♥❤✳ ❚õ ➜Þ♥❤ ❧ý ✷✳✶✳✶✶ ✈➭ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✷✳✷ t❛ ❝ã ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ✷✳✷✳✺ ➜Þ♥❤ ❧ý ✭❬✼❪✮✳ ❍Ư ❤➭♠ ❧➷♣ tr♦♥❣ ➜Þ♥❤ ❧ý ✷✳✷✳✹ ❦❤➠♥❣ t❤á❛ ♠➲♥ ❙❙❈ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ tå♥ t➵✐ ♠ét ➤➢ê♥❣ ➤✐ ✈➠ ❤➵♥ tr♦♥❣ ● ①✉✃t ♣❤➳t tõ ♠ét ➤Ø♥❤ t❤✉é❝ ❈❤ø♥❣ ♠✐♥❤✳ Λ✳ ❚õ ➜Þ♥❤ ❧ý ✷✳✶✳✶✹ ✈➭ ➜Þ♥❤ ♥❣❤Ü❛ ✷✳✷✳✷ t❛ ❝ã ➤✐Ị✉ ♣❤➯✐ ❝❤ø♥❣ ♠✐♥❤✳ ❚õ ❝➳❝ ❦Õt q✉➯ tr×♥❤ ❜➭② ë tr➟♥ t❛ ➤✐ ➤Õ♥ t❤✉❐t t♦➳♥ ❦✐Ó♠ tr❛ ①❡♠ ❤Ö ❤➭♠ ❧➷♣ −1 {λpi x + bi }m > ❧➭ P✳❱ sè ❝è ➤Þ♥❤ ✈➭ (p1 , , pm ) ∈ Nm ❀ (b1 , , bm ) ∈ Qm i=1 ✈í✐ λ ❧➭ ❝➳❝ t❤❛♠ sè ❝ã t❤á❛ ♠➲♥ ➤✐Ị✉ ❦✐Ư♥ ❖❙❈ ❤❛② ❦❤➠♥❣✳ ✷✳✷✳✻ ❚❤✉❐t t♦➳♥ ❇➢í❝ ✶✳ ◆Õ✉ ✭❬✼❪✮✳ (b1 , , bm ) ∈ Qm sè ❝❤✉♥❣ ❜Ð ♥❤✃t ❝ñ❛ b1 , , bm ✳ ♥❤➢♥❣ ❦❤➠♥❣ t❤✉é❝ ❑❤✐ ➤ã✱ (b1 , , bm ) ∈ Zm i = 1, , m✳ ✷✺ Zm ✱ ✈í✐ t❛ ❧✃② a bi = bi a✱ ỗ {pi x + bi }m i=1 t❤á❛ ♠➲♥ ❖❙❈ ❦❤✐ ✈➭ ❑❤✐ ➤ã✱ ❞Ô ❝❤ø♥❣ ♠✐♥❤ ➤➢ỵ❝ ❤Ư ❤➭♠ ❧➷♣ ❝❤Ø ❦❤✐ ❤Ư ❤➭♠ ❧➷♣ {λpi x + bi }m i=1 t❤á❛ ♠➲♥ ❖❙❈✳ ❉♦ ➤ã✱ t ét tr trờ ợ ì t ➤Ø♥❤ C ={x ∈ R : |x| ≤ (b1 , , bm ) ∈ Zm ✳ Ξ = {n ∈ Z : |n| ≤ T − 1} × C ✱ t❤ù❝ ❝❤✃t ❧➭ t×♠ t❐♣ (1+λ−T +1 ) b0 1−λ ; x = λT −2 R λ−1 ❱í✐ ❝➳❝ ➤➳♥❤ ❣✐➳ ♥❤➢ ë ◆❤❐♥ ①Ðt ✷✳✷✳✸ t❛ t×♠ ❱× C = λT −2 y : y ∈ D C ✈í✐ R ∈ Z − Z }✳ ♥❤➢ s❛✉✳ ✱ Dk = {y : |y| ≤ d}∩ λ−1 Dk−1 + (B − B) ♥➟♥ ✈í✐ k = 0✱ t❛ ❝ã D0 = {y : |y| ≤ d}∩ {B − B} ❉♦ ➤ã✱ ❧✃② y ∈B−B ✈➭ ❦✐Ó♠ tr❛ ①❡♠ |y| ≤ λ−T +2 ❝ã ➤ó♥❣ ❤❛② ❦❤➠♥❣✳ ◆Õ✉ ➤ó♥❣ t❤× ❉♦ ➤ã✱ ❧✃② ✈í✐ (1+λ−T +1 ).b0 1−λ y ∈ D0 ợ tì y / D0 y ∈ λ−1 D0 + B − B ❤❛② y = λ−1 (bi − bj ) + bk − bl |bi − bj | ≤ d; bi , bj , bk , bl ∈ {b1 , , bm } ❦❤➠♥❣ ➤Ĩ t×♠ |y| ≤ d ❝ã ➤ó♥❣ D1 ✳ ❈ø tế tụ ể tì ế tì ợ D2 , D3 , ✳ k0 ❜Ð ♥❤✃t s❛♦ ❝❤♦ Dk0 +1 = Dk0 ó t tì ợ ✈➭ ❦✐Ĩ♠ tr❛ ①❡♠ t❤× ❧✃② D = Dk0 ✳ C tứ tì ợ t ỉ ẽ tt tr tị G tì ➤➢ỵ❝ t❐♣ ❝➳❝ ➤Ø♥❤ Ξ ë ❇➢í❝ ✷✳ ❚❛ ❝ã Ξ = ([−T + 1, T − 1] ∩ Z) × C ✳ ▲✃② ❤❛✐ ➤Ø♥❤ (p, b) , (p , b ) ∈ Ξ✱ ❦✐Ó♠ tr❛ ①❡♠ ❝ã tå♥ t➵✐ α, β ∈ Σ∗ ✈í✐ α = i1 i2 ik , β = j1 j2 jl s❛♦ ❝❤♦ ✷✻ ≤ pi1 + pi2 + + pik , pj1 + pj2 + + pjl ≤ 2T − tø❝ ≤ pα , pβ ≤ 2T − ♠➭ (p , b ) = (pα + p − pβ , λ−pβ +p bα + λ−pβ b − λ−pβ bβ )✱ ❤❛② pα − pβ = p − p λ−pβ [λp bα + b − bβ ] = b ế tì ợ , () tỏ ♠➲♥ ✭✯✮ t❤× ❝ã ♠ét ❝➵♥❤ ♥è✐ ❤❛✐ ➤Ø♥❤ (p, b) ✈➭ (p , b )✳ ❉➱♥ ➤Õ♥✱ ✈✐Õt ➤➢ỵ❝ ❤Õt ❝➳❝ ➤➢ê♥❣ ➤✐ tõ ➤Ø♥❤ ♥➭② ➤Õ♥ ➤Ø♥❤ ❦✐❛ tr ợ C G ì ự ữ ❑✐Ĩ♠ tr❛ sù tå♥ t➵✐ ❝đ❛ ➤➢ê♥❣ ➤✐ tõ ♠ét ➤Ø♥❤ tr♦♥❣ Λ ➤Õ♥ ➤Ø♥❤ (0, 0)✳ ◆Õ✉ tå♥ t➵✐ tì ệ tỏ ợ t❤× ❤Ư ❤➭♠ ❧➷♣ t❤á❛ ♠➲♥ ❖❙❈ ✳ ❚➢➡♥❣ tù✱ ♥Õ✉ tå♥ t➵✐ ➤➢ê♥❣ ➤✐ ✈➠ ❤➵♥ tr♦♥❣ G✱ ①✉✃t ♣❤➳t tõ ♠ét ➤Ø♥❤ tr♦♥❣ G t❤× ❤Ư ❤➭♠ ❧➷♣ ❦❤➠♥❣ t❤á❛ ♠➲♥ ❙❙❈✳ ◆❣➢ỵ❝ ❧➵✐ ❤Ư ❤➭♠ ❧➷♣ t❤á❛ ♠➲♥ ❙❙❈ s✉② r❛ t❤á❛ ♠➲♥ ❖❙❈✳ ✷✳✷✳✼ ◆❤❐♥ ①Ðt ✭❬✼❪✮✳ ✶✮ ❑❤✐ T = 1✱ tø❝ max {p1 , , pm } = ✈í✐ pi ∈ N∗ s✉② r❛ p1 = = pm = 1✳ ❳Ðt ❱× F = {σ : ≤ pσ ≤ 2T − = 1} t❤× F = {1, , m}✳ Λ = ỗ ỉ ì 0, (bj bi ) : i = j πZ (1, bi )−1 (0, c) (1, bj ) = (q, c) ♣❤➯✐ ❝ã q = 0✳ F = {1, , m} ♥➟♥ pσk ◦ pτk = k ∈ {1, , m}✱ ❞♦ ➤ã t❛ ❝ã πR (pτk , bτk )−1 (pσk , bσk ) ✈í✐ ✈➭ max |b| = |x1 − x2 | ≤ x1 , x2 ∈ E ✳ ❉♦ ➤ã✱ t❐♣ ➤Ø♥❤ tr♦♥❣ tr➢ê♥❣ ❤ỵ♣ T = ❧➭ ✷✼ b∈B−B 1−λ ♥➟♥ max |b| Ξ = {0} × C = {0} ×{x : |x| ≤ ✈í✐ b∈B−B ✈➭ 1−λ x = λ−1 R λ−1 R ∈ Z − Z✳ ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ ♥➭② ✈✐Ư❝ ❦✐Ĩ♠ tr❛ ①❡♠ ❤Ư ❤➭♠ ❧➷♣ ❝ã t❤á❛ ♠➲♥ ❖❙❈ ❤❛② ❦❤➠♥❣ ❧➭ ❦❤➳ ➤➡♥ ❣✐➯♥ ❞♦ t❛ ❝ã t❤Ĩ ➤➳♥❤ ❣✐➳ ➤➢ỵ❝ ❧ù❝ ❧➢ỵ♥❣ C ❧➭ max |b| #C ≤ b∈B−B 1−λ + ✈➭ t❐♣ {p1 , , pm } ❝è ➤Þ♥❤✳ ✷✮ N tì t ợ #C = #Dk0 ≤ 2d + = ✷✳✷✳✽ ▼Ö♥❤ ➤Ò ✭❬✼❪✮✳ max |bi | i=1,m (1−λ)(λT −2 +λ2T −3 ) + 1✳ ❱í✐ ❚❤✉❐t t♦➳♥ ✷✳✷✳✻ t❤× t❤ê✐ ❣✐❛♥ ❝❤➵② ❝ñ❛ t❤✉❐t t♦➳♥ ✭➤é ♣❤ø❝ t➵♣ ❝ñ❛ t❤✉❐t t♦➳♥✮ ➤➢ỵ❝ ➢í❝ ❧➢ỵ♥❣ ❧➭ ✐✮ ❚❤ê✐ ❣✐❛♥ ❦✐Ĩ♠ tr❛ ❤Ư ❤➭♠ ❧➷♣ t❤á❛ ♠➲♥ ❖❙❈ ❧➭ t ≤ c.