Robustez da dinâmica sob perturbações: da semicontinuidade superior à estabilidade...

Robustez da dinâmica sob perturbações: da

semicontinuidade superior à estabilidade

estrutural

SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP

Data de Depósito:

Assinatura:_______________________

Arthur Geromel Fischer

Robustez da dinâmica sob perturbações: da

semicontinuidade superior à estabilidade estrutural

Dissertação apresentada ao Instituto de Ciências Matemáticas e de Computação - ICMC-USP, como parte dos requisitos para obtenção do título de Mestre em Ciências – Matemática. VERSÃO REVISADA

Área de Concentração: Matemática

Orientador: Prof. Dr. Hildebrando Munhoz Rodrigues

Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP,

com os dados fornecidos pelo(a) autor(a)

F529r

Fischer, Arthur Geromel

Robustez da dinâmica sob perturbações: da

semicontinuidade superior à estabilidade estrutural / Arthur Geromel Fischer; orientador Hildebrando Munhoz Rodrigues. -- São Carlos, 2015.

126 p.

Dissertação (Mestrado - Programa de Pós-Graduação em Matemática) -- Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, 2015.

1. Semigrupos e Atratores Globais. 2. Semicontinuidade inferior e superior. 3.

Arthur Geromel Fischer

Robustness of the dynamics under perturbation: from the

upper semicontinuity to the structural stability

Master dissertation submitted to the Instituto de Ciências Matemáticas e de Computação - ICMC-USP, in partial fulfillment of the requirements for the degree of the Master Program in Mathematics. FINAL VERSION

Concentration Area: Mathematics

Advisor: Prof. Dr. Hildebrando Munhoz Rodrigues

❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r❛♠❡♥t❡✱ ❣♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛♦s ♠❡✉s ♣❛✐s✳

❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢❡ss♦r ❍✐❧❞❡❜r❛♥❞♦✱ ❡ ❛♦ Pr♦❢❡ss♦r ❆❧❡①❛♥❞r❡ ♣♦r t♦❞❛ ❛ ❛❥✉❞❛ ❡ s✉♣♦rt❡ ❞✉r❛♥t❡ t♦❞♦s ❡ss❡s ❛♥♦s✱ ❞❡s❞❡ ♦ ♣r✐♠❡✐r♦ ❛♥♦ ❞❛ ♠✐♥❤❛ ❣r❛❞✉❛çã♦✳

❆❣r❛❞❡ç♦ t❛♠❜é♠ à ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦ ❡ ❛♦s ♣r♦❢❡ss♦r❡s q✉❡ ❝♦❧❛❜♦r❛r❛♠✱ ❞❡ ✉♠ ♠♦❞♦ ♦✉ ❞❡ ♦✉tr♦✱ ♣❛r❛ ❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳

P♦r ✜♠✱ ❛❣r❛❞❡ç♦ à ❋❆P❊❙P ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

❘❡s✉♠♦

❖ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦ é ♦ ❡st✉❞♦ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❡str✉t✉r❛❧ ❞♦s ❛tr❛t♦r❡s ❞❡ s❡♠✐❣r✉♣♦s✳

❈♦♠❡ç❛♠♦s ❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❡ ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞❡ s❡♠✐❣r✉✲ ♣♦s q✉❡ ♣♦ss✉❡♠ ❛tr❛t♦r❡s ❣❧♦❜❛✐s✳

❊st✉❞❛♠♦s✱ ❡♥tã♦✱ s❡♠✐❣r✉♣♦s ❣r❛❞✐❡♥t❡s ❡ ❞✐♥❛♠✐❝❛♠❡♥t❡ ❣r❛❞✐❡♥t❡s✱ ♠♦str❛♥❞♦ q✉❡ ❡❧❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s ❡ q✉❡ ✉♠❛ ♣❡q✉❡♥❛ ♣❡rt✉r❜❛çã♦ ❛✉tô♥♦♠❛ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❣r❛❞✐✲ ❡♥t❡ ❝♦♥t✐♥✉❛ s❡♥❞♦ ❣r❛❞✐❡♥t❡✳

❊st✉❞❛♠♦s ❛s ✈❛r✐❡❞❛❞❡s ❡stá✈❡❧ ❡ ✐♥stá✈❡❧ ❞❡ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❤✐♣❡r❜ó❧✐❝♦ ❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s s♦❜ ♣❡rt✉r❜❛çã♦✳

❈♦♥❝❧✉í♠♦s ❡st❡ tr❛❜❛❧❤♦ ❝♦♠ ♦ ❡st✉❞♦ ❞♦s s❡♠✐❣r✉♣♦s ▼♦rs❡✲❙♠❛❧❡✳

P❛❧❛✈r❛s✲❝❤❛✈❡✿ ❙❡♠✐❣r✉♣♦s✱ ❛tr❛t♦r❡s ❣❧♦❜❛✐s✱ ❡st❛❜✐❧✐❞❛❞❡ ❡str✉t✉r❛❧✱ s❡♠✐❣r✉♣♦s ❣r❛❞✐❡♥t❡s✱ s❡♠✐❣r✉♣♦s ▼♦rs❡✲❙♠❛❧❡✳

❆❜str❛❝t

❚❤❡ ♠❛✐♥ ❣♦❛❧ ♦❢ t❤✐s ✇♦r❦ ✐s t❤❡ st✉❞② ♦❢ str✉❝t✉r❛❧ st❛❜✐❧✐t② ♦❢ ❣❧♦❜❛❧ ❛ttr❛❝t♦rs✳ ❲❡ st❛rt t❤✐s ✇♦r❦ ❜② ♣r❡s❡♥t✐♥❣ t❤❡ ❝♦♥❝❡♣t ❛♥❞ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ s❡♠✐❣r♦✉♣s ❛♥❞ ❣❧♦❜❛❧ ❛ttr❛❝t♦rs✳

❲❡ t❤❡♥ st✉❞✐❡❞ ❣r❛❞✐❡♥t ❛♥❞ ❞✐♥❛♠✐❝❛❧❧② ❣r❛❞✐❡♥t s❡♠✐❣r♦✉♣s✱ s❤♦✇✐♥❣ t❤❛t t❤❡s❡ ❝♦♥❝❡♣ts ❛r❡ ❡q✉✐✈❛❧❡♥t ❛♥❞ t❤❛t ❛ s♠❛❧❧ ❛✉t♦♥♦♠♦✉s ♣❡rt✉❜❛t✐♦♥ ♦❢ ❛ ❣r❛❞✐❡♥t s❡♠✐❣r♦✉♣ r❡♠❛✐♥s ❛ ❣r❛❞✐❡♥t s❡♠✐❣r♦✉♣✳

❲❡ st✉❞✐❡❞ t❤❡ st❛❜❧❡ ❛♥❞ ✉♥st❛❜❧❡ ♠❛♥✐❢♦❧❞s ✐♥ t❤❡ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ❛ ❤②♣❡r❜♦❧✐❝ ❡q✉✐❧✐❜r✐✉♠ ♣♦✐♥t ❛♥❞ t❤❡ ❜❡❤❛✈✐♦r ♦❢ ♣❡r✐♦❞✐❝ s♦❧✉t✐♦♥s ✉♥❞❡r ♣❡rt✉r❜❛t✐♦♥✳

❋✐♥❛❧❧②✱ ✇❡ st✉❞✐❡❞ t❤❡ ▼♦rs❡✲❙♠❛❧❡ s❡♠✐❣r♦✉♣s✳

❑❡②✇♦r❞s✿ ❙❡♠✐❣r♦✉♣s✱ ❣❧♦❜❛❧ ❛ttr❛❝t♦rs✱ str✉❝t✉r❛❧ st❛❜✐❧✐t②✱ ❣r❛❞✐❡♥t s❡♠✐❣r♦✉♣s✱ ▼♦rs❡✲❙♠❛❧❡ s❡♠✐❣r♦✉♣s✳

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ ❙❡♠✐❣r✉♣♦s ❡ ❆tr❛t♦r❡s ●❧♦❜❛✐s ✸

✶✳✶ ◆♦çõ❡s ❡ ❢❛t♦s ❜ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❊①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r❡s ❣❧♦❜❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷ ❙❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r ✶✼

✸ ❊st❛❜✐❧✐❞❛❞❡ ❊str✉t✉r❛❧ ❚♦♣♦❧ó❣✐❝❛ ❞❡ ❆tr❛t♦r❡s ✷✸ ✸✳✶ ❙❡♠✐❣r✉♣♦s ❣r❛❞✐❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✸✳✷ ❙❡♠✐❣r✉♣♦s ❞✐♥❛♠✐❝❛♠❡♥t❡ ❣r❛❞✐❡♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✸✳✸ ❊str✉t✉r❛s ❤♦♠♦❝❧í♥✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✹ ❊q✉✐✈❛❧ê♥❝✐❛ ❡♥tr❡ s❡♠✐❣r✉♣♦s ❞✐♥❛♠✐❝❛♠❡♥t❡ ❣r❛❞✐❡♥t❡s ❡ ❣r❛❞✐❡♥t❡s ✳ ✳ ✳ ✸✺ ✸✳✹✳✶ ❉❡❝♦♠♣♦s✐çã♦ ❞❡ ▼♦rs❡ ♣❛r❛ s❡♠✐❣r✉♣♦s ❞✐♥❛♠✐❝❛♠❡♥t❡ ❣r❛❞✐❡♥t❡s ✸✼ ✸✳✹✳✷ ❋✉♥çõ❡s ❞❡ ▲②❛♣✉♥♦✈ ♣❛r❛ s❡♠✐❣r✉♣♦s ❞✐♥❛♠✐❝❛♠❡♥t❡ ❣r❛❞✐❡♥t❡s ✳ ✹✺ ✸✳✺ ❊st❛❜✐❧✐❞❛❞❡ ❡str✉t✉r❛❧ t♦♣♦❧ó❣✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

✹ ❱✐③✐♥❤❛♥ç❛ ❞❡ ✉♠ ❊❧❡♠❡♥t♦ ❈rít✐❝♦ ✺✼

✹✳✶ ❱❛r✐❡❞❛❞❡s ✐♥✈❛r✐❛♥t❡s ♣ró①✐♠❛s ❛ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✶✳✶ ❱❛r✐❡❞❛❞❡s ✐♥✈❛r✐❛♥t❡s ❝♦♠♦ ❣rá✜❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✷ ❉✐❢❡r❡♥❝✐❛❜✐❧✐❞❛❞❡ ❞❡ ✈❛r✐❡❞❛❞❡s ✐♥✈❛r✐❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✳✸ ❱❛r✐❡❞❛❞❡s ✐♥✈❛r✐❛♥t❡s s♦❜ ♣❡rt✉r❜❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✹✳✹ ❖ λ✲▲❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼

✺ P❡r♠❛♥ê♥❝✐❛ ❡ ❈♦♥t✐♥✉✐❞❛❞❡ ❞❡ ❊❧❡♠❡♥t♦s ❈rít✐❝♦s ✽✾ ✺✳✶ P❡r♠❛♥ê♥❝✐❛ ❡ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾ ✺✳✷ P❡r♠❛♥ê♥❝✐❛ ❡ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✺✳✷✳✶ ❙♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✺✳✷✳✷ ❊st❛❜✐❧✐❞❛❞❡ ❡ ✐♥st❛❜✐❧✐❞❛❞❡ ❞❡ ór❜✐t❛s ♣❡r✐ó❞✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷ ✺✳✷✳✸ P❡r♠❛♥ê♥❝✐❛ ❞❡ ór❜✐t❛s ♣❡r✐ó❞✐❝❛s s♦❜ ♣❡rt✉r❜❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✾

✻ ❙❡♠✐❣r✉♣♦s ▼♦rs❡✲❙♠❛❧❡ ✶✶✺

✻✳✶ ◆♦çõ❡s ❜ás✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✺ ✻✳✷ ❊st❛❜✐❧✐❞❛❞❡ ❡str✉t✉r❛❧ ❞❡ s❡♠✐❣r✉♣♦s ▼♦rs❡✲❙♠❛❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✵

①✐✈ ❙❯▼➪❘■❖

✻✳✸ ❊①❡♠♣❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✹

■♥tr♦❞✉çã♦

❯♠ s✐st❡♠❛ ❞✐♥â♠✐❝♦ é ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ♦♣❡r❛❞♦r❡s {S(t, s) :t ≥ s}✱ ❞❡✜♥✐❞❛ ❡♠ ✉♠

❡s♣❛ç♦ X ❡ t♦♠❛♥❞♦ ✈❛❧♦r❡s ♥❡❧❡ ♠❡s♠♦ ❞❡ ❢♦r♠❛ q✉❡✱ s❡ ♦ ✈❛❧♦r ❞❡ ✉♠❛ ❝❡rt❛ ✈❛r✐á✈❡❧

