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Almost Totally Minimal Systems; Periodicity in Hyperspaces Mate Puljiz joint with L. Fernndez & C. Good University of Birmingham 31 st Summer Conference on Topology and its Applications Leicester, 5 th August 2016 Mate Puljiz ( University of

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Almost Totally Minimal Systems; Periodicity in Hyperspaces

Mate Puljiz

joint with L. Fernández & C. Good

University of Birmingham

31st Summer Conference on Topology and its Applications Leicester, 5th August 2016

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 1 / 13

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Co-authors

Chris Good Leobardo Fernández

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 2 / 13

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Co-authors

Chris Good Leobardo Fernández

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 2 / 13

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Almost minimal systems

Definition

Let X be a compact metric space and T : X → X a homeomorphism. We say that (X, T) is almost minimal if: (1) There exists a unique fixed point x0 ∈ X s.t. T (x0) = x0 (2) The full orbit of every other point y ∈ X \ {x0} is dense

{T i(y) | i ∈ Z} = X

N.B. (X \ {x0}, T |X\{x0}) is a well defined minimal non-compact system.

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 3 / 13

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Almost minimal systems

Definition

Let X be a compact metric space and T : X → X a homeomorphism. We say that (X, T) is almost minimal if: (1) There exists a unique fixed point x0 ∈ X s.t. T (x0) = x0 (2) The full orbit of every other point y ∈ X \ {x0} is dense

{T i(y) | i ∈ Z} = X

N.B. (X \ {x0}, T |X\{x0}) is a well defined minimal non-compact system. Do such systems exist?

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 3 / 13

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Almost minimal systems

Definition

Let X be a compact metric space and T : X → X a homeomorphism. We say that (X, T) is almost minimal if: (1) There exists a unique fixed point x0 ∈ X s.t. T (x0) = x0 (2) The full orbit of every other point y ∈ X \ {x0} is dense

{T i(y) | i ∈ Z} = X

N.B. (X \ {x0}, T |X\{x0}) is a well defined minimal non-compact system. Do such systems exist? The trivial example How about non-trivial?

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 3 / 13

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Almost minimal systems

Definition

Let X be a compact metric space and T : X → X a homeomorphism. We say that (X, T) is almost minimal if: (1) There exists a unique fixed point x0 ∈ X s.t. T (x0) = x0 (2) The full orbit of every other point y ∈ X \ {x0} is dense

{T i(y) | i ∈ Z} = X

N.B. (X \ {x0}, T |X\{x0}) is a well defined minimal non-compact system. Do such systems exist? The trivial example How about non-trivial? (Z ∪ {∞}, +1)

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 3 / 13

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Almost minimal systems

Definition

Let X be a compact metric space and T : X → X a homeomorphism. We say that (X, T) is almost minimal if: (1) There exists a unique fixed point x0 ∈ X s.t. T (x0) = x0 (2) The full orbit of every other point y ∈ X \ {x0} is dense

{T i(y) | i ∈ Z} = X

N.B. (X \ {x0}, T |X\{x0}) is a well defined minimal non-compact system. Do such systems exist? The trivial example How about non-trivial? (Z ∪ {∞}, +1)

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 3 / 13

Historical note

(1992) R. Herman, I. Putnam, C. Skau — relate K-theory and topological dynamics (2001) A. Danilenko — extends their theory to non-compact setting by looking at almost minimal systems

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ATMs

Definition

(X, T) is almost totally minimal if (X, T k) is almost minimal for every k ∈ N.

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 4 / 13

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ATMs

Definition

(X, T) is almost totally minimal if (X, T k) is almost minimal for every k ∈ N. I.e. (1) There exists a unique fixed point x0 ∈ X s.t. T (x0) = x0 (2) The full T k-orbit of every other point y ∈ X \ {x0} is dense for every k ∈ N

{T ki(y) | i ∈ Z} = X

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 4 / 13

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ATMs

Definition

(X, T) is almost totally minimal if (X, T k) is almost minimal for every k ∈ N. I.e. (1) There exists a unique fixed point x0 ∈ X s.t. T (x0) = x0 (2) The full T k-orbit of every other point y ∈ X \ {x0} is dense for every k ∈ N

{T ki(y) | i ∈ Z} = X

Do these exist?

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 4 / 13

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ATMs

Definition

(X, T) is almost totally minimal if (X, T k) is almost minimal for every k ∈ N. I.e. (1) There exists a unique fixed point x0 ∈ X s.t. T (x0) = x0 (2) The full T k-orbit of every other point y ∈ X \ {x0} is dense for every k ∈ N

{T ki(y) | i ∈ Z} = X

Do these exist? The trivial system How about non-trivial?