(#Ξ)2 + m4T (#Ξ) + (m + 1)2 ✳ ✐✐✮ ❚❤ê✐ ❣✐❛♥ ❦✐Ó♠ tr❛ ■❋❙ t❤á❛ ♠➲♥ ❙❙❈ ❧➭ t ≤ c.(#Ξ)3 + m4T (#Ξ) + (m + 1)2 ✳ ❚r♦♥❣ ➤ã c ❧➭ ❤➺♥❣ sè tr♦♥❣ t❤✉❐t t♦➳♥ ♥❣✉②➟♥ t❤ñ② ❝ñ❛ ❉✐❥❦str❛ ✈➭ #Ξ = (2T − 1) #C ≤ (2T − 1) 2l (1+λ−T +1 ).bl+1 o l λT −2 (1−λ) (1−|ηi |) i=1 ❈❤ø♥❣ ♠✐♥❤✳ ❚r♦♥❣ ❇➢í❝ ✷✱ tờ ể tì t ỉ ợ ko t1 = #Ξ ≤ # (B − B) # (Dk \Dk−1 ) i=0 ≤ # (B − B) #C ≤ (m + 1)2 #C ✷✽ ❚❤ê✐ ❣✐❛♥ t×♠ ❝➳❝ ❝➵♥❤ ë ❇➢í❝ ✸ ❧➭ ✿ ❚❛ ❝➬♥ t×♠ (pα + p − pβ ; λ−pβ +p bα + λ−pβ b − λ−pβ bβ ) ✈í✐ ♠ä✐ ♥➟♥ (p, b) ∈ Ξ ✈➭ α, β ∈ F = {σ : ≤ pσ ≤ 2T − 1} #F ≤ m + m2 + + m2T −1 < m2T ✳ ❉♦ ➤ã✱ sè ❜➢í❝ ❝➬♥ t❤ù❝ ❤✐Ư♥ ë ✈✐Ư❝ t×♠ ❝➵♥❤ ❧➭ t2 ≤ #Ξ.(#F )2 ≤ m4T #Ξ✳ ❚❤ê✐ ❣✐❛♥ t×♠ ➤➢ê♥❣ ➤✐ ë ❇➢í❝ ✹ ➤Ĩ ❦✐Ĩ♠ tr❛ t❤á❛ ♠➲♥ ❖❙❈ ❧➭ t3 ≤ c.(#Ξ)2 ✳ ✈➭ ➤Ó ❦✐Ó♠ tr❛ ❙❙❈ ❧➭ t3 ≤ c.(#Ξ)3 ✳ ❉♦ ✈❐②✱ t❛ t❤✉ ➤➢ỵ❝ ➤➳♥❤ ❣✐➳ ✈Ị ➤é ♣❤ø❝ t➵♣ t❤✉❐t t♦➳♥ ♥❤➢ tr➟♥✳ ✷✳✸ ▼ét sè ✈Ý ❞ô ♠✐♥❤ ❤ä❛ ✷✳✸✳✶ ❱Ý ❞ô ✶ ❳Ðt λ−1 = > ❧➭ P✳❱ sè✱ m = 3❀ p = (1; 1; 1) ✈➭ b = ❚❤❡♦ ❜➢í❝ ✶✱ t❤❛② ✈× ①Ðt (b1 , b2 , b3 ) = 2 ; 3n+1 ; ∈ Q3 t❛ ①Ðt 2 ; 3n+1 ; ✳ (b1 , b2 , b3 ) = (2.3n ; 2; 0) ∈ Z3 ✳ ❑❤✐ ➤ã✱ B − B = {−2.3n ; (1 − 3n ) ; −2; 0; 2; (3n − 1) ; 2.3n }✱ ✈× p1 = p2 = p3 = ♥➟♥ T = s✉② r❛ E = {1, 2, 3}✳ ❚❤❡♦ ◆❤❐♥ ①Ðt ✷✳✷✳✼✱ t❛ ❝ã C ∩ λ−1 (bi − bj ) i=j C ={m : m ❝❤➼♥ ✈➭ m ∈ −3n+1 , 3n+1 ∩ Z} = {−6; 6}✳ ❉♦ ➤ã✱ Λ = {(0; −6) , (0; 6)}✳ ❚õ ❜➢í❝ ✸✱ ❜➢í❝ ✹ ❝đ❛ t❤✉❐t t♦➳♥ t❛ ❝ã ➤➢ê♥❣ ➤✐ ❧➭ ✷✾ ✈➭ (0; 6) → 0; 2.32 → 0; 2.33 → → (0; 2.3n ) → (0; 0) → (0; 0) → ①✉✃t ♣❤➳t tõ (0; 6)✳ ❉♦ ➤ã✱ ❤Ö ❤➭♠ ❧➷♣ ♥➭② ❦❤➠♥❣ t❤á❛ ♠➲♥ ❙❙❈ ✈➭ ❖❙❈✳ ❱× t❤Õ✱ t❐♣ ❜✃t ❜✐Õ♥ ❝đ❛ ❤Ư ❤➭♠ ❧➷♣ ♥➭② ❝ã ❝❤✐Ị✉ ❍❛✉s❞♦r❢❢ ❧➭ ❱× ❝➳❝ ❤Ư sè ➤å♥❣ ❞➵♥❣ ♣❤➢➡♥❣ tr×♥❤✿ s E dimH E = = dims E ✳ c1 = c2 = c3 = ♥➟♥ dims E = s ❧➭ ♥❣❤✐Ư♠ ❝đ❛ = 1✱ s✉② r❛ s = 1✳ o ❑❤✐ ➤ã✱ t❛ ❝ò♥❣ ❝ã E = ∅✳ ✷✳✸✳✷ ❱Ý ❞ô ✷ ▲✃② λ= − ❜ë✐ ✈➭ ①Ðt ❤Ư ❤➭♠ ❧➷♣ ✈í✐ p = (1; 1; 2) , b = 11 18 ; 0; ✳ ▲✃② a = 18 ✈➭ ❜ b = (11; 0; 12)✳ ❙✉② r❛ B = {0; 11; 12}✱ tõ ➤ã t❛ ❝ã B−B = {−12; −11; −1; 0; 1; 11; 12}✳ λ−T +2 (1+λ−T +1 ) max|b| ❱í✐ T = 2✱ ❦❤✐ ➤ã t❛ ❝ã d = = 72✳ ❉ï♥❣ ❜➢í❝ ✷ ❝đ❛ t❤✉❐t 1−λ t♦➳♥ t❛ ❝ã✿ ❚❛ ❝ã✿ C = [−72; 72] ∩ Z ✈➭ Ξ = {−1, 0, 1} × C ✳ Λ = {(−1, 9) ; (0; −36) ; (0; 36) ; (1; −3) ; (1; 33)}✳ ❚❤ù❝ ❤✐Ư♥ ❜➢í❝ ✸✱ ✹ t❛ t❤✃② ➤➢ê♥❣ ➤✐ trù❝ t✐Õ♣ ❦❤➠♥❣ ①✉✃t ♣❤➳t tõ ❜✃t ❦ú ➤Ø♥❤ ♥➭♦ tr♦♥❣ Λ ✈➭ ❦Õt t❤ó❝ t➵✐ (0; 0)✳ ❉♦ ✈❐②✱ ❤Ö ❤➭♠ ❧➷♣ t❤á❛ ♠➲♥ ❖❙❈✳ ❚✉② ♥❤✐➟♥✱ t❛ ❝ã ➤➢ê♥❣ ➤✐ trù❝ t✐Õ♣ ✈➠ ❤➵♥ ❧➭✿ (−1; 9) → (−1; −9) → (−1; −27) → (0; 