♥♦ ✐♥st❛♥t❡s ❢♦rx✱ ❡♥tã♦S(t, s)xr❡♣r❡s❡♥t❛ ♦ ✈❛❧♦r ❞❡ss❛ ✈❛r✐á✈❡❧ ♥✉♠ ✐♥st❛♥t❡ ♣♦st❡r✐♦r t✳ ❈♦♥❤❡❝❡♥❞♦ ♦ s✐st❡♠❛ ❞✐♥â♠✐❝♦✱ ♣♦❞❡♠♦s s❛❜❡r✱ ♥♦ ❢✉t✉r♦✱ ♦s ✈❛❧♦r❡s ❞❡ ✈❛r✐á✈❡✐s q✉❡

❝♦♥❤❡❝❡♠♦s ♥♦ ♣r❡s❡♥t❡✳ ❆ ✈❛r✐á✈❡❧ x ♣♦❞❡ r❡♣r❡s❡♥t❛r ❞✐✈❡rs♦s t✐♣♦s ❞❡ q✉❛♥t✐❞❛❞❡s✱

s❡❥❛♠ ❡❧❛s ❢ís✐❝❛s✱ ❜✐♦❧ó❣✐❝❛s ♦✉ ❡♠ q✉❛❧q✉❡r ♦✉tr❛ ár❡❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✳ P♦❞❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡r❡r r❡♣r❡s❡♥t❛r ❛ ♣♦s✐çã♦ ❞❡ ✉♠ ❝♦r♣♦ ♥♦ ❡s♣❛ç♦✱ ♦ ❝r❡s❝✐♠❡♥t♦ ❞❡ ✉♠❛ ♣♦♣✉❧❛çã♦ ♦✉ ❛ ❞✐str✐❜✉✐çã♦ ❞❛ t❡♠♣❡r❛t✉r❛ ❡♠ ✉♠ ❝♦r♣♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♦ ❡s♣❛ç♦ X ♣♦❞❡

s❡r ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ ♦✉ ✐♥✜♥✐t❛✳

P♦❞❡♠♦s ✉t✐❧✐③❛r ❛ t❡♦r✐❛ ❞♦s s✐st❡♠❛s ❞✐♥â♠✐❝♦s ♣❛r❛ r❡♣r❡s❡♥t❛r s✐st❡♠❛s ❢ís✐❝♦s ❡①✐st❡♥t❡s ♥♦ ♥♦ss♦ ♠✉♥❞♦✳ ◆♦ ❡♥t❛♥t♦✱ ❛♦ ❢❛③❡r♠♦s ♥♦ss♦ ♠♦❞❡❧♦✱ ❞❡✈❡♠♦s ♥♦s ❧❡♠❜r❛r q✉❡ ❡❧❡ é ✉♠❛ s✐♠♣❧✐✜❝❛çã♦ ❞♦ s✐st❡♠❛ r❡❛❧ ❡✱ ♣♦rt❛♥t♦✱ ♣♦❞❡rá ❝♦♥t❡r ❡rr♦s✳ P♦r ✐ss♦✱ ♦ ❡st✉❞♦ ❞❡ ♣❡rt✉r❜❛çõ❡s ❞♦s s✐st❡♠❛s ❞✐♥â♠✐❝♦s é ❡ss❡♥❝✐❛❧✳

❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛ ❛♥á❧✐s❡ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❛ss✐♥tót✐❝♦ ❞♦s s❡♠✐✲ ❣r✉♣♦s✳ P❛r❛ ✐ss♦✱ é ❢✉♥❞❛♠❡♥t❛❧ ❢♦❝❛r♠♦s ♥❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦s ❛tr❛t♦r❡s ❣❧♦❜❛✐s ❞❡st❡s s❡♠✐❣r✉♣♦s✳

◆❡st❡ tr❛❜❛❧❤♦✱ tr❛❜❛❧❤❛♠♦s ❛♣❡♥❛s ❝♦♠ s✐st❡♠❛s ❛✉tô♥♦♠♦s ❡ ♦s s❡♠✐❣r✉♣♦s s❡rã♦ ❞❡♥♦t❛❞♦s s♦❜ ❛ ❢♦r♠❛ {S(t) :t ≥0}✳

❈♦♠❡ç❛♠♦s ❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♥❞♦ ♣r♦♣r✐❡❞❛❞❡ ❜ás✐❝❛s ❞♦s s❡♠✐❣r✉♣♦s ❡ ❝♦♥❝❡✐t♦s ❝♦♠♦ ✐♥✈❛r✐â♥❝✐❛✱ ❛tr❛çã♦✱ ❛❜s♦rçã♦ ❡ s♦❧✉çõ❡s ❣❧♦❜❛✐s✳

P❛r❛ ♠❡❞✐r♠♦s ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s✱ ❞❡✜♥✐♠♦s ❛ s❡♠✐✲❞✐stâ♥❝✐❛ ❞❡ ❍❛✉s✲ ❞♦r✛ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s A ❡ B✿

❞✐stH(A, B) := sup x∈A

inf

y∈Bd(x, y)✳

P♦r ❡❧❛✱ ♦ ❝♦♥❥✉♥t♦ A ❡stá ✧♣ró①✐♠♦✧❞♦ ❝♦♥❥✉♥t♦ B s❡ t♦❞♦ ♣♦♥t♦ ❞❡ A ❡st✐✈❡r ✧♣ró✲

①✐♠♦✧❞❡ ❛❧❣✉♠ ♣♦♥t♦ ❞❡ B✳

❈♦♠ ❡ss❛s ♥♦çõ❡s✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ ✉♠ ❛tr❛t♦r ❣❧♦❜❛❧ A ♣❛r❛ ✉♠ s❡♠✐✲

❣r✉♣♦ {T(t) :t ≥0} ❝♦♠♦ s❡♥❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ ❡ ✐♥✈❛r✐❛♥t❡ t❛❧ q✉❡

lim

✷ ❙❯▼➪❘■❖

♣❛r❛ t♦❞♦ ❝♦♥❥✉♥t♦B ❧✐♠✐t❛❞♦✳

P♦❞❡♠♦s✱ ❛❣♦r❛✱ ❞❡s❡♥✈♦❧✈❡r ❛ ♣❛rt❡ ♣r✐♥❝✐♣❛❧ ❞❡st❡ tr❛❜❛❧❤♦✿ ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦s ❛tr❛✲ t♦r❡s ❞❡ s❡♠✐❣r✉♣♦s s♦❜ ✉♠❛ ♣❡rt✉r❜❛çã♦ ❛✉tô♥♦♠❛✳

❈♦♠❡ç❛♠♦s ♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❝♦♠ ♦ ❡st✉❞♦ ❞❛ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r ❞♦s ❛tr❛✲ t♦r❡s ❡ ♠♦str❛♠♦s q✉❡ s❡ {Tη(t) :t≥0}η∈[0,1] ❢♦r ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s❡♠✐❣r✉♣♦s ❝♦♥tí♥✉❛ ❡♠

η = 0 ❞❡ ❢♦r♠❛ q✉❡ ❝❛❞❛ {Tη(t) :t≥0} ♣♦ss✉✐ ❛tr❛t♦r ❣❧♦❜❛❧Aη✱ ❡♥tã♦

lim

η→0❞✐stH(Aη,A0) = 0✳

❋❡✐t♦ ✐ss♦✱ ❡st✉❞❛♠♦s ❛ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ ✐♥❢❡r✐♦r ❞♦s ❛tr❛t♦r❡s ❡ ♠♦str❛♠♦s q✉❡✱ s♦❜ ❝❡rt❛s ❝♦♥❞✐çõ❡s✱ ♣♦❞❡♠♦s ❣❛r❛♥t✐r q✉❡

lim

η→0❞✐stH(A0,Aη) = 0✳

➱ ✐♠♣♦rt❛♥t❡ ♥♦t❛r q✉❡ ❛ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r ❝♦st✉♠❛ s❡r ♦❜t✐❞❛ ❞❡ ♠❛♥❡✐r❛ ♠❛✐s s✐♠♣❧❡s q✉❡ ❛ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ ✐♥❢❡r✐♦r✳ ❉❡ss❛ ❢♦r♠❛✱ ♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s ❛♣r❡✲ s❡♥t❛♠ ❝♦♥❞✐çõ❡s ♣❛r❛ ♦❜t❡r♠♦s ❛ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ ✐♥❢❡r✐♦r✳

❖ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ é ❞❡❞✐❝❛❞♦ ❛ ✉♠ t✐♣♦ ✐♠♣♦rt❛♥t❡ ❞❡ s❡♠✐❣r✉♣♦✿ ♦s s❡♠✐❣r✉♣♦s ❣r❛❞✐❡♥t❡s✱ ♦✉ s❡❥❛✱ s❡♠✐❣r✉♣♦s q✉❡ ♣♦ss✉❡♠ ✉♠❛ ❢✉♥çã♦ ❞❡ ▲②❛♣✉♥♦✈ ❛ss♦❝✐❛❞❛✳

❊st✉❞❛♠♦s✱ ❛✐♥❞❛ ♥♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ♦s s❡♠✐❣r✉♣♦s ❞✐♥❛♠✐❝❛♠❡♥t❡ ❣r❛❞✐❡♥t❡s ✭s❡✲ ♠✐❣r✉♣♦s q✉❡ ♣♦ss✉❡♠ ❛s ♠❡s♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞✐♥â♠✐❝❛s ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❣r❛❞✐❡♥t❡✮✳ ❉❡✜♥✐♠♦s ✉♠ ♣❛r ❛tr❛t♦r✲r❡♣✉❧s♦r ❡ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ▼♦rs❡ ♣❛r❛ ❝♦♥str✉✐r♠♦s ✉♠❛ ❢✉♥✲ çã♦ ❞❡ ▲②❛♣✉♥♦✈ ❛ss♦❝✐❛❞❛ ❛ ✉♠ s❡♠✐❣r✉♣♦ ❞✐♥❛♠✐❝❛♠❡♥t❡ ❣r❛❞✐❡♥t❡✱ ♠♦str❛♥❞♦ ❛ ❡q✉✐✲ ✈❛❧ê♥❝✐❛ ❡♥tr❡ s❡♠✐❣r✉♣♦s ❣r❛❞✐❡♥t❡s ❡ ❞✐♥❛♠✐❝❛♠❡♥t❡ ❣r❛❞✐❡♥t❡s✳

❆♣ós ❝❛r❛❝t❡r✐③❛r♠♦s ♦s ❛tr❛t♦r❡s ❞❡st❡s s❡♠✐❣r✉♣♦s✱ ❜✉s❝❛♠♦s ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ✉♠❛ ♣❡rt✉r❜❛çã♦ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ❣r❛❞✐❡♥t❡ ❝♦♥t✐♥✉❡ s❡♥❞♦ ❣r❛❞✐❡♥t❡✳

❉❡✜♥✐♠♦s ❡ ❡st✉❞❛♠♦s✱ ♥♦ q✉❛rt♦ ❝❛♣ít✉❧♦✱ ❛s ✈❛r✐❡❞❛❞❡s ❡stá✈❡❧ ❡ ✐♥stá✈❡❧ ❞❡ ✉♠ ❡❧❡✲ ♠❡♥t♦ ❝rít✐❝♦s ✭✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❤✐♣❡r❜ó❧✐❝♦ ♦✉ ✉♠❛ ór❜✐t❛ ♣❡r✐ó❞✐❝❛ ♥♦r♠❛❧♠❡♥t❡ ❤✐♣❡r❜ó❧✐❝❛✮✳ P❛r❛ ✐ss♦✱ ✉t✐❧✐③❛♠♦s r❡s✉❧t❛❞♦s r❡❧❛❝✐♦♥❛❞♦s à ❞❡❝♦♠♣♦s✐çã♦ ❡s♣❡❝tr❛❧ ❞❡ ♦♣❡r❛❞♦r❡s ❧✐♥❡❛r❡s ✭♣❛r❛ ♠❛✐s ❞❡t❛❧❤❡s✱ ❝♦♥s✉❧t❛r ❬✻❪ ❡ ❬✽❪✮✳

❈♦♠❡ç❛♠♦s ♦ q✉✐♥t♦ ❝❛♣ít✉❧♦ ♣r♦✈❛♥❞♦✱ s♦❜ ❤✐♣ót❡s❡s ♥❛t✉r❛✐s✱ ❛s s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡s s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r ❞♦s ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❞❡ ❛♣❧✐❝❛çõ❡s✳

