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Topological Entropy of Compact Subsystems of Transitive Real Line Maps Martha cka joint work with: Dominik Kwietniak (UJ) North Bay, July 23, 2013 M. cka Topological Entropy of Compact Subsystems... Canovas-Rodriguez Entropy CR

  1. Topological Entropy of Compact Subsystems of Transitive Real Line Maps Martha Łącka joint work with: Dominik Kwietniak (UJ) North Bay, July 23, 2013 M. Łącka Topological Entropy of Compact Subsystems...

  2. Canovas-Rodriguez Entropy CR Definition of Entropy for non-compact spaces h CR ( f , X ) := sup � h ( f , K ) K ⊂ X − compact, invariant � | M. Łącka Topological Entropy of Compact Subsystems...

  3. Canovas-Rodriguez Entropy CR Definition of Entropy for non-compact spaces h CR ( f , X ) := sup � h ( f , K ) K ⊂ X − compact, invariant � | Question (CR) f ∈ F � =? inf � h CR ( f , X ) | M. Łącka Topological Entropy of Compact Subsystems...

  4. Canovas-Rodriguez Entropy CR Definition of Entropy for non-compact spaces h CR ( f , X ) := sup � h ( f , K ) K ⊂ X − compact, invariant � | Question (CR) f ∈ F � =? inf � h CR ( f , X ) | X = R M. Łącka Topological Entropy of Compact Subsystems...

  5. Canovas-Rodriguez Entropy CR Definition of Entropy for non-compact spaces h CR ( f , X ) := sup � h ( f , K ) K ⊂ X − compact, invariant � | Question (CR) f ∈ F � =? inf � h CR ( f , X ) | X = R F = { f : X → X , transitive , non − bitransitive , continuous } M. Łącka Topological Entropy of Compact Subsystems...

  6. Canovas-Rodriguez Entropy CR Definition of Entropy for non-compact spaces h CR ( f , X ) := sup � h ( f , K ) K ⊂ X − compact, invariant � | Question (CR) f ∈ F � =? inf � h CR ( f , X ) | X = R F = { f : X → X , transitive , non − bitransitive , continuous } F = { f : X → X , bitransitive , continuous } M. Łącka Topological Entropy of Compact Subsystems...

  7. Canovas-Rodriguez Entropy CR Definition of Entropy for non-compact spaces h CR ( f , X ) := sup � h ( f , K ) K ⊂ X − compact, invariant � | Question (CR) f ∈ F � =? inf � h CR ( f , X ) | X = R F = { f : X → X , transitive , non − bitransitive , continuous } F = { f : X → X , bitransitive , continuous } X = [ 0 , ∞ ) M. Łącka Topological Entropy of Compact Subsystems...

  8. Question (CR) f - transitive, non-bitransitive � =? inf � h ( f , [ 0 , ∞ )) | M. Łącka Topological Entropy of Compact Subsystems...

  9. Question (CR) f - transitive, non-bitransitive � =? inf � h ( f , [ 0 , ∞ )) | Theorem Let f be a transitive map of a real interval J. Then, exactly one of the following statements holds: 1 f 2 is transitive, 2 there exist intervals K , L ⊂ J, with K ∩ L = { c } and K ∪ L = J, such that c is the unique fixed point for f, f ( K ) = L and f ( L ) = K. M. Łącka Topological Entropy of Compact Subsystems...

  10. Question (CR) f - transitive, non-bitransitive � =? inf � h ( f , [ 0 , ∞ )) | Theorem Let f be a transitive map of a real interval J. Then, exactly one of the following statements holds: 1 f 2 is transitive, 2 there exist intervals K , L ⊂ J, with K ∩ L = { c } and K ∪ L = J, such that c is the unique fixed point for f, f ( K ) = L and f ( L ) = K. Corollary If f is a transitive map of a half-open interval, then f is bitransitive. M. Łącka Topological Entropy of Compact Subsystems...

  11. Horseshoes Let f be a map from a real interval L to R . M. Łącka Topological Entropy of Compact Subsystems...

  12. Horseshoes Let f be a map from a real interval L to R . Horseshoe An s-horseshoe for f is a compact interval J ⊂ L, and a collection C = { A 1 , . . . , A s } of s ≥ 2 nonempty compact intervals of J fulfilling the following two conditions: M. Łącka Topological Entropy of Compact Subsystems...

  13. Horseshoes Let f be a map from a real interval L to R . Horseshoe An s-horseshoe for f is a compact interval J ⊂ L, and a collection C = { A 1 , . . . , A s } of s ≥ 2 nonempty compact intervals of J fulfilling the following two conditions: the interiors of the sets from C are pairwise disjoint, 1 M. Łącka Topological Entropy of Compact Subsystems...

  14. Horseshoes Let f be a map from a real interval L to R . Horseshoe An s-horseshoe for f is a compact interval J ⊂ L, and a collection C = { A 1 , . . . , A s } of s ≥ 2 nonempty compact intervals of J fulfilling the following two conditions: the interiors of the sets from C are pairwise disjoint, 1 J ⊂ f ( A ) for every A ∈ C . 2 M. Łącka Topological Entropy of Compact Subsystems...