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 4 / 13

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ATMs

Definition

(X, T) is almost totally minimal if (X, T k) is almost minimal for every k ∈ N. I.e. (1) There exists a unique fixed point x0 ∈ X s.t. T (x0) = x0 (2) The full T k-orbit of every other point y ∈ X \ {x0} is dense for every k ∈ N

{T ki(y) | i ∈ Z} = X

Do these exist? The trivial system How about non-trivial? X has to be perfect, and hence uncountable (why?) X cannot be an interval (why?)

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 4 / 13

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ATMs

Definition

(X, T) is almost totally minimal if (X, T k) is almost minimal for every k ∈ N. I.e. (1) There exists a unique fixed point x0 ∈ X s.t. T (x0) = x0 (2) The full T k-orbit of every other point y ∈ X \ {x0} is dense for every k ∈ N

{T ki(y) | i ∈ Z} = X

Do these exist? The trivial system How about non-trivial? X has to be perfect, and hence uncountable (why?) X cannot be an interval (why?) How about the Cantor Set?

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 4 / 13

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— Yes!

We use graph covers devised by: (2006) J.-M. Gambaudo, M. Martens (2008) E. Akin, E. Glassner, B. Weiss (2014) T. Shimomura

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 5 / 13

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— Yes!

We use graph covers devised by: (2006) J.-M. Gambaudo, M. Martens (2008) E. Akin, E. Glassner, B. Weiss (2014) T. Shimomura

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 5 / 13

∗ G0 :

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— Yes!

We use graph covers devised by: (2006) J.-M. Gambaudo, M. Martens (2008) E. Akin, E. Glassner, B. Weiss (2014) T. Shimomura

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 5 / 13

∗ G0 : 0∗ G1 : 1∗

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— Yes!

We use graph covers devised by: (2006) J.-M. Gambaudo, M. Martens (2008) E. Akin, E. Glassner, B. Weiss (2014) T. Shimomura

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 5 / 13

∗ G0 : 0∗ G1 : 1∗ 00∗ G2 : 01∗ 10∗ 11∗

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— Yes!

We use graph covers devised by: (2006) J.-M. Gambaudo, M. Martens (2008) E. Akin, E. Glassner, B. Weiss (2014) T. Shimomura

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 5 / 13

∗ G0 : 0∗ G1 : 1∗ 00∗ G2 : 01∗ 10∗ 11∗ . . .

{0, 1}N

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— Yes!

We use graph covers devised by: (2006) J.-M. Gambaudo, M. Martens (2008) E. Akin, E. Glassner, B. Weiss (2014) T. Shimomura

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 5 / 13

∗ G0 : 0∗ G1 : 1∗ 00∗ G2 : 01∗ 10∗ 11∗ . . . ({0, 1}N, σ)

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— Yes!

We use graph covers devised by: (2006) J.-M. Gambaudo, M. Martens (2008) E. Akin, E. Glassner, B. Weiss (2014) T. Shimomura

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 5 / 13

∗ G0 : 0∗ G1 : 1∗ 00∗ G2 : 01∗ 10∗ 11∗ . . . ({0, 1}N, σ)

Compare

(1963) J. Mioduszewski

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The construction

G0 :

  • Mate Puljiz (University of Birmingham)

ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 6 / 13

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The construction

G0 : G1 :

  • Mate Puljiz (University of Birmingham)

ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 6 / 13

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The construction

G0 : G1 :

  • Mate Puljiz (University of Birmingham)

ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 6 / 13

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The construction

G0 : G1 : 1!

  • Mate Puljiz (University of Birmingham)

ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 6 / 13

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The construction

G0 : G1 : 1! G2 :

  • Mate Puljiz (University of Birmingham)

ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 6 / 13

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The construction

G0 : G1 : 1! G2 :

  • Mate Puljiz (University of Birmingham)

ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 6 / 13

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The construction

G0 : G1 : 1! G2 : 2!

  • Mate Puljiz (University of Birmingham)

ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 6 / 13

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The construction

G0 : G1 : 1! G2 : 2!

  • Mate Puljiz (University of Birmingham)

ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 6 / 13

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Any questions so far?

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 7 / 13

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An Application

Theorem

Let (Y, S) be a 0-dimensional minimal system. There exist a system ( ˆ Y, ˆ S) on the Cantor set ˆ Y such that: (1) (Y, S) dynamically embeds into ( ˆ Y, ˆ S) as a nowhere dense set, (2) Every full ˆ Sk-orbit of any point y ∈ ˆ Y \ Y is dense in ˆ Y for every k ∈ N. N.B. The nowhere density in (1) is redundant as it follows from (2).

