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▲✐♠✐t s❡ts ✐♥ t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡ ❝②❧✐♥❞r✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ❏❛♥ ❑✇✐❛t❦♦✇s❦✐ ❚❤❡ t❛❧❦ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❆rt✉r ❙✐❡♠❛s③❦♦✳ ❇➛❞❧❡✇♦✱ ❏✉♥❡✱ ✷✵✶✽

❈②❧✐♥❞r✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥s X ✕ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ ♠♦♥♦t❤❡t✐❝ ❣r♦✉♣✱ → X ✕ ❛ ♠✐♥✐♠❛❧ ❤♦♠❡♦♠♦r♣❤✐s♠❀ T : X − → R ✕ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ f : X − → X × R ✱ T f : X × R − T f ( x , r ) = ( Tx , r + f ( x )) − ❛ ❝②❧✐♥❞r✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥ ( T f ) n ( x , r ) = ( T n x , r + f ( n ) ( x )) ✱ ✇❤❡r❡ f ( n ) ( x ) =  f ( x ) + f ( Tx ) + . . . + f ( T n − ✶ x ) , n ≥ ✶ ,   ✵ , n = ✵ ,   − ( T − ✶ x ) − . . . − f ( T − n x ) , n ≥ ✶ . f ( · ) ( · ) : Z × X − → R ✱ s❛t✐s✜❡s ❛ ❝♦❝②❝❧❡ ❝♦♥❞✐t✐♦♥✿ f ( n + m ) ( x ) = f ( n ) ( x ) + f ( m ) ( T n x )

❚❤❡♦r❡♠ ✭❇❡s✐❝♦✈✐t❝❤ ✶✾✺✶✮ ✶ ■❢ T f ✐s t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡ ✭✐✳❡✳ ❛❞♠✐ts ❞❡♥s❡ ♦r❜✐ts✮✱ t❤❡♥ ✐t ♣♦ss❡ss❡s ❛ ❜♦✉♥❞❡❞ ❛❜♦✈❡ ♦r❜✐t ✭❛♥❞ ❛ ❜♦✉♥❞❡❞ ❜❡❧♦✇ ✐♥ ♣❛rt✐❝✉❧❛r✱ T f ❝❛♥♥♦t ❜❡ ♠✐♥✐♠❛❧ ✳ ♦r❜✐t✮❀ ✷ ❚❤❡r❡ ❡①✐sts ❛♥ ✐rr❛t✐♦♥❛❧ ❝✐r❝❧❡ r♦t❛t✐♦♥ ❛♥❞ ✐ts t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡ ❝♦❝②❝❧❡ ✇✐t❤ ❛ ❞✐s❝r❡t❡ ♦r❜✐t✳ ❚❤❡♦r❡♠ ✭●♦tts❝❤❛❧❦✲❍❡❞❧✉♥❞ ✶✾✺✺✱ ▲❡♠❛➠❝③②❦✲▼❡♥t③❡♥ ✷✵✵✷✮ ▲❡t µ ❜❡ t❤❡ ❍❛❛r ♠❡❛s✉r❡ ♦♥ X ✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ tr✐❝❤♦t♦♠② ❤♦❧❞s✿ � f d µ � = ✵ ≡ ❡✈❡r② ♦r❜✐t ✐s ❛ ❞✐s❝r❡t❡ s❡t❀ ✶ ✕ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❡✈❡r② ♦r❜✐t ✐s ✐ts❡❧❢ ❛ ♠✐♥✐♠❛❧ s❡t❀ ✷ f ✐s ❛ ❝♦❜♦✉♥❞❛r② ≡ T f ❛❞♠✐ts ❜♦✉♥❞❡❞ s❡♠✐✲♦r❜✐ts ✭✐✳❡✳ t❤❡r❡ ❡①✐sts ∃ g ∈ C ( X ) f = g ◦ T − g ✮❀ ✕ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❡✈❡r② ♠✐♥✐♠❛❧ s❡t ✐s ❛ s❤✐❢t❡❞ ❣r❛♣❤ ♦❢ g ❀ ✸ T f ✐s t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡✳

P♦ss✐❜❧❡ ♣r♦❜❧❡♠s ✶ ❉♦ t❤❡r❡ ❡①✐st t♦♣✳ tr❛♥s✳ tr❛♥s❢♦r♠❛t✐♦♥s ✇✐t❤♦✉t ♠✐♥✐♠❛❧ s❡ts❄ ✷ ●✐✈❡♥ ❛ ♠✐♥✐♠❛❧ r♦t❛t✐♦♥✱ ❝♦♥str✉❝t ✐ts t♦♣✳ tr❛♥s✳ ❝♦❝②❝❧❡ ✇✐t❤♦✉t ❞✐s❝r❡t❡ ♦r❜✐ts✳ ✸ ●✐✈❡♥ ❛ ♠✐♥✐♠❛❧ r♦t❛t✐♦♥✱ ❝♦♥str✉❝t ✐ts t♦♣✳ tr❛♥s✳ ❝♦❝②❝❧❡ ✇✐t❤ ♣♦ss✐❜❧② ♠❛♥② ❞✐s❝r❡t❡ ♦r❜✐ts✳ ✹ ❲❤❛t ❦✐♥❞ ♦❢ ♦r❜✐ts ❝❛♥ ❝♦❡①✐st ❢♦r ❛ t♦♣✳ tr❛♥s✳ tr❛♥s❢♦r♠❛t✐♦♥ ✭❧♦❝❛❧❧② ❝♦♠♣❛❝t σ ✲❈❛♥t♦r s❡ts ❢♦r ✐♥st❛♥❝❡✮❄ ✺ ❉♦❡s t❤❡r❡ ❡①✐st ❛ t♦♣✳ tr❛♥s✳ ❝♦❝②❧❡ ❤❛✈✐♥❣ ❡✐t❤❡r ❞❡♥s❡ ♦r ❞✐s❝r❡t❡ ♦r❜✐ts ♦♥❧②❄ ✕ ✭❖P❊◆✮ ✻ ❲❤❛t ❛r❡ ♣♦ss✐❜❧❡ ♠✐♥✐♠❛❧ s❡ts ❢♦r ❝②❧✐♥❞r✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥s❄ ✭❖P❊◆✳ ❊✈❡r② ♥♦♥ ❞✐s❝r❡t❡ ♠✐♥✐♠❛❧ s❡t ✐s ❛ ❧♦❝❛❧❧② ❝♦♠♣❛❝t ❈❛♥t♦r s❡t✳ ❖♥❧② ❞✐s❝r❡t❡ ♦r❜✐ts ❤❛✈❡ ❜❡❡♥ r❡❝♦❣♥✐③❡❞ s♦ ❢❛r ❛s ♠✐♥✐♠❛❧ s❡ts✳✮

❚❤❡♦r❡♠ ✭▼❛ts✉♠♦t♦✲❙❤✐s❤✐❦✉r❛ ✷✵✵✷✱ ▼❡♥t③❡♥✲❙✐❡♠❛s③❦♦ ✷✵✵✹✮ ▲❡t ( X , T ) ❜❡ ❡✐t❤❡r ❛♥ ✐rr❛t✐♦♥❛❧ ❝✐r❝❧❡ r♦t❛t✐♦♥ ♦r ❛♥ ♦❞♦♠❡t❡r✳ ❆ t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡✱ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥ ❝♦❝②❝❧❡ ♦✈❡r ( X , T ) ❛❞♠✐ts ♥♦ ♠✐♥✐♠❛❧ s❡ts✳ ❚❤❡♦r❡♠ ✭▼❡♥t③❡♥✱ ❙✐❡♠❛s③❦♦ ✷✵✵✹✮ ✶ ❊✈❡r② ♠✐♥✐♠❛❧ r♦t❛t✐♦♥ ♦♥ ❛♥ ✐♥✜♥✐t❡ ❝♦♠♣❛❝t ♠❡tr✐❝ ❣r♦✉♣ ❛❞♠✐ts ❛ t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡ ❝♦❝②❝❧❡✳ ✷ ■❢ ( X , T ) ✐s ❡✐t❤❡r ❛♥ ✐rr❛t✐♦♥❛❧ ❝✐r❝❧❡ r♦t❛t✐♦♥ ♦r ❛♥ ♦❞♦♠❡t❡r t❤❡♥ ( X , T ) ❛❞♠✐ts ❛ t♦♣♦❧♦❣✐❝❛❧❧② tr❛♥s✐t✐✈❡ ❝♦❝②❝❧❡ t❤❛t ✐s ♦❢ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✳

❚❤❡♦r❡♠ ✭❋r→❝③❡❦✲▲❡♠❛➠❝③②❦✱ ✷✵✵✾✮ ❊✈❡r② ✐rr❛t✐♦♥❛❧ ❝✐r❝❧❡ r♦t❛t✐♦♥ ❛❞♠✐ts ❛ t♦♣✳ tr❛♥s✳ tr❛♥s❢♦r♠❛t✐♦♥ ✇✐t❤ ❛ ❧♦t ♦❢ ❞✐s❝r❡t❡ ♦r❜✐ts✿ t❤❡ s✉❜s❡t D := { x ∈ S ✶ : ( x , ✵ ) ❤❛s ❛ ❞✐s❝r❡t❡ ♦r❜✐t } ✇✐t❤ t❤❡ ❍❛✉s❞♦r✛ ❞✐♠❡♥s✐♦♥ ✐♥ ( ✵ , ✶ ✷ ) ✳ ❚❤❡♦r❡♠ ✭❉②♠❡❦✱ ✷✵✶✸✱ ♣r❡♣r✐♥t✮ ❊✈❡r② ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥ ❛❞♠✐ts ❛ t♦♣✳ tr❛♥s✳ ❝♦❝②❝❧❡ ✇✐t❤ t❤❡ s❡t D ♦❢ ❢✉❧❧ ❍❛✉s❞♦r✛ ❞✐♠❡♥s✐♦♥✳ ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖ ▲❡t ✉s ❞❡♥♦t❡ D − , − = { x ∈ X : ❧✐♠ j →±∞ f ( j ) ( x ) = −∞} ✱ D + , + = { x ∈ X : ❧✐♠ j →±∞ f ( j ) ( x ) = + ∞} ✱ D − , + = { x ∈ X : ❧✐♠ j →±∞ f ( j ) ( x ) = ±∞} ✱ D + , − = { x ∈ X : ❧✐♠ j →±∞ f ( j ) ( x ) = ∓∞} ❚❤❡♦r❡♠ ✭❑✇✐❛t❦♦✇s❦✐✱ ❙✐❡♠❛s③❦♦ ✷✵✶✵✮ ❊✈❡r② ♦❞♦♠❡t❡r ❛❞♠✐ts ❛ t♦♣✳ tr❛♥s✳ ❝♦❝②❝❧❡ ✇✐t❤ ❛❧❧ D ′ s ❤❛✈✐♥❣ t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ ❝♦♥t✐♥✉✉♠✳