❋✐♥❛❧✐③❛♠♦s ❡st❡ ❝❛♣ít✉❧♦ ❝♦♠ ✉♠ ❡st✉❞♦ ❞❛ ❡st❛❜✐❧✐❞❛❞❡ ❛ss✐♥tót✐❝❛ ❞❡ s♦❧✉çõ❡s ♣❡✲ r✐ó❞✐❝❛s✳

P♦r ✜♠✱ ❡st✉❞❛♠♦s✱ ♥♦ s❡①t♦ ❝❛♣ít✉❧♦✱ ♦s s❡♠✐❣r✉♣♦s ▼♦rs❡✲❙♠❛❧❡✳ ❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ ✐♠♣♦rt❛♥t❡ ❞❡st❡s s❡♠✐❣r✉♣♦s é q✉❡ ❡❧❡s sã♦ ❞✐♥❛♠✐❝❛♠❡♥t❡ ❣r❛❞✐❡♥t❡s ❡ ❛ ❢❛♠í❧✐❛ ❞❡ ✐♥✈❛r✐❛♥t❡s ✐s♦❧❛❞♦s é ❢♦r♠❛❞❛ ❛♣❡♥❛s ♣♦r ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ ❡ ór❜✐t❛s ♣❡r✐ó❞✐❝❛s✳ ❉❡ss❛ ❢♦r♠❛✱ ♦s r❡s✉❧t❛❞♦s ❞♦ q✉✐♥t♦ ❝❛♣ít✉❧♦s sã♦ ❡ss❡♥❝✐❛✐s ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s s❡♠✐❣r✉♣♦s ▼♦rs❡✲❙♠❛❧❡✳

❈♦♥❝❧✉í♠♦s ❡st❡ tr❛❜❛❧❤♦ ❞❡♠♦♥str❛♥❞♦ ✉♠ r❡s✉❧t❛❞♦ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ♠❛✐s ❢♦rt❡ ❞♦ q✉❡ ♦ ♦❜t✐❞♦ ♣❛r❛ s❡♠✐❣r✉♣♦s ❣r❛❞✐❡♥t❡s✿ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ♦ ❞✐❛❣r❛♠❛ ❞❡ ❢❛s❡ ❞❡ ✉♠ s❡♠✐❣r✉♣♦ ▼♦rs❡✲❙♠❛❧❡ ❡ ♦ ❞♦ s❡♠✐❣r✉♣♦ ♣❡rt✉r❜❛❞♦✳

❈❛♣ít✉❧♦

1

❙❡♠✐❣r✉♣♦s ❡ ❆tr❛t♦r❡s ●❧♦❜❛✐s

◆❡st❡ ❝❛♣ít✉❧♦✱ ✐♥tr♦❞✉③✐♠♦s ♥♦çõ❡s ❜ás✐❝❛s q✉❡ s❡rã♦ ✉t✐❧✐③❛❞❛s ♥♦ r❡st❛♥t❡ ❞❡st❡ tr❛❜❛❧❤♦✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❞❡✜♥✐♠♦s ❡ ❡st✉❞❛♠♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ s❡♠✐❣r✉♣♦✱ s♦❧✉çõ❡s ❣❧♦❜❛✐s ❡ ❛tr❛t♦r❡s ❣❧♦❜❛✐s✳

✶✳✶ ◆♦çõ❡s ❡ ❢❛t♦s ❜ás✐❝♦s

❙❡❥❛X ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦ ❝♦♠ ❛ ♠étr✐❝❛d:X×X →R+_{✱ ♦♥❞❡R}+ _{:= [0,∞)❡ ❞❡♥♦✲}

t❛r❡♠♦s ♣♦r C(X) ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❞❡ X ❡♠ X✳ ❉❛❞♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ A ⊂X✱ ❞❡✜♥✐♠♦s ❛ǫ✲✈✐③✐♥❤❛♥ç❛ ❞❡ A ❡♠ X ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥❥✉♥t♦

Oǫ(A) ={x∈X :d(x, a)< ǫ ♣❛r❛ ❛❧❣✉♠ a∈A}

❈♦♠❡ç❛r❡♠♦s✱ ❛❣♦r❛✱ ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❡ s✐st❡♠❛s ❞✐♥â♠✐❝♦s ❛✉tô♥♦♠♦s ✭♦✉ s❡♠✐❣r✉✲ ♣♦s✮ ❡✱ ❞❡ ❛❣♦r❛ ❡♠ ❞✐❛♥t❡✱ t r❡♣r❡s❡♥t❛ ✉♠❛ ✈❛r✐á✈❡❧ r❡❛❧✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❯♠ s❡♠✐❣r✉♣♦ é ✉♠❛ ❢❛♠í❧✐❛ {T(t) :t≥0} ⊂ C(X) ❝♦♠ ❛s s❡❣✉✐♥t❡s

♣r♦♣r✐❡❞❛❞❡s✿

✭❛✮ T(0) =IX✱ ♦♥❞❡ IX é ♦ ♦♣❡r❛❞♦r ✐❞❡♥t✐❞❛❞❡ ❡♠ X✱

✭❜✮ T(t+s) = T(t)T(s)✱ ♣❛r❛ t♦❞♦ t, s≥0✱

✭❝✮ ❛ ❢✉♥çã♦ R+_{×X ∋(t, x)7→T(t)x∈X é ❝♦♥tí♥✉❛✳}

✹ ❙❡♠✐❣r✉♣♦s ❡ ❆tr❛t♦r❡s ●❧♦❜❛✐s

✐♥✈❛r✐❛♥t❡✱ ❞✐③❡♠♦s q✉❡ a∗ _{é ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ✉♠ ❡q✉✐❧í❜r✐♦}

♣❛r❛ {T(t) :t≥0}✳

❉✐③❡♠♦s q✉❡ A é ♣♦s✐t✐✈❛♠❡♥t❡ ✐♥✈❛r✐❛♥t❡ s❡ T(t)A ⊆ A✱ ♣❛r❛ t♦❞♦ t ≥ 0✳ P♦r

✜♠✱ ❞✐③❡♠♦s q✉❡ A é ♥❡❣❛t✐✈❛♠❡♥t❡ ✐♥✈❛r✐❛♥t❡ s❡ A ⊆T(t)A✱ ♣❛r❛ t♦❞♦ t≥0✳

❱❛♠♦s ❞❡✜♥✐r ❛s ♥♦çõ❡s ❞❡ ❛tr❛çã♦ ❡ ❛❜s♦rçã♦ ♣❡❧❛ ❛çã♦ ❞❡ ✉♠ s❡♠✐❣r✉♣♦✳ P❛r❛ ✐ss♦✱ ♣r❡❝✐s❛♠♦s ❞❛ s❡♠✐✲❞✐stâ♥❝✐❛ ❞❡ ❍❛✉s❞♦r✛ ❞✐stH(A, B) ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B ❞❡ X✱ ❞❡✜♥✐❞❛ ♣♦r✿

❞✐stH(A, B) := sup x∈A

inf

y∈Bd(x, y)✳

❉❡✜♥✐♠♦s✱ t❛♠❜é♠✱ ❛ ❞✐stâ♥❝✐❛ s✐♠étr✐❝❛ ❞❡ ❍❛✉s❞♦r✛ ❡♥tr❡ ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B

♣♦r✿

dH(A, B) = ❞✐stH(A, B) +❞✐stH(B, A)✳

❯t✐❧✐③❛♥❞♦ ❡ss❛ ❞✐stâ♥❝✐❛✱ ❞♦✐s ❝♦♥❥✉♥t♦s ❡stã♦ ♣❡rt♦ s❡ t♦❞♦ ♣♦♥t♦ ❞♦ ♣r✐♠❡✐r♦ ❝♦♥✲ ❥✉♥t♦ ❡stá ♣❡rt♦ ❞❡ ❛❧❣✉♠ ♣♦♥t♦ ❞♦ s❡❣✉♥❞♦ ❝♦♥❥✉♥t♦✳

P♦r ✜♠✱ ❞❡♥♦t❛♠♦s ♣♦rdist(A, B) ❛ ❞✐stâ♥❝✐❛ ✉s✉❛❧ ❡♥tr❡ ❝♦♥❥✉♥t♦s✿

❞✐st(A, B) := inf

x∈Ayinf∈Bd(x, y).

❖❜s❡r✈❛çã♦ ✶✳✶✳✸✳ ◆♦t❡♠♦s q✉❡ ❞✐stH(A, B) = 0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A⊂B✳ ❉❡ ❢❛t♦✱ s❡ A⊂B✱ ❡♥tã♦✱ ♣❡❧❛ ❞❡✜♥✐çã♦✱ é ✐♠❡❞✐❛t♦ q✉❡ ❞✐stH(A, B) = 0✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ✈❛♠♦s s✉♣♦r q✉❡ ❞✐stH(A, B) = 0✳ ❚❡♠♦s✱ ❡♥tã♦✱ q✉❡ infy∈Bd(x, y) =

0,∀x∈A✳ ❆ss✐♠✱ ✜①❛❞♦s x∈A ❡ n ∈N∗_{✱ ❡①✐st❡ y∈B t❛❧ q✉❡ d(x, y)<} 1 2n✳ ❙❡❥❛ a ∈ A✳ ❉❛❞♦ n ∈ N∗_{✱ ❡①✐st❡ a}

n ∈ A t❛❧ q✉❡ d(a, an)< ₂1_n✳ P❛r❛ ❝❛❞❛ an✱ ❡①✐st❡

bn∈B t❛❧ q✉❡ d(an, bn)< ₂1_n✳ ❉❡ss❛ ❢♦r♠❛✱ d(a, bn)≤d(a, an) +d(an, bn)< _n1✳ ❆ss✐♠✱ ❛ s❡q✉ê♥❝✐❛ {bn}n∈N⊂B ❝♦♥✈❡r❣❡ ♣❛r❛ a✳ ▲♦❣♦ a∈B✳

❉❡✜♥✐çã♦ ✶✳✶✳✹ ✭❆tr❛çã♦ ❡ ❛❜s♦rçã♦✮✳ ❙❡❥❛♠ A ❡ B s✉❜❝♦♥❥✉♥t♦s ❞❡ X✳

✭■✮ ❉✐③❡♠♦s q✉❡ A ❛tr❛✐ B ♣❡❧❛ ❛çã♦ ❞♦ s❡♠✐❣r✉♣♦ {T(t) :t≥0} s❡

lim

t→∞❞✐stH(T(t)B, A) = 0

✭■■✮ ❉✐③❡♠♦s q✉❡ A ❛❜s♦r✈❡ B s❡ ❡①✐st❡ t0 ≥ 0 t❛❧ q✉❡ T(t)B ⊂ A✱ ♣❛r❛ t♦❞♦ t ≥ t0✳

✶✳✶ ◆♦çõ❡s ❡ ❢❛t♦s ❜ás✐❝♦s ✺

❉❡✜♥✐çã♦ ✶✳✶✳✺ ✭❆tr❛t♦r ❣❧♦❜❛❧✮✳ ❯♠ ❝♦♥❥✉♥t♦ A é ❝❤❛♠❛❞♦ ❞❡ ❛tr❛t♦r ❣❧♦❜❛❧ ♣❛r❛ ♦

s❡♠✐❣r✉♣♦ {T(t) :t≥0} s❡ A ❢♦r ❝♦♠♣❛❝t♦✱ ✐♥✈❛r✐❛♥t❡ ❡ ❛tr❛✐ ♦s s✉❜❝♦♥❥✉♥t♦s ❧✐♠✐t❛❞♦s

❞❡ X ♣❡❧❛ ❛çã♦ ❞❡ {T(t) :t≥0}✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✻✳ ❙❡ ✉♠ s❡♠✐❣r✉♣♦ {T(t) : t ≥ 0} ♣♦ss✉✐ ❛tr❛t♦r ❣❧♦❜❛❧✱ ❡♥tã♦ ❡st❡

❛tr❛t♦r ❣❧♦❜❛❧ é ú♥✐❝♦✳

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠ A ❡ Aˆ ❛tr❛t♦r❡s ❣❧♦❜❛✐s ♣❛r❛ ♦ s❡♠✐❣r✉♣♦ {T(t) : t ≥ 0}✳ P❡❧❛

✐♥✈❛r✐â♥❝✐❛ ❞❡ A✱ t❡♠♦s q✉❡ T(t)A =A ♣❛r❛ t♦❞♦ t ≥0✳ ❆ss✐♠✱ ❝♦♠♦ Aˆ❛tr❛✐ t♦❞♦s ♦s

❧✐♠✐t❛❞♦s✱ _Aˆ_{❛tr❛✐ A ❡}

0 = lim

t→∞❞✐stH(T(t)A, ˆ

A) = lim

t→∞❞✐stH(A, ˆ

A) =❞✐stH(A,Aˆ)

