Topological Entropy of Compact Subsystems of Transitive Real Line - - PowerPoint PPT Presentation

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Topological Entropy of Compact Subsystems of Transitive Real Line - - PowerPoint PPT Presentation

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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

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Topological Entropy of Compact Subsystems

  • f Transitive Real Line Maps

Martha Łącka joint work with: Dominik Kwietniak (UJ) North Bay, July 23, 2013

  • M. Łącka

Topological Entropy of Compact Subsystems...

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Canovas-Rodriguez Entropy

CR Definition of Entropy for non-compact spaces hCR(f, X) := sup h(f, K) | K ⊂ X − compact, invariant

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Topological Entropy of Compact Subsystems...

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Canovas-Rodriguez Entropy

CR Definition of Entropy for non-compact spaces hCR(f, X) := sup h(f, K) | K ⊂ X − compact, invariant Question (CR) inf hCR(f, X) | f ∈ F =?

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Topological Entropy of Compact Subsystems...

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Canovas-Rodriguez Entropy

CR Definition of Entropy for non-compact spaces hCR(f, X) := sup h(f, K) | K ⊂ X − compact, invariant Question (CR) inf hCR(f, X) | f ∈ F =? X = R

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Topological Entropy of Compact Subsystems...

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Canovas-Rodriguez Entropy

CR Definition of Entropy for non-compact spaces hCR(f, X) := sup h(f, K) | K ⊂ X − compact, invariant Question (CR) inf hCR(f, X) | f ∈ F =? X = R F = {f : X → X, transitive, non − bitransitive, continuous}

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Topological Entropy of Compact Subsystems...

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Canovas-Rodriguez Entropy

CR Definition of Entropy for non-compact spaces hCR(f, X) := sup h(f, K) | K ⊂ X − compact, invariant Question (CR) inf hCR(f, X) | f ∈ F =? X = R F = {f : X → X, transitive, non − bitransitive, continuous} F = {f : X → X, bitransitive, continuous}

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Canovas-Rodriguez Entropy

CR Definition of Entropy for non-compact spaces hCR(f, X) := sup h(f, K) | K ⊂ X − compact, invariant Question (CR) inf hCR(f, X) | f ∈ F =? X = R F = {f : X → X, transitive, non − bitransitive, continuous} F = {f : X → X, bitransitive, continuous} X = [0, ∞)

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Question (CR) inf h(f, [0, ∞)) | f - transitive, non-bitransitive =?

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Question (CR) inf h(f, [0, ∞)) | f - transitive, non-bitransitive =? Theorem Let f be a transitive map of a real interval J. Then, exactly one

  • f the following statements holds:

1 f2 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.

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Question (CR) inf h(f, [0, ∞)) | f - transitive, non-bitransitive =? Theorem Let f be a transitive map of a real interval J. Then, exactly one

  • f the following statements holds:

1 f2 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.

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Horseshoes

Let f be a map from a real interval L to R.

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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 = {A1, . . . , As} of s ≥ 2 nonempty compact intervals

  • f J fulfilling the following two conditions:
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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 = {A1, . . . , As} of s ≥ 2 nonempty compact intervals

  • f J fulfilling the following two conditions:

1

the interiors of the sets from C are pairwise disjoint,

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Topological Entropy of Compact Subsystems...

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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 = {A1, . . . , As} of s ≥ 2 nonempty compact intervals

  • f J fulfilling the following two conditions:

1

the interiors of the sets from C are pairwise disjoint,

2

J ⊂ f(A) for every A ∈ C.

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Topological Entropy of Compact Subsystems...

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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 = {A1, . . . , As} of s ≥ 2 nonempty compact intervals

  • f J fulfilling the following two conditions:

1

the interiors of the sets from C are pairwise disjoint,

2

J ⊂ f(A) for every A ∈ C. 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.

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Topological Entropy of Compact Subsystems...

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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 = {A1, . . . , As} of s ≥ 2 nonempty compact intervals

  • f J fulfilling the following two conditions:

1

the interiors of the sets from C are pairwise disjoint,

2

J ⊂ f(A) for every A ∈ C. 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.

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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.

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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.

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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.

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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 hCR(f) ≥ h(f|K) > log s.

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inf h(f, [0, ∞)) | f − bitransitive, continuous =?

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inf h(f, [0, ∞)) | f − bitransitive, continuous =? 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 hCR(g) > log 3.

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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 hCR(f) > log 2.

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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 hCR(f) > log 2.

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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 hCR(f) > log 2.

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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 hCR(f) > log 2.

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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 hCR(f) > log 2.

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Proof

AK, KC form a loose 2-horseshoe for f.

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Theorem (DK, MŁ) If a transitive map f of the real line has a unique fixed point, then f2 has a loose 3-horseshoe, hence hCR(f) > log √ 3.

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Theorem Let f be a transitive map of a real interval J. Then, exactly one

  • f the following statements holds:
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Theorem Let f be a transitive map of a real interval J. Then, exactly one

  • f the following statements holds:

1 f2 is transitive,

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Theorem Let f be a transitive map of a real interval J. Then, exactly one

  • f the following statements holds:

1 f2 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.

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Proof

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Proof

Z− = (−∞, Z], Z+ = [Z, +∞)

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Proof

Z− = (−∞, Z], Z+ = [Z, +∞) Z+ ⊂ f(Z−), Z− ⊂ f(Z+).

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Proof

Z− = (−∞, Z], Z+ = [Z, +∞) Z+ ⊂ f(Z−), Z− ⊂ f(Z+).

1 f(Z−) = Z+,

f(Z+) = Z−,

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Proof

Z− = (−∞, Z], Z+ = [Z, +∞) Z+ ⊂ f(Z−), Z− ⊂ f(Z+).

1 f(Z−) = Z+,

f(Z+) = Z−,

2 Z+ f(Z−) ( ⇐⇒ f is bitransitive).

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Proof

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Proof

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Proof

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Proof

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Proof

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Proof

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Proof

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[Z, S], [S, B], [B, T] form a loose 3−horseshoe for f2.

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inf hCR(f, R) | f : X → X bitransitive = log √ 3

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inf hCR(f, R) | f : X → X transitive, non-bitransitive = log √ 3

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Bibliography

1 Ll. Alsed`

a, J. Llibre, and M. Misiurewicz, „Combinatorial dynamics and entropy in dimension one”, second ed., World Scientific, River Edge, NJ, 2000.

2 J.S. C´

anovas, J.M. Rodr´ ıguez, „Topological entropy of maps on the real line”, Topology Appl. 153 (2005), no. 5-6, 735–746. MR 2201485 (2006i:37086)

3 D. Kwietniak, M. Ubik

„Topological entropy of compact subsystems of transitive real line maps”, Volume 28, Issue 1, 2013, p. 62-75

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