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 8 / 13

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An Application

Theorem

Let (Y, S) be a 0-dimensional minimal system. There exist a system ( ˆ Y, ˆ S) on the Cantor set ˆ Y such that: (1) (Y, S) dynamically embeds into ( ˆ Y, ˆ S) as a nowhere dense set, (2) Every full ˆ Sk-orbit of any point y ∈ ˆ Y \ Y is dense in ˆ Y for every k ∈ N. N.B. The nowhere density in (1) is redundant as it follows from (2).

Proof.

(X, T) is the ATM constructed before with the fixed point x0 A ⊔ B = X is a separation s.t. x0 is in A Fix an arbitrary point y0 ∈ Y so that (x0, y0) acts as an origin On X × Y, consider ˆ S = π ◦ (T × S) ◦ π where π: (x, y) →      (x, y), if x ∈ A (x, y0), if x ∈ B Let ˆ Y ⊂ X × Y be minimal (w.r.t. ⊆) closed ˆ S-invariant set containing B × {y0} Then ( ˆ Y, ˆ S| ˆ

Y ) works!

  • Mate Puljiz (University of Birmingham)

ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 8 / 13

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A question remains

What about the non-minimal case?

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 9 / 13

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A question remains

What about the non-minimal case? — It is true for some trivial non-minimal systems (if they are a power of a minimal system)

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 9 / 13

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Periodicity in Hyperspaces

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 10 / 13

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We start with a problem

X is compact metric T : X → X is continuous 2X is the set of all compact subsets of X with the Hausdorff distance dH(F, G) = inf{ε ≥ 0 | F ⊆ Gε and G ⊆ Fε} 2T : 2X → 2X is given by 2T (F) = T (F) Per(T) = {k ∈ N | ∃ periodic point for T with fundamental period k} Per(2T) as above but for (2X, 2T)

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 11 / 13

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We start with a problem

X is compact metric T : X → X is continuous 2X is the set of all compact subsets of X with the Hausdorff distance dH(F, G) = inf{ε ≥ 0 | F ⊆ Gε and G ⊆ Fε} 2T : 2X → 2X is given by 2T (F) = T (F) Per(T) = {k ∈ N | ∃ periodic point for T with fundamental period k} Per(2T) as above but for (2X, 2T)

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 11 / 13

Šarkovs′ki˘ ı’s theorem (1964)

For any interval map T: 3 ∈ Per(T) =⇒ Per(T) = N.

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We start with a problem

X is compact metric T : X → X is continuous 2X is the set of all compact subsets of X with the Hausdorff distance dH(F, G) = inf{ε ≥ 0 | F ⊆ Gε and G ⊆ Fε} 2T : 2X → 2X is given by 2T (F) = T (F) Per(T) = {k ∈ N | ∃ periodic point for T with fundamental period k} Per(2T) as above but for (2X, 2T) Characterise all admissible pairs (Per(T), Per(2T))?

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 11 / 13

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We start with a problem

X is compact metric T : X → X is continuous 2X is the set of all compact subsets of X with the Hausdorff distance dH(F, G) = inf{ε ≥ 0 | F ⊆ Gε and G ⊆ Fε} 2T : 2X → 2X is given by 2T (F) = T (F) Per(T) = {k ∈ N | ∃ periodic point for T with fundamental period k} Per(2T) as above but for (2X, 2T) Characterise all admissible pairs (Per(T), Per(2T))? Is there a system (X, T) for which Per(2T) = {1, 2, 3}?

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 11 / 13

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Per(2T) = {1, 2, 3}

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 12 / 13

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SLIDE 41

Per(2T) = {1, 2, 3}

z w

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 12 / 13

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Per(2T) = {1, 2, 3}

z w

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 12 / 13

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Per(2T) = {1, 2, 3}

z w z1 w1 z2 w2 z3 w3 ↑ ↑

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 12 / 13

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Per(2T) = {1, 2, 3}

z w z1 w1 z2 w2 z3 w3 ↑ ↑ z w

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 12 / 13

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Thank you for your attention!

Almost Totally Minimal Systems; Periodicity in Hyperspaces

Mate Puljiz

joint with L. Fernández & C. Good

University of Birmingham

31st Summer Conference on Topology and its Applications Leicester, 5th August 2016

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 13 / 13

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References I

[1] Ethan Akin, Eli Glasner, and Benjamin Weiss, Generically there is but one self homeomorphism of the Cantor set, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3613–3630. MR2386239 [2] Louis Block and Ethan M. Coven, Maps of the interval with every point chain recurrent, Proc. Amer. Math.