❉❡st❛ ❢♦r♠❛✱ A =A ⊂Aˆ= ˆA✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ A ⊂ Aˆ ✱ ♠♦str❛♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡

♦s ❞♦✐s ❛tr❛t♦r❡s✳

❉❡✜♥✐çã♦ ✶✳✶✳✼✳ ❯♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❞❡ {T(t) : t ≥ 0} ♣♦r x ∈ X é ✉♠❛ ❢✉♥çã♦

❝♦♥tí♥✉❛ φ :R→X t❛❧ q✉❡ φ(0) =x ❡ T(t)(φ(s)) =φ(t+s)✱ ♣❛r❛ t♦❞♦ s∈R ❡ t≥0✳

❯♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❝♦♥st❛♥t❡ s❡rá ❝❤❛♠❛❞❛ ❞❡ s♦❧✉çã♦ ❡st❛❝✐♦♥ár✐❛ ❡ é ❢á❝✐❧ ✈❡r q✉❡ s❡✉ ✈❛❧♦r é ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✳

❈♦♠♦ T(t) ♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✐♥❥❡t✐✈❛✱ ♣♦❞❡♠ ❡①✐st✐r ❞✐❢❡r❡♥t❡s s♦❧✉çõ❡s ❣❧♦❜❛✐s

♣♦r x✳

◗✉❛♥❞♦ ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧φ:R→X ♣♦r x ❡①✐st❡✱ ❞❡✜♥✐♠♦s ❛ ór❜✐t❛ ❞❡x r❡❧❛t✐✈❛

à s♦❧✉çã♦ ❣❧♦❜❛❧ φ ♣♦r γφ(x) :={φ(t) :t ∈R}✳

❚❛♠❜é♠ ❡s❝r❡✈❡♠♦s (γφ)−t (x) :={φ(s) :s≤ t} ❡ ❞❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦ α✲❧✐♠✐t❡ ❞❡

x ❝♦♠ r❡❧❛çã♦ ❛ φ ❝♦♠♦

αφ(x) = \

t∈R−

(γφ)−t (x)

▲❡♠❛ ✶✳✶✳✽✳ ❙❡❥❛{T(t) :t≥0}✉♠ s❡♠✐❣r✉♣♦ ❝♦♠ ❛tr❛t♦r ❣❧♦❜❛❧A✳ ❊♥tã♦✱ ❞❛❞♦x∈ A✱

❡①✐st❡ ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❧✐♠✐t❛❞❛ φx :R→ A t❛❧ q✉❡ φx(0) =x✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡

✻ ❙❡♠✐❣r✉♣♦s ❡ ❆tr❛t♦r❡s ●❧♦❜❛✐s

❉❡♠♦♥str❛çã♦✳ P❡❧❛ ✐♥✈❛r✐â♥❝✐❛ ❞❡ A✱ t❡♠♦s q✉❡✱ ♣❛r❛ ❝❛❞❛ x ∈ A✱ R+ _{∋ t 7→ φ(t) :=}

T(t)x é s❡♠♣r❡ ❜❡♠ ❞❡✜♥✐❞❛ ❡ r❡❧❛t✐✈❛♠❡♥t❡ ❝♦♠♣❛❝t❛✳

❈♦♠♦ A =T(1)A✱ ♣❛r❛ ❝❛❞❛ x∈ A✱ ❡①✐st❡ x−1 ∈ A t❛❧ q✉❡ T(1)x−1 =x✳ P♦❞❡♠♦s✱

❡♥tã♦✱ ❝♦♥str✉✐r ✉♠❛ s❡q✉ê♥❝✐❛ {x−n : n ∈ N} ⊂ A t❛❧ q✉❡ x0 = x ❡ T(1)x−n−1 = x−n✱ ♣❛r❛ t♦❞♦ n∈N✳ ◆♦t❡♠♦s q✉❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❝♦♠ ❡ss❛s ♣r♦♣r✐❡❞❛❞❡s ♣♦❞❡ ♥ã♦ s❡r ú♥✐❝❛✳

❙❡❥❛♠ j, n∈N✱ ❝♦♠j ≤n✳ ❊♥tã♦

T(j)x−n =T(j−1)T(1)x−n=T(j −1)x−n+1

=T(j−2)T(1)x−n+1 =T(j−2)x−n+2 =...=T(j −j)x−n+j =x−n+j

❉❡✜♥❛♠♦s

φx(t) =  



T(t)x t ≥0

T(j+t)x−j t∈[−j,−j+ 1), j = 1,2,3, ...

➱ ❝❧❛r♦ q✉❡ φx(0) =x ❡ q✉❡ φx :R→ A é ❧✐♠✐t❛❞❛✱ ❥á q✉❡ ❡stá ❝♦♥t✐❞❛ ❡♠A✳ ▼♦str❡♠♦s q✉❡ ❡❧❛ é ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧✳ P❛r❛ ✐ss♦✱ t♦♠❡♠♦st≥0✳

✶✳ ❙❡❥❛ s ≥0✳ ❊♥tã♦ T(t)(φx(s)) =T(t)T(s)x=T(t+s)x=φx(t+s)

✷✳ ❙❡❥❛ s <0 ❡ t+s≥0✳ ❊♥tã♦ ❡①✐st❡j ∈N t❛❧ q✉❡ s∈[−j,−j+ 1)✳ ❆ss✐♠✱

T(t)(φx(s)) = T(t)T(j +s)x−j = T(t+j +s)x−j = T(t+s)T(j)x−j = T(t+s)x =

φx(t+s)

✸✳ ❙❡❥❛ s < 0 ❡ t+s < 0✳ ❊♥tã♦ ❡①✐st❡♠ j, n ∈ N t❛✐s q✉❡ s ∈ [−j,−j + 1) ❡ t+s ∈

[−n,−n+ 1)✳

❆ss✐♠✱ T(t)(φx(s)) =T(t)T(j+s)x−j =T(t+j+s)x−j =T([t+n+s] + [j−n])x−j =

T(t+n+s)T(j−n)x−j =T([t+s+n)x−n =φx(t+s)

❆ss✐♠✱ φx :R→ A é ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❧✐♠✐t❛❞❛ ♣♦r x ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱

✶✳✷ ❊①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r❡s ❣❧♦❜❛✐s ✼

P❛r❛ ❛ ♦✉tr❛ ✐♥❝❧✉sã♦✱ ❞❛❞❛ ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❧✐♠✐t❛❞❛φ :R→X ♣❛r❛ ♦ s❡♠✐❣r✉♣♦ {T(t) :t≥0}✱ ❞♦ ❢❛t♦ q✉❡φ(R)é ✉♠ ❝♦♥❥✉♥t♦ ✐♥✈❛r✐❛♥t❡ ❧✐♠✐t❛❞♦ ❡ q✉❡A❛tr❛✐ ❝♦♥❥✉♥t♦s

❧✐♠✐t❛❞♦s✱ t❡♠♦s q✉❡

0 = lim

t→∞❞✐stH(T(t)φ(R),A) =❞✐stH(φ(R),A).

❙❡♥❞♦ A ❢❡❝❤❛❞♦✱ φ(R)⊂ A✳ ❉✐ss♦✱

{x∈X :❡①✐st❡ ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❧✐♠✐t❛❞❛ ♣♦r x} ⊂ A.

❡ ♦ ❧❡♠❛ ❡stá ♣r♦✈❛❞♦✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✾✳ P❡❧❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞❛ ❞❡♠♦♥str❛çã♦ ❛❝✐♠❛✱ ♣♦❞❡♠♦s ✈❡r q✉❡ s❡ A é

✉♠ ❝♦♥❥✉♥t♦ ✐♥✈❛r✐❛♥t❡ ♣❛r❛ ♦ s❡♠✐❣r✉♣♦ {T(t) :t ≥0}✱ ❡♥tã♦ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ φ :R→A ♣♦r x✱ ♣❛r❛ t♦❞♦ x∈A✳

✶✳✷ ❊①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r❡s ❣❧♦❜❛✐s

◆❡st❛ s❡çã♦✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r r❡s✉❧t❛❞♦s q✉❡ ❝❛r❛❝t❡r✐③❛♠ s❡♠✐❣r✉♣♦s q✉❡ ♣♦ss✉❡♠ ✉♠ ❛tr❛t♦r ❣❧♦❜❛❧✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❖ ❝♦♥❥✉♥t♦ ω✲❧✐♠✐t❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ B ⊂X é ❞❡✜♥✐❞♦ ♣♦r

ω(B) = \

t∈R+

γ_t+(B)✱

♦♥❞❡

γ_t+(B) := [

s≥t

T(s)B✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✷✳ ❙❡❥❛ B ⊂X✳ ❊♥tã♦ ω(B) é ❢❡❝❤❛❞♦ ❡

ω(B) ={y∈X : ❡①✐st❡♠ s❡q✉ê♥❝✐❛s {tn}n∈N ❡♠ R+ ❡ {xn}n∈N❡♠ B

✽ ❙❡♠✐❣r✉♣♦s ❡ ❆tr❛t♦r❡s ●❧♦❜❛✐s

❙❡ φ:R→X é ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ♣♦r x∈X✱ ❡♥tã♦ αφ(x) é ❢❡❝❤❛❞♦ ❡

αφ(x) ={v ∈X :❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ {tn}n∈N ❡♠ R+ t❛❧ q✉❡ tn n→∞ −−−→ ∞

❡ φ(−tn) n→∞ −−−→v}

❉❡♠♦♥str❛çã♦✳ ❖ ❝♦♥❥✉♥t♦ ω(B) é ❢❡❝❤❛❞♦ ♣♦✐s é ❛ ✐♥t❡rs❡çã♦ ❞❡ ❝♦♥❥✉♥t♦s ❢❡❝❤❛❞♦s✳

❙❡y ∈ω(B)✱ ❡♥tã♦y∈T

t∈R+γ_t+(B)❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱y∈γ_t+(B)✱ ♣❛r❛ t♦❞♦t≥0✳

❆ss✐♠✱ ♣❛r❛ ❝❛❞❛ n∈N✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ {yn

k}k∈N⊂γ_n+(B)t❛❧ q✉❡ y_kn−−−→k→∞ y✳

❈♦♠♦ yn

k ∈ γn+(B) ♣❛r❛ t♦❞♦s n, k ∈N✱ ❡①✐st❡♠ {xnk}n,k∈N ⊂B ❡ {qkn}n,k∈N ⊂R+ t❛✐s

q✉❡

y_kn=T(n+q_kn)xn_k.

❚❛♠❜é♠✱ ❞❛❞♦sn ∈N❡ ǫ >0✱ ❡①✐st❡ k(n, ǫ)∈N t❛❧ q✉❡

d(y_kn, y)< ǫ✱ s❡♠♣r❡ q✉❡ k≥k(n, ǫ),

♦✉ s❡❥❛✱ d(T(n+qn

k)xnk, y)< ǫ s❡ k≥k(n, ǫ)✳ ❉❡✜♥❛♠♦s tn:=n+qn_k(n,1

n) ❡

xn :=xn_k(n,1

n) ❚❡♠♦s q✉❡

d(T(tn)xn, y)<

1

n

n→∞ −−−→0

❆ss✐♠✱ s❡ y ∈ ω(B)✱ y = limn→∞T(tn)xn✱ ♦♥❞❡ tn n→∞

−−−→ ∞ ❡ xn ∈ B✱ ♣❛r❛ t♦❞♦

n ∈N✳

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❥❛♠ y ∈ X ❡ {tn}n∈N ⊂ R+✱ {xn}n∈N ⊂ B s❡q✉ê♥❝✐❛s s❛t✐s❢❛③❡♥❞♦ tn

n→∞

−−−→ ∞ ❡ y = limn→∞T(tn)xn✳ ❊♥tã♦✱ ✜①❛♥❞♦ τ ∈ R+✱ t❡♠♦s q✉❡ {T(tn)xn}tn≥τ ⊂

γ+

τ(B) ❡ y ∈ γ+τ(B)✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ y ∈ ω(B) ❡ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❞❡ ω(B) ❡stá ♣r♦✈❛❞❛✳

❖s r❡s✉❧t❛❞♦s ♣❛r❛αφ(x)sã♦ ❛♥á❧♦❣♦s✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✸✳ ❙❡❥❛♠K ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ ❝♦♥t✐❞♦ ❡♠X ❡{xn}n∈N✉♠❛ s❡q✉ê♥✲

✶✳✷ ❊①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r❡s ❣❧♦❜❛✐s ✾

lim

n→∞d(xn, K) = 0

❊♥tã♦ {xn}n∈N t❡♠ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ ❛❧❣✉♠ ♣♦♥t♦ ❞❡ K✳