  15. Horseshoes Let f be a map from a real interval L to R . Horseshoe An s-horseshoe for f is a compact interval J ⊂ L, and a collection C = { A 1 , . . . , A s } of s ≥ 2 nonempty compact intervals of J fulfilling the following two conditions: the interiors of the sets from C are pairwise disjoint, 1 J ⊂ f ( A ) for every A ∈ C . 2 Tightness vs Looseness A horseshoe ( J , C ) is tight if J is the union of elements of C and f ( A ) = J for every A ∈ C . M. Łącka Topological Entropy of Compact Subsystems...

  16. Horseshoes Let f be a map from a real interval L to R . Horseshoe An s-horseshoe for f is a compact interval J ⊂ L, and a collection C = { A 1 , . . . , A s } of s ≥ 2 nonempty compact intervals of J fulfilling the following two conditions: the interiors of the sets from C are pairwise disjoint, 1 J ⊂ f ( A ) for every A ∈ C . 2 Tightness vs Looseness A horseshoe ( J , C ) is tight if J is the union of elements of C and f ( A ) = J for every A ∈ C . A horseshoe ( J , C ) is loose if the union of elements of C is a proper subset of J. M. Łącka Topological Entropy of Compact Subsystems...

  17. A horseshoe ( J , C ) is tight if J is the union of elements of C and f ( A ) = J for every A ∈ C . A horseshoe ( J , C ) is loose if the union of elements of C is a proper subset of J. M. Łącka Topological Entropy of Compact Subsystems...

  18. A horseshoe ( J , C ) is tight if J is the union of elements of C and f ( A ) = J for every A ∈ C . A horseshoe ( J , C ) is loose if the union of elements of C is a proper subset of J. M. Łącka Topological Entropy of Compact Subsystems...

  19. A horseshoe ( J , C ) is tight if J is the union of elements of C and f ( A ) = J for every A ∈ C . A horseshoe ( J , C ) is loose if the union of elements of C is a proper subset of J. M. Łącka Topological Entropy of Compact Subsystems...

  20. Horseshoes imply entropy Theorem If a transitive map f of a real interval L has a loose s-horseshoe then there exists a compact invariant subset K such that h CR ( f ) ≥ h ( f | K ) > log s. M. Łącka Topological Entropy of Compact Subsystems...

  21. f − bitransitive, continuous � =? inf � h ( f , [ 0 , ∞ )) | M. Łącka Topological Entropy of Compact Subsystems...

  22. f − bitransitive, continuous � =? inf � h ( f , [ 0 , ∞ )) | Theorem (DK, MŁ) If a map g from the half-open interval [ 0 , ∞ ) to itself is transitive, then g has a loose 3-horseshoe, hence h CR ( g ) > log 3. M. Łącka Topological Entropy of Compact Subsystems...

  23. Theorem (DK, MŁ) If a transitive map of the real line f has at least two fixed points, then f has a loose 2-horseshoe, hence h CR ( f ) > log 2. M. Łącka Topological Entropy of Compact Subsystems...

  24. Proof Theorem (DK, MŁ) If a transitive map of the real line f has at least two fixed points, then f has a loose 2-horseshoe, hence h CR ( f ) > log 2. M. Łącka Topological Entropy of Compact Subsystems...

  25. Proof Theorem (DK, MŁ) If a transitive map of the real line f has at least two fixed points, then f has a loose 2-horseshoe, hence h CR ( f ) > log 2. M. Łącka Topological Entropy of Compact Subsystems...

  26. Proof Theorem (DK, MŁ) If a transitive map of the real line f has at least two fixed points, then f has a loose 2-horseshoe, hence h CR ( f ) > log 2. M. Łącka Topological Entropy of Compact Subsystems...

  27. Proof Theorem (DK, MŁ) If a transitive map of the real line f has at least two fixed points, then f has a loose 2-horseshoe, hence h CR ( f ) > log 2. M. Łącka Topological Entropy of Compact Subsystems...

  28. Proof AK, KC form a loose 2-horseshoe for f. M. Łącka Topological Entropy of Compact Subsystems...

  29. Theorem (DK, MŁ) If a transitive map f of the real line has a unique fixed point, √ then f 2 has a loose 3-horseshoe, hence h CR ( f ) > log 3. M. Łącka Topological Entropy of Compact Subsystems...

  30. Theorem Let f be a transitive map of a real interval J. Then, exactly one of the following statements holds: M. Łącka Topological Entropy of Compact Subsystems...

  31. Theorem Let f be a transitive map of a real interval J. Then, exactly one of the following statements holds: 1 f 2 is transitive, M. Łącka Topological Entropy of Compact Subsystems...

  32. Theorem Let f be a transitive map of a real interval J. Then, exactly one of the following statements holds: 1 f 2 is transitive, 2 there exist intervals K , L ⊂ J, with K ∩ L = { c } and K ∪ L = J, such that c is the unique fixed point for f, f ( K ) = L and f ( L ) = K. M. Łącka Topological Entropy of Compact Subsystems...

  33. Proof M. Łącka Topological Entropy of Compact Subsystems...

  34. Proof Z − = ( −∞ , Z ] , Z + = [ Z , + ∞ ) M. Łącka Topological Entropy of Compact Subsystems...

  35. Proof Z − = ( −∞ , Z ] , Z + = [ Z , + ∞ ) Z + ⊂ f ( Z − ) , Z − ⊂ f ( Z + ) . M. Łącka Topological Entropy of Compact Subsystems...

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