L x , y = { s ∈ R : s = ❧✐♠ f ( j ) ( x ) , T j ( x ) → y ✐♥ X } , x , y ∈ X . P♦✐♥❝❛ré ❛tt❡♠♣t❡❞ t♦ ❝❧❛ss✐❢② ♣♦ss✐❜❧❡ ❢♦r♠s ♦❢ S = L ( ✵ , ✵ ) ✳ P♦✐♥❝❛ré ❝❧❛✐♠❡❞ t❤❛t ✶ S = ∅ ✕ ❛ ❞✐s❝r❡t❡ ♦r❜✐t❀ ✷ S = { ✵ } ✕ ❝♦❜♦✉♥❞❛r②❀ ✸ S = R ✕ ❞❡♥s❡ ♦r❜✐t❀ ✹ S = τ Z , τ > ✵✳

❚❤❡♦r❡♠ ✭❑r②❣✐♥✱ ✶✾✼✺✮ ✶ ❊✐t❤❡r S = ∅ ♦r S ✐s ❛ ❝❧♦s❡❞ s❡♠✐✲❣r♦✉♣ ✐♥ R ✳ ✷ ■❢ S ✐s ❛ ❣r♦✉♣✱ t❤❡♥ ❡✐t❤❡r S = { ✵ } ♦r S = R ✳ ✸ ■❢ K ✐s ❛ ❝❧♦s❡❞ s❡♠✐✲❣r♦✉♣ ♦❢ R t❤❛t ✐s ♥♦t ❛ ❣r♦✉♣✱ t❤❡♥ ❢♦r ❡❛❝❤ ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥ ✇✐t❤ ✉♥❜♦✉♥❞❡❞ ♣❛rt✐❛❧ q✉♦t✐❡♥ts t❤❡r❡ ❡①✐sts ❛ ❝♦❝②❝❧❡ f : S ✶ − → R ✇✐t❤ S = K ✳ ❚❤❡♦r❡♠ ✭❑r②❣✐♥✱✶✾✼✽✮ ▲❡t ✉s ❛ss✉♠❡ t❤❛t f ✐s ❞✐✛❡r❡♥t✐❛❜❧❡✳ ❚❤❡♥ ✶ S = { ✵ } ✕ ❝♦❜♦✉♥❞❛r②❀ ✷ S = R ✕ ❞❡♥s❡ ♦r❜✐t❀ ✸ S = R + ❀ ✹ S = R − ✳ f ❤❛s t❤❡ ❜♦✉♥❞❡❞ ✈❛r✐❛t✐♦♥✱ s♦ S = ∅ ✐s ❡①❝❧✉❞❡❞ ❜② ▼✳✲❙✳

❖❞♦♠❡t❡r ❙❡t λ t ≥ ✷✱ p − ✶ = ✶ , p t = λ t p t − ✶ ✱ t = ✵ , ✶ , ✷ , . . . ✳ � � � ∞ � � x t p t − ✶ : x t ∈ { ✵ , ✶ , . . . , λ t − ✶ } ✭✶✮ λ = x = Z t = ✵ � � ❋♦r x ∈ Z ❛♥❞ s ≥ ✵ ✇❡ ✇✐❧❧ ✇r✐t❡ λ s � X − ✶ = ✵ , X s = x t p t − ✶ . t = ✵ ❙❡t Tx = x + ¯ ✶✱ ¯ ✶ = ✶ + ✵ + ✵ + . . . ✳ � � T ✕ ❛ ♠✐♥✐♠❛❧ ❤♦♠❡♦♠♦r♣❤✐s♠ ♦❢ Z λ T ❛❝ts ❛s ❛ ❝②❝❧❡ ♦♥ ❧❡✈❡❧s ♦❢ ❡✈❡r② T ( t ) ✳