❆❧é♠ ❞✐ss♦✱ ❞❛❞♦ ✉♠ s❡♠✐❣r✉♣♦{T(t) :t≥0} ❡K ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦✱ s❡ K ❛tr❛✐

♦✉tr♦ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ K1✱ ❡♥tã♦ γ0+(K1) é r❡❧❛t✐✈❛♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡ ∅ 6=ω(K1)⊂K✳

❉❡♠♦♥str❛çã♦✳ ❉❛❞♦ m ∈ N✱ ❡①✐st❡♠ nm ∈ N ❡ ynm ∈ K✱ t❛✐s q✉❡ d(xnm, ynm) < 1

m✳ ❙❡♥❞♦ K ❝♦♠♣❛❝t♦ ❡ t♦♠❛♥❞♦ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ s❡ ♥❡❝❡ssár✐♦✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ ynm

m→∞

−−−→y0✱ ♣❛r❛ ❛❧❣✉♠ y0 ∈K✳

❉❡ss❛ ❢♦r♠❛✱ ♦❜t❡♠♦s

d(xnm, y0)≤d(xnm, ynm) +d(ynm, y0)

m→∞ −−−→0

■ss♦ ♠♦str❛ q✉❡ {xn}n∈N ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ ✉♠ ♣♦♥t♦ ❞❡K✱

♣r♦✈❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞❛ ♣r♦♣♦s✐çã♦✳

Pr♦✈❡♠♦s ❛ s❡❣✉♥❞❛ ♣❛rt❡✳ ❉❛❞♦ ǫ >0✱ ❡①✐st❡ t0 ∈R+ t❛❧ q✉❡

T(t)K1 ⊂ Oǫ/2(K),∀t ≥t0

❙❡♥❞♦K ❝♦♠♣❛❝t♦✱ ❡①✐st❡♠ ♣♦♥t♦s y1, ..., yp ∈K t❛✐s q✉❡K ❡stá ♥❛ r❡✉♥✐ã♦ ❞❡ ❜♦❧❛s ❞❡ r❛✐♦ ǫ/2 ❡ ❝❡♥tr♦ yj✱ 1≤ j ≤p✳ ❈♦♠♦ ♣❛r❛ ❝❛❞❛ x∈T(t)K1 ❡①✐st❡ ✉♠ y∈K t❛❧ q✉❡

d(x, y)< ǫ₂✱ ❡♥tã♦ ❡①✐st❡j ∈N t❛❧ q✉❡ d(x, yj)≤d(x, y) +d(y, yj)< ǫ₂ +₂ǫ =ǫ✳

❉❡ss❛ ❢♦r♠❛✱ s❡t≥t0✱T(t)K1 ❡stá ❝♦♥t✐❞♦ ♥❛ r❡✉♥✐ã♦ ✜♥✐t❛ ❞❡ ❜♦❧❛s ❞❡ r❛✐♦ǫ✳ ❈♦♥s❡✲

q✉❡♥t❡♠❡♥t❡✱ S

t≥t0T(t)K1 ❡stá ♥❡st❛ ♠❡s♠❛ r❡✉♥✐ã♦ ✜♥✐t❛ ❞❡ ❜♦❧❛s ❡✱ ❧♦❣♦✱ é t♦t❛❧♠❡♥t❡

❧✐♠✐t❛❞♦✳ ❙❡♥❞♦ S

0≤t≤t0T(t)K1 ❝♦♠♣❛❝t♦✱ s❡❣✉❡ q✉❡ γ

+_{(K1)∪K é t♦t❛❧♠❡♥t❡ ❧✐♠✐t❛❞♦✳}

❆ss✐♠✱ γ+_{(K1)∪K é ❝♦♠♣❛❝t♦✳ ❉✐ss♦✱ s❡❣✉❡ q✉❡ γ}+_{(K1)é r❡❧❛t✐✈❛♠❡♥t❡ ❝♦♠♣❛❝t♦✳}

❋✐♥❛❧♠❡♥t❡✱ γ_t+(K1) é ♥ã♦✲✈❛③✐♦ ❡ ❝♦♠♣❛❝t♦ ♣❛r❛ ♣❛r❛ ❝❛❞❛ t ∈ (R)+ _{❡ γ}+

t (K1) ⊂

γ+

s(K1)s❡s≤t✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ ❛ ❢❛♠í❧✐❛{γt+(K1)}t∈R+ t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ✐♥t❡rs❡çã♦

✶✵ ❙❡♠✐❣r✉♣♦s ❡ ❆tr❛t♦r❡s ●❧♦❜❛✐s

ω(K1) = \

t∈R+

γt+(K1)6=∅.

❉❛❞♦y ∈ω(K1) ❡ǫ >0✱ ❡①✐st❡ t0 ∈R+ t❛❧ q✉❡

y ∈γt+0(K1)⊂ Oǫ(K).

❊♥tã♦✱ d(y, K)≤ǫ ❡✱ s❡♥❞♦ǫ ❛r❜✐trár✐♦✱ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡✳

▲❡♠❛ ✶✳✷✳✹✳ ❙❡❥❛ {T(t) : t ≥ 0} ✉♠ s❡♠✐❣r✉♣♦ ❡♠ X✳ ❙❡ B ⊂ X✱ ❡♥tã♦ T(t)ω(B) ⊂ ω(B)♣❛r❛ t♦❞♦t ≥0✳ ❙❡B é t❛❧ q✉❡ω(B)é ❝♦♠♣❛❝t♦ ❡ ❛tr❛✐B✱ ❡♥tã♦ω(B)é ✐♥✈❛r✐❛♥t❡✳

❉❡♠♦♥str❛çã♦✳ ❙❡ ω(B) = ∅✱ ♥ã♦ ❤á ♥❛❞❛ ♣❛r❛ ♣r♦✈❛r✳ ❱❛♠♦s✱ ❡♥tã♦✱ ❛ss✉♠✐r q✉❡ ω(B) 6= ∅✳ ❋✐①❡♠♦s t ∈ R+_{✳ ❉❛ Pr♦♣♦s✐çã♦ ✭✶✳✷✳✷✮✱ s❡ y ∈ ω(B)✱ ❡①✐st❡♠ s❡q✉ê♥❝✐❛s}

{tn}n∈N ⊂R+ ❡ {xn}n∈N ⊂B t❛✐s q✉❡ y= limn→∞T(tn)xn✳

P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡T(t)✱ t❡♠♦s q✉❡T(t)y= limn→∞T(t+tn)xn❡ q✉❡ T(t)y ∈ω(B)✳ ▲♦❣♦✱ T(t)ω(B)⊂ω(B)✳

❱❛♠♦s s✉♣♦r q✉❡ ω(B) s❡❥❛ ❝♦♠♣❛❝t♦ ❡ ❛tr❛✐❛ B✳ ❙❡❥❛ x ∈ ω(B)✳ ❊♥tã♦ ❡①✐st❡♠

s❡q✉ê♥❝✐❛s {tn}n∈N ⊂ R+❡ {xn}n∈N ⊂ B✱ t❛✐s q✉❡ tn n→∞

−−−→ ∞ ❡ T(tn)xn n→∞

−−−→ x✳ P❛r❛

❝❛❞❛ t ∈ R+ _{✜①❛❞♦✱ ❡①✐st❡ n}

0 ∈ N t❛❧ q✉❡ tn > t✱ s❡ n ≥ n0✳ ❆ss✐♠✱ T(t)T(tn−t)xn =

T(tn)xn n→∞ −−−→x✳

❈♦♠♦ ω(B) é ❝♦♠♣❛❝t♦ ❡ ❛tr❛✐ B✱ t❡♠♦s q✉❡ d(T(tn − t)xn, ω(B)) n→∞

−−−→ 0✳ ❉❛

Pr♦♣♦s✐çã♦ ✭✶✳✷✳✸✮✱ ♦❜t❡♠♦s q✉❡ {T(tn−t)xn}n∈N ♣♦ss✉✐ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✱

q✉❡ s❡rá ❞❡♥♦t❛❞❛ ❞❛ ♠❡s♠❛ ❢♦r♠❛✳

❙❡ T(tn −t)xn → y✱ t❡♠♦s q✉❡ y ∈ ω(B) ❡ q✉❡ T(t)y = x✱ ♦✉ s❡❥❛✱ q✉❡ ω(B) ⊂

T(t)ω(B)✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳

▲❡♠❛ ✶✳✷✳✺✳ ❙❡❥❛ x ∈ X ❡ s✉♣♦♥❤❛ q✉❡ ❡①✐st❛ ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ φ : R→ X ♣♦r x t❛❧

q✉❡ φ(R−₎ s❡❥❛ ❝♦♠♣❛❝t♦✳ ❊♥tã♦ _α

φ(x) é ♥ã♦✲✈❛③✐♦✱ ❝♦♠♣❛❝t♦ ❡ ✐♥✈❛r✐❛♥t❡✳

✶✳✷ ❊①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r❡s ❣❧♦❜❛✐s ✶✶

❙❡❥❛ t ∈ R+_{✳ P❡❧❛ Pr♦♣♦s✐çã♦ ✭✶✳✷✳✷✮✱ ❞❛❞♦ y ∈ α}

φ(X)✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s r❡❛✐s {tn}n∈N✱ t❛❧ q✉❡tn

n→∞

−−−→ ∞ ❡ φ(−tn) n→∞ −−−→y✳

P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ T(t)✱ t❡♠♦s q✉❡ T(t)φ(−tn) = φ(t −tn) n→∞

−−−→ T(t)y ❡ q✉❡

T(t)y∈αφ(x)✱ ♦✉ s❡❥❛✱ T(t)αφ(x)⊂αφ(x)✳

P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ w ∈ αφ(x)✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ {tn}n∈N✱ t❛❧ q✉❡ tn n→∞ −−−→ ∞ ❡

φ(−tn) n→∞

−−−→w✳ ❈♦♠♦{φ(−tn−t) :n∈N}é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦✱ ❡❧❡ é r❡❧❛t✐✈❛♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡✱ t♦♠❛♥❞♦ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ s❡ ♥❡❝❡ssár✐♦✱ ❛ss✉♠✐r q✉❡ ❡①✐st❡ z ∈X t❛❧ q✉❡ φ(−tn−t)

n→∞

−−−→z ❡z ∈αφ(x)✳ P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ T(t)✱ ♦❜t❡♠♦s

q✉❡ T(t)z =w ❡ αφ(x)⊂T(t)αφ(x)✱ ♣r♦✈❛♥❞♦ ❛ ✐♥✈❛r✐â♥❝✐❛ ❞❡αφ(x)✳

▲❡♠❛ ✶✳✷✳✻✳ ❙❡❥❛ {T(t) :t≥0} ✉♠ s❡♠✐❣r✉♣♦ ❡♠X ❡ B ⊂X t❛❧ q✉❡ ω(B)é ❝♦♠♣❛❝t♦

❡ ❛tr❛✐ B✳ ❙❡ B ❢♦r ❝♦♥❡①♦✱ ❡♥tã♦ ω(B) s❡rá t❛♠❜é♠ ❝♦♥❡①♦✳

❉❡♠♦♥str❛çã♦✳ ❆ ❛♣❧✐❝❛çã♦ [0,∞)×X ∋(s, x)7→T(s)x∈X é ❝♦♥tí♥✉❛ ❡ ❛ ✐♠❛❣❡♠ ❞♦

❝♦♥❥✉♥t♦ ❝♦♥❡①♦ [t,∞)×B éγt+(B)✱ q✉❡ t❛♠❜é♠ s❡rá ❝♦♥❡①♦ ♣❛r❛ t≥0✳ ❖ r❡s✉❧t❛❞♦ ❡♥tã♦ s❡❣✉❡✱ ❧❡♠❜r❛♥❞♦ q✉❡ ω(B) = T

t≥0γ +

t (B)✳

▲❡♠❛ ✶✳✷✳✼✳ ❙❡❥❛ B ✉♠ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞❡ X t❛❧ q✉❡ γ+

t0(B) é ❝♦♠♣❛❝t♦ ♣❛r❛

❛❧❣✉♠ t0 ≥0✳ ❊♥tã♦ ω(B) é ♥ã♦✲✈❛③✐♦✱ ❝♦♠♣❛❝t♦✱ ✐♥✈❛r✐❛♥t❡ ❡ ❛tr❛✐ B✳

❉❡♠♦♥str❛çã♦✳ P❛r❛ ❝❛❞❛ t ≥ t0✱ γt+(B) é ♥ã♦✲✈❛③✐♦ ❡ ❝♦♠♣❛❝t♦✳ ❈♦♠♦ ❛ ❢❛♠í❧✐❛

{γ+

t (B) : t ≥ t0} t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❛ ✐♥t❡rs❡çã♦ ✜♥✐t❛✱ ω(B) = Tt≥t0γ +

t (B) é ♥ã♦✲ ✈❛③✐♦ ❡ ❝♦♠♣❛❝t♦✳

❱❛♠♦s ♣r♦✈❛r ❛❣♦r❛ ❛ ❛tr❛çã♦✳ ❱❛♠♦s s✉♣♦r✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡ ω(B) ♥ã♦ ❛tr❛✐❛ B✳