18) → (0; 18) → (0; 18) → ♣❤➳t tõ ①✉✃t (−1; 9) ∈ Λ✳ ❱❐② ❙❙❈ ❦❤➠♥❣ t❤á❛ ♠➲♥✳ ✷✳✸✳✸ ❱Ý ❞ô ✸ ▲✃② ❳Ðt λ= √ − ⇒ λ−1 = p = (2; 1; 1) , b = √ + ❧➭ P✳❱ sè ✈➭ ❣ä✐ ❧➭ tØ sè ❜➵❝ ✭s✐❧✈❡r✮ ✳ 5; 2; → a = 10 − ✈➭ t❤❛② ❜ ❜ë✐ b = (4; 5; 0) → B = {0; 4; 5} → B − B = {±5, ±4, ±10}✳ ❚❛ ❝ã✿ T = 2✳ ❙✉② r❛ √ d = 15 + 10 2✳ ♣❤➬♥ tö✳ ❑❤✐ ➤ã ✸✵ ❚❤ù❝ ❤✐Ư♥ ❜➢í❝ ✷ s✉② r❛ E ❝❤ø❛ ✶✵✺✾ Λ= √ √ √ 1, + , 0, + , −1, −12 − , √ √ √ −1, + 2 , 0, −5 − , 1, −1 − ✳ ❚❤❡♦ ♣❤➢➡♥❣ ♣❤➳♣ ❦✐Ĩ♠ tr❛✱ t❤ù❝ ❤✐Ư♥ ❜➢í❝ ✸✱ ✹ ❝đ❛ t❤✉❐t t♦➳♥ t❛ ❝ã ➤➢ê♥❣ ✿ √ √ √ −1; + 2 → −1; −9 − → −1; −6 − → (0; 0) → (0; 0) √ ①✉✃t ♣❤➳t tõ −1; + 2 ∈ Λ✳ ❱❐② ❖❙❈ ✈➭ ❙❙❈ ➤Ò✉ ❦❤➠♥❣ t❤á❛ ♠➲♥✳ ✸✶ ❑Õt ❧✉❐♥ ▲✉❐♥ ✈➝♥ ➤➲ ➤➵t ➤➢ỵ❝ ❝➳❝ ❦Õt q✉➯ s❛✉ ✶✳ ❍Ư t❤è♥❣ ➤➢ỵ❝ ❝➳❝ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥✱ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ✱ ➤✐Ị✉ ❦✐Ư♥ ➤đ ➤Ĩ ❤Ư ❤➭♠ ❧➷♣ t❤á❛ ♠➲♥ ❖❙❈✳ ✷✳ ❚r×♥❤ ❜➭② ✈➭ ❝❤ø♥❣ ♠✐♥❤ ❝❤✐ t✐Õt ❇ỉ ➤Ị ✷✳✶✳✷✱ ▼Ư♥❤ ➤Ị ✷✳✶✳✾✱ ❝➳❝ ❝➡ së t♦➳♥ ❤ä❝ ➤Ĩ ①➞② ❞ù♥❣ t❤✉❐t t♦➳♥ ❦✐Ĩ♠ tr❛ ♠ét ❧í♣ ❤Ư ❤➭♠ ❧➷♣ {λpi x + bi }m i=1 ✈í✐ λ−1 > ❧➭ P✳❱ sè✱ pi ∈ N, bi ∈ Q, i = 1, m t❤á❛ ♠➲♥ ❖❙❈✱ ❙❙❈ ❞ù❛ ✈➭♦ ➤å t❤Þ ✈➭ ➤➢ê♥❣ ➤✐✳ ✸✳ ▲✃② ❝➳❝ ✈Ý ❞ơ ♠✐♥❤ ❤ä❛ t❤✉❐t t♦➳♥ tr♦♥❣ ❝➳❝ tr➢ê♥❣ ❤ỵ♣ ❧➭✿ ❍Ö ❤➭♠ ❧➷♣ ❦❤➠♥❣ t❤á❛ ♠➲♥ ❖❙❈✱ ❙❙❈✱ ✈➭ ❍Ö ❤➭♠ ❧➷♣ t❤á❛ ♠➲♥ ❖❙❈ ♥❤➢♥❣ ❦❤➠♥❣ t❤á❛ ♠➲♥ ❙❙❈✳ ✸✷ t➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ❆✳ ❈❤✐❡❢ ✭✶✾✾✹✮✱ ❙❡♣❛r❛t✐♦♥ ♣r♦♣❡rt✐❡s ❢♦r s❡❧❢✲s✐♠✐❧❛r s❡t✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✷✷✱ ✾✾✺ ✲✶✵✵✶✳ ❬✷❪ ❆✳ ▼✳ ●❛rs✐❛ ✭✶✾✻✷✮✱ ❆r✐t❤ ♣r♦♣❡rt✐❡s ♦❢ ❇❡r♥♦✉❧❧✐ ❝♦♥✈♦❧✉t✐♦♥s✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✵✷✱ ✹✵✾ ✲✹✸✷✳ ❬✸❪ ❈✳ ❇❛♥❞t✱ ◆✳❱✳ ❍✉♥❣ ❛♥❞ ❍✳ ❘❛♦ ✭✷✵✵✺✮✱ s✐♠✐❧❛r ❢r❛❝t❛❧s✱ ❬✹❪ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✸✹✱ ✶✸✻✾ ✲✶✸✼✹✳ ❈✳ ❇❛♥❞t ❛♥❞ ◆✳❱✳ ❍✉♥❣ ✭✷✵✵✽✮✱ ❣r❡❛t ✈❛r✐❡t② ♦❢ ♦✈❡r❧❛♣s✱ ❬✺❪ ❖♥ t❤❡ ♦♣❡♥ s❡t ❝♦♥❞✐t✐♦♥ ❢♦r s❡❢❧✲ ❙❡❧❢✲s✐♠✐❧❛r s❡ts ✇✐t❤ ♦♣❡♥ s❡t ❝♦♥❞✐t✐♦♥ ❛♥❞ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✸✻✱ ✸✽✾✺ ✲✸✾✵✸✳ ❈✳ ❇❛♥❞t ❛♥❞ ❙✳ ●r❛❢ ✭✶✾✾✷✮✱ ❙❡❧❢ ✲ ❙✐♠✐❧❛r s❡ts ❱■■✳ ❆ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ s❡❧❢✲ s✐♠✐❧❛r ❢r❛❝t❛❧ ✇✐t❤ ♣♦s✐t✐✈❡ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✶✹✱ ✾✾✺ ✲✶✵✵✶✳ ❬✻❪ ❑✳ ❙✳ ▲❛✉✱ ❙✳ ▼✳ ◆❣❛✐ ✭✷✵✵✼✮✱ ❢✉♥❝t✐♦♥ s②st❡♠s✱ ❬✼❪ ❆ ❣❡♥❡r❛❧✐③❡❞ ❢✐♥✐t❡ t②♣❡ ❝♦♥❞✐t✐♦♥ ❢♦r ✐t❡r❛t❡❞ ❆❞✈✳ ▼❛t❤✳✱ ✷✵✽✱ ✭✷✮ ✻✹✼ ✲ ✻✼✶✳ ❍✳ ◗✳ ●✉♦✱ ◗✳ ❲❛♥❣ ❛♥❞ ▲✳ ❋✳ ❳✐ ✭✷✵✶✹✮✱ ❢♦r s❡❧❢ ✲ s✐♠✐❧❛r s❡t r❡❧❛t❡❞ t♦ ♣✳✈✳♥✉♠❜❡rs✱ ❬✽❪ ❏✳ ❊✳ ❍✉t❝❤✐♥s♦♥ ✭✶✾✽✶✮✱ ❆❧❣♦r✐t❤♠s t♦ t❡st ♦♣❡♥ s❡t ❝♦♥❞✐t✐♦♥ ▼❛t❤ ❙✉❜❥❡❝t ❈❧❛s✐❢✐❝❛t✐♦♥✳ ✷✽❆✽✵✳ ❋r❛❝t❛❧s ❛♥❞ s❡❧❢ s✐♠✐❧❛r✐t②✱ ■♥❞✐❛♥❛✳ ❯♥✐✈✳ ▼❛t❤✳ ❏✳✱ ✸✵✱ ✼✶✸✲✼✹✼✳ ❬✾❪ ❑✳ ❏✳ ❋❛❧❝♦♥❡r ✭✶✾✾✵✮✱ ❝❛t✐♦♥s✱ ❬✶✵❪ ❋r❛❝t❛❧ ●❡♦♠❡tr②✲ ▼❛t❤❡♠❛t✐❝❛❧ ❋♦✉♥❞❛t✐♦♥s ❛♥❞ ❆♣♣❧✐✲ ❏♦❤♥ ❲✐❧❡② ❙♦♥s✳ ◆✳ ❉✳ ▲❛✉ ❛♥❞ ❚✳ ◆✳ ❱✐❡t ✭✷✵✶✷✮✱ ❚❤✉❛t t♦❛♥ ❉✐❥❦str❛✳✳✳✱ ❤♦❝ ❍✉❡✱ ✺✱ ✽✶✲✾✷✳ ✸✸ ❚❛♣ ❝❤✐ ❦❤♦❛ ❤♦❝✲ ❉❛✐ ❬✶✶❪ P✳ ❆✳ P✳ ▼♦r❛♥ ✭✶✾✹✻✮✱ ❆❞❞✐t✐✈❡ ❢✉♥❝t✐♦♥ ♦❢ ✐♥t❡r✈❛❧s ❛♥❞ ❍❛✉s❞♦r❢❢ ♠❡❛s✉r❡✱ Pr♦❝✳ ❈❛♠❜r✐❞❣❡ P❤✐❧♦s✳ ❙♦❝✳✱ ✹✷✱ ✶✺✲✷✸✳ ❬✶✷❪ ◗✳ ❘✳ ❉❡♥❣ ❛♥❞ ❑✳ ❙✳ ▲❛✉✭✷✵✵✽✮✱ ❖❙❈ ❛♥❞ ♣♦st ❝r✐t✐❝❛❧❧② ❢✐♥✐t❡ s❡❧❢✲ s✐♠✐❧❛r s❡ts✱ ◆♦♥❧✐♥❡❛r✐t② ✷✶✱ ✶✷✷✼✲ ✶✷✸✷✳ ❬✶✸❪ ❙✳ P✳ ▲❛❧❧❡② ✭✶✾✾✼✮✱ β ✲❡①♣❛♥s✐♦♥s ✇✐t❤ ❞❡❧❡t❡❞ ❞✐❣✐ts ❢♦r P✐s♦t ♥✉♠❜❡rs✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✸✹✾✱ ✹✸✺✺✲✹✸✻✺✳ ❬✶✹❪ ❙✳ ▼✳ ◆❣❛✐ ❛♥❞ ❨✳ ❲❛♥❣ ✭✷✵✵✶✮✱ ♦✈❡r❧❛♣s✱ ❍❛✉s❞♦r❢❢ ❞✐♠❡♥s✐♦♥ ♦❢ s❡❧❢✲s✐♠✐❧❛r s❡ts ✇✐t❤ ❏✳ ▲♦♥❞✳ ▼❛t❤✳ ❙♦❝✳✱ ✻✸✱ ✻✺✺✲✼✷✳ ✸✹ ... ✶✳✶✳✼ ➜Þ♥❤ ♥❣❤Ü❛ ✶✮ ❚❐♣ t❐♣ ❤ót F ✭❬✽❪✮✳ ❈❤♦ ❤Ư ❤➭♠ ❧➷♣ {Si }m i=1 tr D ó ợ ị tr ị ý ợ ọ (attractor) t❐♣ ❋r❛❝t❛❧ ❝đ❛ ❤Ư ❤➭♠ ❧➷♣ {Si }m i=1 ✳ ✻ t❐♣ ❜✃t ❜✐Õ♥ ❤❛② ✷✮ ◆Õ✉ Si (1 ≤ i ≤ m) ❧➭ ➳♥❤