Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theorem Ugur G. Abdulla FIT Colloquium May 30, 2014 1 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Introduction Let f : I → I be a continuous map, and I be an interval. Interval is a connected subset of the real line which contains more than one point. < a, b > is a closed interval with endpoints a and b . f 1 = f, f n +1 = f ◦ f n , n ≥ 1 2 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Introduction Let f : I → I be a continuous map, and I be an interval. Interval is a connected subset of the real line which contains more than one point. < a, b > is a closed interval with endpoints a and b . f 1 = f, f n +1 = f ◦ f n , n ≥ 1 y y = x x 0 x 1 x 3 x 2 x 0 2 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Sharkovski’s Theorem { f n ( x ) : n ≥ 0 } be an orbit of x. 3 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Sharkovski’s Theorem { f n ( x ) : n ≥ 0 } be an orbit of x. c ∈ I is a fixed point if f ( c ) = c . 3 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Sharkovski’s Theorem { f n ( x ) : n ≥ 0 } be an orbit of x. c ∈ I is a fixed point if f ( c ) = c . If I = [ a, b ] is compact ⇒ ∃ a fixed point c ∈ I , since f ( a ) − a ≥ 0 ≥ f ( b ) − b 3 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Sharkovski’s Theorem { f n ( x ) : n ≥ 0 } be an orbit of x. c ∈ I is a fixed point if f ( c ) = c . If I = [ a, b ] is compact ⇒ ∃ a fixed point c ∈ I , since f ( a ) − a ≥ 0 ≥ f ( b ) − b A point c ∈ I is said to be periodic point of f with period m if f m ( c ) = c , f k ( c ) � = c for 1 ≤ k < m . 3 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Sharkovski’s Theorem { f n ( x ) : n ≥ 0 } be an orbit of x. c ∈ I is a fixed point if f ( c ) = c . If I = [ a, b ] is compact ⇒ ∃ a fixed point c ∈ I , since f ( a ) − a ≥ 0 ≥ f ( b ) − b A point c ∈ I is said to be periodic point of f with period m if f m ( c ) = c , f k ( c ) � = c for 1 ≤ k < m . Periodic m -orbit is c, f ( c ) , f 2 ( c ) , · · · , f m − 1 ( c ) 3 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Sharkovski’s Theorem { f n ( x ) : n ≥ 0 } be an orbit of x. c ∈ I is a fixed point if f ( c ) = c . If I = [ a, b ] is compact ⇒ ∃ a fixed point c ∈ I , since f ( a ) − a ≥ 0 ≥ f ( b ) − b A point c ∈ I is said to be periodic point of f with period m if f m ( c ) = c , f k ( c ) � = c for 1 ≤ k < m . Periodic m -orbit is c, f ( c ) , f 2 ( c ) , · · · , f m − 1 ( c ) Theorem 1 (Sharkovsky, 1964) Let the positive integers be totally ordered in the following way: 1 ≻ 2 ≻ 2 2 ≻ 2 3 ≻ ... ≻ 2 2 · 5 ≻ 2 2 · 3 ≻ ... ≻ 2 · 5 ≻ 2 · 3 ≻ ... ≻ 7 ≻ 5 ≻ 3 If f has a cycle of period n and m ≻ n , then f also has a periodic orbit of period m . 3 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Proof of Sharkovski’s Theorem References ◮ A.N.Sharkovsky, Coexistence of Cycles of a Continuous Transformation of a Line into itself Ukrain. Mat. Zhurn. , 16 , 1(1964), 61-71. 4 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Proof of Sharkovski’s Theorem References ◮ A.N.Sharkovsky, Coexistence of Cycles of a Continuous Transformation of a Line into itself Ukrain. Mat. Zhurn. , 16 , 1(1964), 61-71. ◮ L.S. Block, J. Guckenheimer, M. Misiurewich, L.S. Young, Periodic Points and Topological Entropy of One-dimensional Maps, Global Theory of Dynamical Systems , Proc. International Conf., Northwestern Univ., Evanston, Ill.,1979, 18-34. 4 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Proof of Sharkovski’s Theorem References ◮ A.N.Sharkovsky, Coexistence of Cycles of a Continuous Transformation of a Line into itself Ukrain. Mat. Zhurn. , 16 , 1(1964), 61-71. ◮ L.S. Block, J. Guckenheimer, M. Misiurewich, L.S. Young, Periodic Points and Topological Entropy of One-dimensional Maps, Global Theory of Dynamical Systems , Proc. International Conf., Northwestern Univ., Evanston, Ill.,1979, 18-34. ◮ L.S.Block, W.A.Coppel, Dynamics in One Dimension, Springer-Verlag, Berlin, 1992. 