  • Soc. 98 (1986), no. 3, 513–515. MR857952 (87j:54059)

[3] Alexandre I. Danilenko, Strong orbit equivalence of locally compact Cantor minimal systems, Internat. J. Math. :2 (2001), no. 1, 113–123. MR1812067 (2002j:37016) [4] Jean-Marc Gambaudo and Marco Martens, Algebraic topology for minimal Cantor sets, Ann. Henri Poincaré 7 (2006), no. 3, 423–446. MR2226743 (2006m:37007) [5] Richard H. Herman, Ian F. Putnam, and Christian F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), no. 6, 827–864. MR1194074 (94f:46096) [6] J. Mioduszewski, Mappings of inverse limits, Colloq. Math. :0 (1963), 39–44. MR0166762 [7] A. N. Sharkovski˘ ı, Coexistence of cycles of a continuous map of the line into itself, Proceedings of the Conference “Thirty Years after Sharkovski˘ ı’s Theorem: New Perspectives” (Murcia, 1994), 1995,

  • pp. 1263–1273. Translated from the Russian [Ukrain. Mat. Zh. :6 (1964), no. 1, 61–71; MR0159905 (28 #3121)]

by J. Tolosa. MR1361914 (96j:58058) [8] Takashi Shimomura, Special homeomorphisms and approximation for Cantor systems, Topology Appl. :6: (2014), 178–195. MR3132360

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 14 / 13

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Fact sheet I

Trivial periods coming from (X, f ) can be characterised as [D(Per(f ))] =

∞

  • l=1
  • [d1, . . . , dl] | di|mi ∈ Per(f ) for 1 ≤ i ≤ l
  • But there could be more!

Theorem

Given a continuous map f : X → X, the set of periods Per(2f ) of the induced map

  • n 2X contains [D(Per(f ))] and is closed under taking prime power divisors.

Theorem

Let f be a continuous map of a compact interval to itself. Then Per(2f ) is either {1}

  • r {1, 2} or N.

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 15 / 13

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Fact sheet II

Theorem (Shimomura)

Any surjective dynamics over a 0-dimensional system is topologically conjugate to G∞ = lim ← − − Gi = G0

ϕ0

← − − G1

ϕ1

← − − G2

ϕ2

← − − · · · where Gi are finite directed graphs (each vertex has at least one in- and out-edge), and bonding maps ϕi : Gi+1 → Gi are graph covers (vertex map that respects edges and +-directional).

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 16 / 13

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A different proof I

A ⊔ B = C , x0 the fixed point of ATM (C , T) is in A ˆ X = A × {0, 1} ⊔ B × {0} f : ˆ X → ˆ X by f (x, i) =      (T (x), 1), if i = 0 and x ∈ T −1(A), (T (x), 0), otherwise. N (x, B) = min{k ∈ N0 | T −k(x) ∈ B} time elapsed since x last visited B U = {x ∈ C | N (x, B) < ∞} dense and open X = {(x, N (x, B) mod 2) | x ∈ U} ⊂ ˆ X unique minimal closed f m-invariant set containing B × {0} for any m ∈ N

Lemma

For any z ∈ X \ π−1(x0) and any m ∈ N we have

{f mk(z) | k ∈ Z} = X

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 17 / 13

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A different proof II

l = (x0, 0) and r = (x0, 1) form a 2-cycle in (X, f ) Z = X × {0, 1, 2}/∼ a quotient space obtained by gluing L = {(l, i) | i = 0, 1, 2} together and likewise R = {(r, i) | i = 0, 1, 2} g : Z → Z, g(x, i) = (f (x), i + 1 mod 3), well-defined

{L} → {R} is a 2-cycle in 2Z

X × {0} → X × {1} → X × {2} is a 3-cycle in 2Z No 6-cycle in 2Z exists! Otherwise let S be it; ∃z = (z1, i) ∈ S other than L or R

{g6k(z) | k ∈ Z} ⊂ S {f 6k(z1) | k ∈ Z} × {i} = X × {i} ⊂ S

Hence S = X × F/∼ for some F ⊆ {0, 1, 2}

  • Mate Puljiz (University of Birmingham)

ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 18 / 13

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SLIDE 51

Example – 2-adic odometer

T : {0, 1}N → {0, 1}N T (x1, x2, . . . ) =      (1, x2, . . . ), if x1 = 0, (0, T (x2, x3, . . . )), o/w.

1 1 1

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 19 / 13

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Example – 2-adic odometer

T : {0, 1}N → {0, 1}N T (x1, x2, . . . ) =      (1, x2, . . . ), if x1 = 0, (0, T (x2, x3, . . . )), o/w.

1 1 1

G0 : G1 : G2 : G3 :

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 19 / 13

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SLIDE 53

Bratteli-Vershik representation

L0 L1 R1 L2 R2 L3 R3 e1

1

e2

1

e3

1

f 1

1

f 1

2

f 1

3

f 2

1

f 2

2

f 2

3

f 2

4

f 2

5

f 2

6

f 2

7

Mate Puljiz (University of Birmingham) ATM systems & Periodicity in Hyperspaces Leicester, 5th August 2016 20 / 13