❊♥tã♦ ❡①✐st❡♠ ǫ0 >0 ❡ s❡q✉ê♥❝✐❛s {xn}n∈N ❡♠ X ❡ {tn}n∈N ❡♠ R+ t❛✐s q✉❡ tn n→∞ −−−→ ∞

❡ d(T(tn)xn, ω(B))> ǫ0 ♣❛r❛ t♦❞♦n∈N✳

❈♦♠♦ γ_t+₀(B) é ❝♦♠♣❛❝t♦ ❡ {T(tn)xn, n ≥ n1} ⊂ γt+0(B) ♣❛r❛ ❛❧❣✉♠ n1 ∈ N✱ ❡①✐st❡♠

s✉❜s❡q✉ê♥❝✐❛s✱ ✐♥❞❡①❛❞❛s ♣♦r {nj}j∈N✱ t❛✐s q✉❡ tnj

j→∞

−−−→ ∞ ❡ xnj

j→∞

−−−→ y✱ ♣❛r❛ ❛❧❣✉♠

y ∈X✳

❉✐ss♦✱ s❡❣✉❡ q✉❡y ∈ω(B) ❡ d(y, ω(B))≥ǫ0✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ ❆ss✐♠✱ ω(B) ❛tr❛✐

✶✷ ❙❡♠✐❣r✉♣♦s ❡ ❆tr❛t♦r❡s ●❧♦❜❛✐s

❈♦♠♦ ω(B) é ❝♦♠♣❛❝t♦ ❡ ❛tr❛✐ B✱ ♦ ▲❡♠❛ ✭✶✳✷✳✹✮ ❣❛r❛♥t❡ ❛ ✐♥✈❛r✐â♥❝✐❛ ❞❡ ω(B)✳

❉❡✜♥✐çã♦ ✶✳✷✳✽✳ ❯♠ s❡♠✐❣r✉♣♦ {T(t) :t≥0}é ❝❤❛♠❛❞♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❝♦♠♣❛❝t♦

q✉❛♥❞♦✱ ♣❛r❛ ❝❛❞❛ ❝♦♥❥✉♥t♦ B ⊂ X ♥ã♦✲✈❛③✐♦✱ ❢❡❝❤❛❞♦ ❡ ❧✐♠✐t❛❞♦ t❛❧ q✉❡ T(t)B ⊂ B✱

♣❛r❛ t♦❞♦ t ≥0✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ J ⊂B q✉❡ ❛tr❛✐ B✳

❉❡✜♥✐çã♦ ✶✳✷✳✾✳ ❯♠ s❡♠✐❣r✉♣♦ {T(t) : t ≥ 0} é ❝❤❛♠❛❞♦ ❡✈❡♥t✉❛❧♠❡♥t❡ ❧✐♠✐t❛❞♦

q✉❛♥❞♦✱ ♣❛r❛ ❝❛❞❛ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦ B ⊂X✱ ❡①✐st❡ t0 ≥0 t❛❧ q✉❡ γt+0(B) é ❧✐♠✐t❛❞♦✳

▲❡♠❛ ✶✳✷✳✶✵✳ ❙❡ {T(t) : t ≥ 0} é ✉♠ s❡♠✐❣r✉♣♦ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❝♦♠♣❛❝t♦ ❡ B é ✉♠

s✉❜❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦ ❞❡ X t❛❧ q✉❡ γ_t+₀(B) é ❧✐♠✐t❛❞♦ ♣❛r❛ ❛❧❣✉♠ t0 ≥ 0✱ ❡♥tã♦ ω(B) é

♥ã♦✲✈❛③✐♦✱ ❝♦♠♣❛❝t♦✱ ✐♥✈❛r✐❛♥t❡ ❡ ❛tr❛✐ B✳

❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ T(t) é ❝♦♥tí♥✉♦ ❡ T(t)γ_t+₀(B) ⊂ γ_t+₀(B)✱ t❡♠♦s q✉❡ T(t)γ_t+₀(B) ⊂ γt+0(B)

❈♦♠♦ ♦ s❡♠✐❣r✉♣♦ é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❝♦♠♣❛❝t♦✱ t❡♠♦s q✉❡ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠✲ ♣❛❝t♦ J ⊂γ_t+₀(B) q✉❡ ❛tr❛✐ γ+_t0(B)✳ ▲♦❣♦✱ ❡①✐st❡♠ s❡q✉ê♥❝✐❛s {ǫn}n∈N ❡ {tn}n∈N t❛✐s q✉❡ ǫn

n→∞ −−−→ 0✱ tn

n→∞

−−−→ ∞ ❡ T(t)(γt+0(B))⊂ Oǫn(J)✱ ♣❛r❛ t♦❞♦ t ≥tn✳ ▲♦❣♦✱ ∅ 6=ω(B)⊂ J✳

❈♦♠♦ ω(B) é ❢❡❝❤❛❞♦ ❡J é ❝♦♠♣❛❝t♦✱ t❡♠♦s q✉❡ ω(B) é ❝♦♠♣❛❝t♦✳

▼♦str❡♠♦s q✉❡ ω(B) ❛tr❛✐ B✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ✐st♦ ♥ã♦ ♦❝♦rr❛✳ ❊♥tã♦ ❡①✐st❡♠ ǫ0 ❡

s❡q✉ê♥❝✐❛s{xn}n∈(N) ❡♠X❡{tn}n∈(N) ❡♠Rt❛✐s q✉❡tn −−−→ ∞n→∞ ❡d(T(tn)xn, ω(B))> ǫ0✳

❉❛ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡ J ❡ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✭✶✳✷✳✸✮✱ ❡①✐st❡♠ s✉❜s❡q✉ê♥❝✐❛s ❞❡♥♦t❛❞❛s ♣♦r {nj}j∈N⊂N ❡ z∈J t❛✐s q✉❡ tnj

j→∞

−−−→ ∞ ❡ T(tnj)xnj

j→∞ −−−→z

❉❡st❛ ❢♦r♠❛✱ z ∈ ω(B) ❡ d(z, ω(B)) ≥ ǫ0✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✱ ♣r♦✈❛♥❞♦ q✉❡ ω(B)

❛tr❛✐ B✳

▲♦❣♦✱ ω(B) é ♥ã♦ ✈❛③✐♦✱ ❝♦♠♣❛❝t♦ ❡ ❛tr❛✐ B✳ ❆ ✐♥✈❛r✐â♥❝✐❛ ❞❡ ω(B) s❡❣✉❡ ❞♦ ▲❡♠❛

✭✶✳✷✳✹✮✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✶✶✳ ❙❡❥❛ {T(t) : t ≥0} ✉♠ s❡♠✐❣r✉♣♦ t❛❧ q✉❡ {T(tn)xn}n∈N é r❡❧❛t✐✈❛✲

♠❡♥t❡ ❝♦♠♣❛❝t♦ s❡♠♣r❡ q✉❡ {T(tn)xn}n∈N ❡ {xn}n∈N sã♦ ❧✐♠✐t❛❞♦s ❡ tn n→∞

−−−→ ∞✳ ❊♥tã♦

✶✳✷ ❊①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r❡s ❣❧♦❜❛✐s ✶✸

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ {T(t) : t ≥ 0} é ✉♠ s❡♠✐❣r✉♣♦ ❡✈❡♥t✉❛❧♠❡♥t❡ ❧✐♠✐t❛❞♦ ❡ ❛ss✐♥t♦✲

t✐❝❛♠❡♥t❡ ❝♦♠♣❛❝t♦✱ ❡♥tã♦ {T(tn)xn}n∈N é r❡❧❛t✐✈❛♠❡♥t❡ ❝♦♠♣❛❝t♦ s❡♠♣r❡ q✉❡ {xn}n∈N

❢♦r ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡♠ X ❡ tn n→∞ −−−→ ∞✳

❉❡♠♦♥str❛çã♦✳

❙❡❥❛ B ⊂ X ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦✱ ❧✐♠✐t❛❞♦ ❡ ♥ã♦✲✈❛③✐♦ t❛❧ q✉❡ T(t)B ⊂ B✱ ♣❛r❛

t♦❞♦ t ≥ 0✳ ❆ss✐♠✱ ω(B) ⊂ B é ♥ã♦✲✈❛③✐♦✱ ❝♦♠♣❛❝t♦✱ ✐♥✈❛r✐❛♥t❡ ❡ ❛tr❛✐ B✳ ❊♥tã♦ {T(t) :t≥0} é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❝♦♠♣❛❝t♦✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ {T(t) :t ≥ 0} é ✉♠ s❡♠✐❣r✉♣♦ ❡✈❡♥t✉❛❧♠❡♥t❡ ❧✐♠✐t❛❞♦ ❡ {xn}n∈N

é ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡♠ X✱ ❡①✐st❡ t0 ≥ 0 t❛❧ q✉❡ B = γt0({xn}n∈N) é ✉♠ ❝♦♥❥✉♥t♦

❧✐♠✐t❛❞♦✳ ❈♦♠♦ B é ♣♦s✐t✐✈❛♠❡♥t❡ ✐♥✈❛r✐❛♥t❡ ❡ ♦ s❡♠✐❣r✉♣♦ ❡♠ q✉❡stã♦ é ❛ss✐♥t♦t✐❝❛✲

♠❡♥t❡ ❝♦♠♣❛❝t♦✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ J ⊂B ❞❡ ❳ q✉❡ ❛tr❛✐ B✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ {T(tn)xn}n∈N ❝♦♥✈❡r❣❡ ♣❛r❛ J q✉❛♥❞♦ n→ ∞ ❡✱ ❧♦❣♦✱ é r❡❧❛t✐✈❛♠❡♥t❡ ❝♦♠♣❛❝t♦✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✷✳ ❯♠ s❡♠✐❣r✉♣♦ {T(t) :t≥0}é ❝❤❛♠❛❞♦ ♣♦♥t♦ ❞✐ss✐♣❛t✐✈♦ s❡ ❡①✐st✐r

✉♠ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦ B ⊂X q✉❡ ❛❜s♦r✈❡ ♣♦♥t♦s ❞❡ X✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✸✳ ❯♠ s❡♠✐❣r✉♣♦ {T(t) : t ≥ 0} é ❝❤❛♠❛❞♦ ❧✐♠✐t❛❞♦ ❞✐ss✐♣❛t✐✈♦ s❡

❡①✐st✐r ✉♠ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦ B ⊂X q✉❡ ❛❜s♦r✈❡ s✉❜❝♦♥❥✉♥t♦s ❧✐♠✐t❛❞♦s ❞❡ X✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✹✳ ❯♠ s❡♠✐❣r✉♣♦ {T(t) : t ≥ 0} é ❝❤❛♠❛❞♦ ❝♦♠♣❛❝t♦ ❞✐ss✐♣❛t✐✈♦ s❡

❡①✐st✐r ✉♠ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦ B ⊂X q✉❡ ❛❜s♦r✈❡ s✉❜❝♦♥❥✉♥t♦s ❝♦♠♣❛❝t♦s ❞❡ X✳

❖❜s❡r✈❛çã♦ ✶✳✷✳✶✺✳ ◆❛s três ❞❡✜♥✐çõ❡s ❛❝✐♠❛✱ ♣♦❞❡rí❛♠♦s t❡r tr♦❝❛❞♦ ❛ ♣❛❧❛✈r❛ ❛❜s♦r✈❡ ♣♦r ❛tr❛✐ s❡♠ ♠✉❞❛r ♦s ❝♦♥❝❡✐t♦s ❡♠ q✉❡stã♦✳

▲❡♠❛ ✶✳✷✳✶✻✳ ❙❡❥❛ {T(t) : t ≥ 0} ✉♠ s❡♠✐❣r✉♣♦ ♣♦♥t♦ ❞✐ss✐♣❛t✐✈♦ ❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡

❝♦♠♣❛❝t♦✳ ❙❡ γ+_{(K)❢♦r ❧✐♠✐t❛❞♦ s❡♠♣r❡ q✉❡ K ❢♦r ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦✱ ❡♥tã♦{T(t) :}

t ≥0} é ❝♦♠♣❛❝t♦ ❞✐ss✐♣❛t✐✈♦✳

❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ {T(t) : t ≥ 0} é ♣♦♥t♦ ❞✐ss✐♣❛t✐✈♦✱ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦

❧✐♠✐t❛❞♦ B q✉❡ ❛❜s♦r✈❡ ♣♦♥t♦s ❞❡ X✳

✶✹ ❙❡♠✐❣r✉♣♦s ❡ ❆tr❛t♦r❡s ●❧♦❜❛✐s

❈♦♠♦ B ❛❜s♦r✈❡ ♣♦♥t♦s✱ U ♥ã♦ é ✈❛③✐♦ ♣♦✐s s❡ x ∈ B✱ ❡♥tã♦ ❡①✐st❡ t0 ≥ 0 t❛❧ q✉❡

T(t)x∈B✱ s❡ t≥t0✳ ❆ss✐♠✱ T(t0)x∈U✳

❈❧❛r❛♠❡♥t❡✱ γ+_{(U) = U ❡ U é ❧✐♠✐t❛❞♦✳}

◆♦t❡♠♦s✱ t❛♠❜é♠✱ q✉❡✱ s❡ x ∈ X✱ ❡♥tã♦ ❡①✐st❡ t0 ≥ 0 t❛❧ q✉❡ T(t)x ∈ B✱ ♣❛r❛ t♦❞♦

t ≥ t0✳ ❉✐ss♦✱ γ+(T(t)x) ⊂ B✱ t ≥ t0✱ ♦✉ s❡❥❛✱ T(t)x ∈ U✱ ♠♦str❛♥❞♦ q✉❡ U ❛❜s♦r✈❡ ♦s

♣♦♥t♦s ❞❡ X✳

❈♦♠♦T(t)γ+_(U)⊂γ+_(U)✱_{t ≥0}❡_{{T(t) :t≥0}}é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❝♦♠♣❛❝t♦✱ ❡①✐st❡

✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❛❝t♦ K ⊂ γ+_{(U) = U} q✉❡ ❛tr❛✐ _U✳ ❊♥tã♦ _K t❛♠❜é♠ ❛tr❛✐ ♦s ♣♦♥t♦s

❞❡ X✳

▼♦str❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ V ❞❡ K t❛❧ q✉❡ γt+(V) é ❧✐♠✐t❛❞♦ ♣❛r❛ ❛❧❣✉♠

t ≥ 0✳ ❱❛♠♦s s✉♣♦r q✉❡ ✐ss♦ ♥ã♦ ♦❝♦rr❛✳ ❊♥tã♦ ❡①✐st❡♠ s❡q✉ê♥❝✐❛s {xn}n∈N ⊂ X ❡

{tn}n∈N ⊂ R+ ❡ y ∈ K t❛✐s q✉❡ xn n→∞ −−−→ y✱ tn

n→∞

−−−→ ∞ ❡ {T(tn)xn}n∈N ♥ã♦ é ❧✐♠✐t❛❞♦✳

❙❡❥❛ A ={xn}n∈N✳ ❊♥tã♦ A é ❝♦♠♣❛❝t♦ ❡✱ ♣❛r❛ t♦❞♦ t ≥ 0✱ γt+(A) ♥ã♦ é ❧✐♠✐t❛❞♦✳ ■ss♦ ❝♦♥tr❛❞✐③ ❛s ❤✐♣ót❡s❡s ❞♦ ❧❡♠❛✳

❋✐①❡♠♦s ✉♠❛ ✈✐③✐♥❤❛♥ç❛V ❞❡K ❡tV ∈R+ t❛✐s q✉❡γt+V(V)é ❧✐♠✐t❛❞♦✳ ❈♦♠♦K ❛tr❛✐

♦s ♣♦♥t♦s ❞❡ X ❡ T(t) é ❝♦♥tí♥✉♦✱ ♣❛r❛ ❝❛❞❛ x∈X✱ ❡①✐st❡ tx > tV ❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ Ox ❞❡ x t❛✐s q✉❡ T(t)x ∈ V✱ t ≥ (tx −tV) ❡ T(tx−tV)Ox ⊂ V✳ ❉✐ss♦✱ T(tx)Ox ⊂ T(tV)V ❡ T(t+tx)Ox ⊂ T(t+tV)V✱ t ≥ 0✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ T(t)Ox ⊂ γt+V(V)✱ s❡ t ≥ tx✳

❆ss✐♠✱ ♣❛r❛ ❝❛❞❛ x∈X✱γ_t+_V(V) ❛❜s♦r✈❡ ✉♠❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ x✳

❈♦♠♦ ♣♦❞❡♠♦s ❝♦❜r✐r ✉♠ ❝♦♠♣❛❝t♦ ♣♦r ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ✈✐③✐♥❤❛♥ç❛s✱γ_t+_V(V)❛❜✲

s♦r✈❡ ❝♦♠♣❛❝t♦s ❞❡X✱ ♣r♦✈❛♥❞♦ q✉❡{T(t) :t≥0}é ✉♠ s❡♠✐❣r✉♣♦ ❝♦♠♣❛❝t♦ ❞✐ss✐♣❛t✐✈♦✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✶✼✳ ❙❡❥❛ {T(t) :t≥0}✉♠ s❡♠✐❣r✉♣♦✳ ❙❡ K ❢♦r ✉♠ ❝♦♠♣❛❝t♦ q✉❡ ❛tr❛✐

❛ s✐ ♠❡s♠♦✱ ❡♥tã♦ ω(K) =T

t≥0T(t)K✳

❉❡♠♦♥str❛çã♦✳ P❡❧❛ ❞❡✜♥✐çã♦✱ é ✐♠❡❞✐❛t♦ q✉❡ T

t≥0T(t)K ⊂ω(K)✳

❯t✐❧✐③❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✭✶✳✷✳✸✮✱ ❝♦♠ K = K1✱ ♦❜t❡♠♦s q✉❡ ω(K) ⊂ K ❡ γ+(K) é

r❡❧❛t✐✈❛♠❡♥t❡ ❝♦♠♣❛❝t♦✳

P❡❧♦ ▲❡♠❛ ✭✶✳✷✳✼✮✱ ω(K)é ♥ã♦✲✈❛③✐♦✱ ❝♦♠♣❛❝t♦✱ ✐♥✈❛r✐❛♥t❡ ❡ ❛tr❛✐ K✳

▲♦❣♦✱ ω(K) = T(t)ω(K)⊂T(t)K✱ ∀t≥0✱ ♣r♦✈❛♥❞♦ q✉❡ω(K)⊂T

✶✳✷ ❊①✐stê♥❝✐❛ ❞❡ ❛tr❛t♦r❡s ❣❧♦❜❛✐s ✶✺

❚❡♦r❡♠❛ ✶✳✷✳✶✽✳ ❯♠ s❡♠✐❣r✉♣♦{T(t) :t≥0}é ❡✈❡♥t✉❛❧♠❡♥t❡ ❧✐♠✐t❛❞♦✱ ♣♦♥t♦✲❞✐ss✐♣❛t✐✈♦

❡ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❝♦♠♣❛❝t♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♣♦ss✉✐ ✉♠ ❛tr❛t♦r ❣❧♦❜❛❧ A✳

❉❡♠♦♥str❛çã♦✳ ❱❛♠♦s s✉♣♦r q✉❡ {T(t) : t ≥ 0} s❡❥❛ ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❝♦♠♣❛❝t♦✱ ♣♦♥t♦

❞✐ss✐♣❛t✐✈♦ ❡ ❡✈❡♥t✉❛❧♠❡♥t❡ ❧✐♠✐t❛❞♦✳

P❡❧♦ ▲❡♠❛ ✭✶✳✷✳✶✻✮✱ ❡st❡ s❡♠✐❣r✉♣♦ é ❝♦♠♣❛❝t♦ ❞✐ss✐♣❛t✐✈♦✳

❙❡❥❛♠ C ✉♠ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦ q✉❡ ❛❜s♦r✈❡ ❝♦♠♣❛❝t♦s ❞❡ X ❡B ={x∈C :γ+_(x)⊂

C}✳ ❙❡❥❛♠ t ≥ 0 ❡ x ∈ T(t)B✳ ❊♥tã♦ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ {xn}n∈N ⊂ B t❛❧ q✉❡

x = limn→∞T(t)xn✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ B✱ ✈❡♠♦s q✉❡ T(t)xn ∈ B ❡ x ∈ B✳ ❆ss✐♠✱

T(t)B ⊂B✳

◆♦t❡♠♦s q✉❡✱ ❞❛❞♦ ✉♠ ❝♦♠♣❛❝t♦ K1 ⊂ X✱ ❡①✐st❡ t0 ≥0 t❛❧ q✉❡ T(t)K1 ⊂C✱ t ≥t0✳

❉✐ss♦✱γ+_{(T(t)K1)⊂C✱t ≥t}

0✱ ♦✉ s❡❥❛✱T(t)K1 ⊂B✱ ♠♦str❛♥❞♦ q✉❡B ❛❜s♦r✈❡ ❝♦♠♣❛❝t♦s

❞❡ X✳

❈♦♠♦ {T(t) : t ≥ 0} é ❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❝♦♠♣❛❝t♦✱ ❡①✐st❡ ✉♠ ❝♦♠♣❛❝t♦ K ⊂ B q✉❡

❛tr❛✐ B✳ ❊♥tã♦ K ❛tr❛✐ ❝♦♠♣❛❝t♦s ❞❡ X✳

❙❡❥❛A =ω(B)✱ ❡♥tã♦ A é ♥ã♦✲✈❛③✐♦✱ ❝♦♠♣❛❝t♦ ❡ ✐♥✈❛r✐❛♥t❡✳

❙❡J ⊂X é ✉♠ ❝♦♠♣❛❝t♦✱ ❡♥tã♦ ω(J)⊂K ❡ω(J) = T(s)ω(J)⊂T(s)K✱s ≥0✳ P❡❧❛

Pr♦♣♦s✐çã♦ ✭✶✳✷✳✶✼✮✱ ω(J)⊂T

s≥0T(s)K =ω(K)❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ω(K) ❛tr❛✐ J✳

❙❡❥❛ D ⊂X ✉♠ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦✳ ❈♦♠♦ {T(t) : t ≥ 0} é ❡✈❡♥t✉❛❧♠❡♥t❡ ❧✐♠✐t❛❞♦ ❡

❛ss✐♥t♦t✐❝❛♠❡♥t❡ ❝♦♠♣❛❝t♦✱ ♣❡❧♦ ▲❡♠❛ ✭✶✳✷✳✶✵✮✱ ω(D)é ♥ã♦✲✈❛③✐♦✱ ❝♦♠♣❛❝t♦✱ ✐♥✈❛r✐❛♥t❡ ❡

❛tr❛✐ D✳ ❙❡♥❞♦ ω(D) ❝♦♠♣❛❝t♦ ❡ ✐♥✈❛r✐❛♥t❡✱ ♣♦❞❡♠♦s t♦♠❛rJ =ω(D)♥❛ ❡t❛♣❛ ❛♥t❡r✐♦r

♣❛r❛ ❝♦♥❝❧✉✐r q✉❡ ω(D) = ω(J) ⊂ A✳ ❆ss✐♠ A ❛tr❛✐ D✱ ♣r♦✈❛♥❞♦ q✉❡ A é ✉♠ ❛tr❛t♦r

❣❧♦❜❛❧✳

P❛r❛ ❛ r❡❝í♣r♦❝❛✱ ✈❛♠♦s s✉♣♦r q✉❡ {T(t) : t ≥ 0} t❡♥❤❛ ❛tr❛t♦r ❣❧♦❜❛❧ A✳ ❉❛❞♦ ✉♠

❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦ B ⊂X✱ ❡①✐st❡ tB ≥0 t❛❧ q✉❡ T(t)B ⊂ O1(A)✱ t ≥tB✳ ❊♥tã♦ γt+B(B) é

❧✐♠✐t❛❞♦ ❡ O1(A) ❛❜s♦r✈❡ ♣♦♥t♦s ❞❡X✳ ❆ss✐♠ {T(t) :t≥0} é ❡✈❡♥t✉❛❧♠❡♥t❡ ❧✐♠✐t❛❞♦ ❡

♣♦♥t♦✲❞✐ss✐♣❛t✐✈♦✳

❙❡❥❛♠ {xn}n∈N ⊂ X ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡ {tn}n∈N ⊂ R+ ❝♦♠ tn n→∞ −−−→ ∞✳

❈♦♠♦ A ❛tr❛✐ ♦ ❝♦♥❥✉♥t♦ {xn : n ∈ N}✱ {T(tn)xn}n∈N é ✉♠❛ s❡q✉ê♥❝✐❛ ❧✐♠✐t❛❞❛ ❡ d(T(tn)xn,A)

n→∞ −−−→0✳

✶✻ ❙❡♠✐❣r✉♣♦s ❡ ❆tr❛t♦r❡s ●❧♦❜❛✐s

❈❛♣ít✉❧♦

2

❙❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r

❊st❡ ❝❛♣ít✉❧♦ é ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞♦ ❛tr❛t♦r q✉❛♥❞♦ ✉♠ s❡♠✐❣r✉♣♦ é ♣❡rt✉r❜❛❞♦✳