4 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Proof of Sharkovski’s Theorem Lemma 2 If J is a compact subinterval such that J ⊆ f ( J ) , then f has a fixed point in J . 5 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Proof of Sharkovski’s Theorem Lemma 2 If J is a compact subinterval such that J ⊆ f ( J ) , then f has a fixed point in J . Proof. If J = [ a, b ] then for some c, d ∈ J we have f ( c ) = a , f ( d ) = b . Thus f ( c ) ≤ c , f ( d ) ≥ d , and by the intermediate value theorem ∃ c ∗ ∈ [ a, b ] f ( c ∗ ) = c ∗ . 5 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Proof of Sharkovski’s Theorem Lemma 2 If J is a compact subinterval such that J ⊆ f ( J ) , then f has a fixed point in J . Proof. If J = [ a, b ] then for some c, d ∈ J we have f ( c ) = a , f ( d ) = b . Thus f ( c ) ≤ c , f ( d ) ≥ d , and by the intermediate value theorem ∃ c ∗ ∈ [ a, b ] f ( c ∗ ) = c ∗ . Lemma 3 If J, K are compact subintervals such that K ⊆ f ( J ) , then there is a compact subinterval L ⊆ J such that f ( L ) = K . 5 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Proof of Sharkovski’s Theorem Lemma 2 If J is a compact subinterval such that J ⊆ f ( J ) , then f has a fixed point in J . Proof. If J = [ a, b ] then for some c, d ∈ J we have f ( c ) = a , f ( d ) = b . Thus f ( c ) ≤ c , f ( d ) ≥ d , and by the intermediate value theorem ∃ c ∗ ∈ [ a, b ] f ( c ∗ ) = c ∗ . Lemma 3 If J, K are compact subintervals such that K ⊆ f ( J ) , then there is a compact subinterval L ⊆ J such that f ( L ) = K . Proof. Let K = [ a, b ] , c = sup { x ∈ J : f ( x ) = a } . If f ( x ) = b for some x ∈ J with x > c , let d be the least and take L = [ c, d ] . 5 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Proof of Sharkovski’s Theorem Lemma 2 If J is a compact subinterval such that J ⊆ f ( J ) , then f has a fixed point in J . Proof. If J = [ a, b ] then for some c, d ∈ J we have f ( c ) = a , f ( d ) = b . Thus f ( c ) ≤ c , f ( d ) ≥ d , and by the intermediate value theorem ∃ c ∗ ∈ [ a, b ] f ( c ∗ ) = c ∗ . Lemma 3 If J, K are compact subintervals such that K ⊆ f ( J ) , then there is a compact subinterval L ⊆ J such that f ( L ) = K . Proof. Let K = [ a, b ] , c = sup { x ∈ J : f ( x ) = a } . If f ( x ) = b for some x ∈ J with x > c , let d be the least and take L = [ c, d ] . Otherwise f ( x ) = b for some x ∈ J with x < c . 5 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Proof of Sharkovski’s Theorem Lemma 2 If J is a compact subinterval such that J ⊆ f ( J ) , then f has a fixed point in J . Proof. If J = [ a, b ] then for some c, d ∈ J we have f ( c ) = a , f ( d ) = b . Thus f ( c ) ≤ c , f ( d ) ≥ d , and by the intermediate value theorem ∃ c ∗ ∈ [ a, b ] f ( c ∗ ) = c ∗ . Lemma 3 If J, K are compact subintervals such that K ⊆ f ( J ) , then there is a compact subinterval L ⊆ J such that f ( L ) = K . Proof. Let K = [ a, b ] , c = sup { x ∈ J : f ( x ) = a } . If f ( x ) = b for some x ∈ J with x > c , let d be the least and take L = [ c, d ] . Otherwise f ( x ) = b for some x ∈ J with x < c . Let c ′ be the greatest and let d ′ ≤ c be the least x ∈ J with x > c ′ for which f ( x ) = a . Then we can take L = [ c ′ , d ′ ] . 5 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Proof of Sharkovski’s Theorem Lemma 4 If J 0 , J 1 , ..., J m are compact subintervals such that J k ⊆ f ( J k − 1 ) (1 ≤ k ≤ m ) , then there is a compact subinterval L ⊆ J 0 such that f m ( L ) = J m and f k ( L ) ⊆ J k (1 ≤ k < m ) . 6 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
Proof of Sharkovski’s Theorem Lemma 4 If J 0 , J 1 , ..., J m are compact subintervals such that J k ⊆ f ( J k − 1 ) (1 ≤ k ≤ m ) , then there is a compact subinterval L ⊆ J 0 such that f m ( L ) = J m and f k ( L ) ⊆ J k (1 ≤ k < m ) . If also J 0 ⊆ J m , then there exists a point y such that f m ( y ) = y and f k ( y ) ∈ J k (0 ≤ k < m ) . 6 Ugur G. Abdulla Topological Dynamics and Universality in Chaos I. Proof of Sharkovski’s Theo
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