❆♣r❡s❡♥t❛♠♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ ✐♥❢❡r✐♦r ❡ s✉♣❡r✐♦r ❡ ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥✲ t✐♥✉✐❞❛❞❡ ♣❛r❛ ♦s ❛tr❛t♦r❡s ❞❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s❡♠✐❣r✉♣♦s✳

❉❡✜♥✐çã♦ ✷✳✵✳✶✾✳ ❙❡❥❛♠X ❡Λ❡s♣❛ç♦s ♠étr✐❝♦s ❡{Aλ}λ∈Λ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s

❞❡ X✳

❉✐③❡♠♦s q✉❡ {Aλ}λ∈Λ é

✭✶✮ s❡♠✐❝♦♥tí♥✉❛ s✉♣❡r✐♦r♠❡♥t❡ ❡♠ λ0 s❡

lim

λ→λ0❞✐stH

(Aλ,Aλ0) = 0❀

✭✷✮ s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♠ λ0 s❡

lim

λ→λ0❞✐st

H(Aλ0,Aλ) = 0❀

▲❡♠❛ ✷✳✵✳✷✵✳ ❙❡❥❛Λ ✉♠ ❡s♣❛ç♦ ♠étr✐❝♦✱ λ0 ∈Λ❡ {Aλ}λ∈Λ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ s✉❜❝♦♥❥✉♥t♦s

❞❡ X✳

✭✶✮ ❙❡ ♣❛r❛ t♦❞❛ s❡q✉ê♥❝✐❛ λn n→∞

−−−→ λ0✱ {xλn}✱ ❝♦♠ xλn ∈ Aλn✱ t❡♠ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛

✶✽ ❙❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r ❡ ✐♥❢❡r✐♦r

❡♠ λ0✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡{Aλ}λ∈Λ❢♦r s❡♠✐❝♦♥tí♥✉❛ s✉♣❡r✐♦r♠❡♥t❡ ❡♠λ0✱ Aλ0 ❢♦r ❝♦♠♣❛❝t♦

❡ λn n→∞

−−−→ λ0✱ ❡♥tã♦ t♦❞❛ s❡q✉ê♥❝✐❛ {xλn}✱ ❝♦♠ xλn ∈ Aλn✱ t❡♠ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛

❝♦♥✈❡r❣❡♥t❡✱ ❝♦♠ ❧✐♠✐t❡ ❡♠ Aλ0✳

✭✷✮ ❙❡ Aλ0 ❢♦r ❝♦♠♣❛❝t♦ ❡✱ ♣❛r❛ t♦❞♦ x ∈ Aλ0 ❡ λn

n→∞

−−−→ λ0✱ ❡①✐st✐r ✉♠❛ s❡q✉ê♥❝✐❛

λnk

k→∞

−−−→ λ0 ❡ {xλ_nk}✱ ❝♦♠ xλ_nk ∈ Aλ_nk✱ q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ x✱ ❡♥tã♦ {Aλ}λ∈Λ é

s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♠ λ0✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ {Aλ}λ∈Λ ❢♦r s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♠ λ0✱ λn −−−→n→∞ λ0 ❡

x ∈Aλ0✱ ❡♥tã♦ ❛ ❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ λnk

k→∞

−−−→ λ0 ❡ {xλ_nk}✱ ❝♦♠ xλ_nk ∈ Aλ_nk✱

q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ x✳ ✭❆ ❝♦♠♣❛❝✐❞❛❞❡ ❞❡ Aλ0 ♥ã♦ é ♥❡❝❡ssár✐❛ ❛q✉✐✮✳

❉❡♠♦♥str❛çã♦✳ ✭✶✮ ❙✉♣♦♥❤❛♠♦s q✉❡✱ ♣❛r❛ t♦❞❛ s❡q✉ê♥❝✐❛ λn n→∞

−−−→ λ0✱ ❛ s❡q✉ê♥❝✐❛

{xλn}✱ ❝♦♠ xλn ∈ Aλn✱ t❡♥❤❛ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ ❝♦♥✈❡r❣❡♥t❡✱ ❝♦♠ ❧✐♠✐t❡ ❡♠ Aλ0

❡ q✉❡{Aλ}λ∈Λ ♥ã♦ s❡❥❛ s❡♠✐❝♦♥tí♥✉❛ s✉♣❡r✐♦r♠❡♥t❡ ❡♠λ0✳ ❊♥tã♦ ❡①✐st❡ǫ >0❡ ✉♠❛

s❡q✉ê♥❝✐❛ {λn}n∈N ⊂ Λ t❛✐s q✉❡ λn −−−→n→∞ λ0 ❡ ❞✐stH(Aλn,Aλ0) ≥ 2ǫ✳ ❆ss✐♠✱ ♣❛r❛

❝❛❞❛ n∈N✱ ❡①✐st❡ xn∈ Aλn t❛❧ q✉❡ d(xn,Aλ0)≥ǫ✳ ■ss♦ ❝♦♥tr❛❞✐③ ♦ ❢❛t♦ ❞❡{xn}n∈N

♣♦ss✉✐r ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ ✉♠ ♣♦♥t♦ ❞❡ Aλ0✳

P❛r❛ ❛ r❡❝í♣r♦❝❛✱ ✈❛♠♦s s✉♣♦r q✉❡ Aλ0 s❡❥❛ ❝♦♠♣❛❝t♦✱ λn

n→∞

−−−→ λ0 ❡ xλ_nk ∈ Aλ_nk✳

❊♥tã♦✱ ❝♦♠♦limn→∞❞✐stH(Aλn,Aλ0) = 0✱ t❡♠♦s q✉❡limn→∞d(xλn,Aλ0) = 0✳ ❆ss✐♠✱

♣❛r❛ ❝❛❞❛n ∈N✱ ❡①✐st❡ zn∈ Aλ0 ❝♦♠ d(xλn, zn)<1/n ✭t♦♠❛♥❞♦ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱

s❡ ♥❡❝❡ssár✐♦✮✳

❙❡♥❞♦ Aλ0 ❝♦♠♣❛❝t♦✱ ❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛ {znk} q✉❡ ❝♦♥✈❡r❣❡ ♣❛r❛ x0 ∈ Aλ0✳

❉❡st❛ ❢♦r♠❛✱d(xλ_nk, x0)≤d(xλ_nk, znk)+d(znk, x0)

k→∞

−−−→0✱ ♣r♦✈❛♥❞♦ ♦ q✉❡ q✉❡rí❛♠♦s✳

✭✷✮ ❙✉♣♦♥❤❛ q✉❡ Aλ0 s❡❥❛ ❝♦♠♣❛❝t♦ ❡ q✉❡✱ ♣❛r❛ t♦❞♦ x ∈ Aλ0 ❡ λn

n→∞

−−−→ λ0✱ ❡①✐st❛♠

s✉❜s❡q✉ê♥❝✐❛s λnk

k→∞

−−−→ λ0 ❡ {xλ_nk}✱ ❝♦♠ xλ_nk ∈ Aλ_nk✱ t❛✐s q✉❡ ♣❛r❛ xλ_nk → x

q✉❛♥❞♦ k→ ∞✳

❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡{Aλ}λ∈Λ ♥ã♦ s❡❥❛ s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♠λ0✳ ❊♥✲

tã♦ ❡①✐st❡♠ǫ >0❡ ✉♠❛ s❡q✉❡♥❝✐❛{λn}n∈N⊂Λt❛✐s q✉❡λn n→∞

−−−→λ0❡ ❞✐stH(Aλ0,Aλn)≥

✶✾

❙❡♥❞♦Aλ0 ❝♦♠♣❛❝t♦✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡{yλn}❝♦♥✈❡r❣❡ ♣❛r❛ ❛❧❣✉♠x∈ Aλ0 ❡ q✉❡ d(x,Aλn)≥ǫ✱ n ∈N✳

❆ss✐♠✱ ❡①✐st❡♠ s✉❜s❡q✉ê♥❝✐❛s λnk

k→∞

−−−→ λ0 ❡ xλ_nk ∈ Aλ_nk t❛✐s q✉❡ xλ_nk k→∞ −−−→ x ❡

ǫ≤d(x,Aλ_nk)≤d(x, xλ_nk)−−−→k→∞ 0✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳

P❛r❛ ❛ r❡❝í♣r♦❝❛✱ ✈❛♠♦s s✉♣♦r q✉❡ {Aλ}λ∈Λ s❡❥❛ s❡♠✐❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡ ❡♠ λ0✱

λn n→∞

−−−→λ0 ❡x∈Aλ0✳

❚♦♠❛♥❞♦ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ ❞✐stH(Aλ0, Aλn) ≤ 1

2n✳ ❆ss✐♠✱

d(x, Aλn) ≤ 1

2n✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ ❝❛❞❛ n✱ ♣♦❞❡♠♦s ❡♥❝♦♥tr❛r xn ∈ Aλn t❛❧ q✉❡ d(x, xn)≤ _n1✳ ❚❡♠♦s✱ ❡♥tã♦ ✉♠❛ s❡q✉ê♥❝✐❛ s❛t✐s❢❛③❡♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡s❡❥❛❞❛s✳

❉❡✜♥✐çã♦ ✷✳✵✳✷✶✳ ❆ ❢❛♠í❧✐❛ ❞❡ s❡♠✐❣r✉♣♦s {Tη(t) : t ≥ 0}η∈[0,1] é ❝♦♥tí♥✉❛ ❡♠ η = 0

s❡ Tη(t)x→T0(t)x q✉❛♥❞♦ η →0✱ ✉♥✐❢♦r♠❡♠❡♥t❡ ♣❛r❛ (t, x) ❡♠ s✉❜❝♦♥❥✉♥t♦s ❝♦♠♣❛❝t♦s ❞❡ R+_×X✳

❚❡♦r❡♠❛ ✷✳✵✳✷✷ ✭❙❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r✮✳ ❙❡❥❛ {Tη(t) :t ≥0}η∈[0,1] ✉♠❛ ❢❛♠í❧✐❛ ❞❡

s❡♠✐❣r✉♣♦s ❝♦♥tí♥✉❛ ❡♠ η= 0✳

❙❡ ❝❛❞❛ {Tη(t) : t ≥ 0} t✐✈❡r ✉♠ ❛tr❛t♦r ❣❧♦❜❛❧ Aη✱ ♣❛r❛ η ∈ [0,1] ❡ S_η∈[0,1]Aη ❢♦r ❝♦♠♣❛❝t♦✱ ❡♥tã♦ ❛ ❢❛♠í❧✐❛ {Aη :η∈[0,1]} é s❡♠✐❝♦♥tí♥✉❛ s✉♣❡r✐♦r♠❡♥t❡ ❡♠ η= 0✳

❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡ ❛s s❡q✉ê♥❝✐❛s {ηk}k∈N ❡{uηk}k∈N✱ t❛✐s q✉❡ηk

k→∞

−−−→0✱uηk ∈ A✳

❈♦♠♦S

η∈[0,1]Aη é ❝♦♠♣❛❝t♦✱ ♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ ❡①✐st❡u0 ∈X t❛❧ q✉❡uηk

k→∞ −−−→u0✳

P❛r❛ ♣r♦✈❛r♠♦s ❛ s❡♠✐❝♦♥t✐♥✉✐❞❛❞❡ s✉♣❡r✐♦r✱ t❡♠♦s q✉❡ ♠♦str❛r q✉❡u0 ∈ A0✳

P❡❧♦ ▲❡♠❛ ✭✶✳✶✳✽✮✱ ❜❛st❛ ♣r♦✈❛r♠♦s q✉❡ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❧✐♠✐t❛❞❛ ❞❡{T0(t) :

t ≥0}♣♦r u0✳

P❛r❛ ❝❛❞❛ k ∈ N✱ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❧✐♠✐t❛❞❛ ❞❡ {Tηk(t) : t ≥ 0}✱ ❞❡♥♦t❛❞❛

♣♦r ψ(ηk) : R → X✱ ♣♦r u

ηk✳ P❛r❛ t ≥ 0✱ ❝♦♠♦ [0,1] ∋ η 7→ Tη(t)x ∈ X é ✉♥✐❢♦r♠❡✲

♠❡♥t❡ ❝♦♥tí♥✉❛ ❡♠ ❝♦♠♣❛❝t♦s ❞❡ R+_{×X✱ t❡♠♦s q✉❡ ψ}(ηk)_{(t) = T}

ηk(t)uηk

k→∞

−−−→ T0(t)u0