Chaos and semiconjugacy arguments

Transkrypt

Title
Piotr Oprocha
Departamento de Matemáticas
Universidad de Murcia, Murcia, Spain
and
Faculty of Applied Mathematics
AGH University of Science and Technology, Kraków, Poland
Będlewo, Poland, June 2009
Piotr Oprocha (UMU & AGH)
Będlewo
1/8
Factor maps
1
Factor map
1
2
2
Some possible properties of π:
1
2
3
4
5
3
(X , f ), (Y , g ) - continuous self-maps of compact metric spaces
π : X → Y - onto, π ◦ f = g ◦ π
π is a homeomorphism (some special
situations...)
π is almost one to one, i.e. Y1 = y : #π −1 ({y }) = 1 is residual
(horseshoes in dimension one, Toeplitz shifts, extensions over
co-cycles...)
one point y0 ∈ Y has unique preimage, possibly with some extra
knowledge about y0 (snap back repellers, isolating segments, covering
relations, horseshoes)
π is finite to one (horseshoes in dimension one, hyperbolic maps, ...)
...
Some possible Y :
1
2
3
4
full shift on n ≥ 2 symbols
shift of finite type
minimal system (e.g. Odometer, rotation, ...)
other well studied spaces/maps
Będlewo
2/8
Factor maps
1
Factor map
1
2
2
1
2
3
4
5
3
π : X → Y - onto, π ◦ f = g ◦ π
situations...)
co-cycles...)
...
Some possible Y :
1
2
3
4
Będlewo
2/8
Factor maps
1
Factor map
1
2
2
1
2
3
4
5
3
π : X → Y - onto, π ◦ f = g ◦ π
situations...)
co-cycles...)
...
Some possible Y :
1
2
3
4
Będlewo
2/8
Factor maps
1
Factor map
1
2
2
1
2
3
4
5
3
π : X → Y - onto, π ◦ f = g ◦ π
situations...)
co-cycles...)
...
Some possible Y :
1
2
3
4
Będlewo
2/8
Factor maps
1
Factor map
1
2
2
1
2
3
4
5
3
π : X → Y - onto, π ◦ f = g ◦ π
situations...)
co-cycles...)
...
Some possible Y :
1
2
3
4
Będlewo
2/8
Factor maps
1
Factor map
1
2
2
1
2
3
4
5
3
π : X → Y - onto, π ◦ f = g ◦ π
situations...)
co-cycles...)
...
Some possible Y :
1
2
3
4
Będlewo
2/8
Factor maps and entropy
π : (X , f ) → (Y , g )
1
Topological entropy: htop (f ) ∈ [0, +∞],
2
htop (f ) > 0 =⇒ chaos,
Positive topological entropy:
3
1
2
4
htop (f ) ≥ htop (g ).
if π is at most k-to-1 for some k > 0 then htop (f ) = htop (g ).
Chaos in the sense of Li and Yorke:
1
there is an uncountable set S ⊂ X such that for any x 6= y , x, y ∈ S:
lim inf d(f n (x), f n (y )) = 0
n→∞
2
,
lim sup d(f n (x), f n (y )) > 0
(> ε).
n→∞
In htop (f ) > 0 then f is LY-chaotic.
Blanchard, Glasner, Kolyada, Maass, J. Reine Angew. Math. 547 (2002) 5168
Będlewo
3/8
π : (X , f ) → (Y , g )
1
2
3
1
2
4
1
n→∞
2
,
(> ε).
n→∞
Będlewo
3/8
π : (X , f ) → (Y , g )
1
2
3
1
2
4
1
n→∞
2
,
(> ε).
n→∞
Będlewo
3/8
π : (X , f ) → (Y , g )
1
2
3
1
2
4
1
n→∞
2
,
(> ε).
n→∞
Będlewo
3/8
Extensions of L-Y definition
π : (X , f ) → (Y , g )
1
distributionally chaotic pair:
1
2
3
4
(n)
Φxy (t) = n1 # i : d(f i (x), f i (y )) < t
(n)
Φxy (t) = lim inf n→∞ Φxy (t)
(n)
Φ∗xy (t) = lim supn→∞ Φxy (t)
a pair is DC if
,0 ≤ i < n ,
Φ∗xy (t) = 1, for all t > 0,
Φxy (s) = 0 for some s > 0.
2
ω-chaotic pair
1
2
3
3
ωf (x) \ ωf (y ) is uncountable,
ωf (x) ∩ ωf (y ) 6= ∅ and
ωf (x) \ Per (f ) 6= ∅.
The above properties are independent of PTE and of each other...
Będlewo
4/8
π : (X , f ) → (Y , g )
1
1
2
3
4
(n)
Φxy (t) = n1 # i : d(f i (x), f i (y )) < t
(n)
(n)
a pair is DC if
,0 ≤ i < n ,
Φ∗xy (t) = 1, for all t > 0,
2
ω-chaotic pair
1
2
3
3
ωf (x) ∩ ωf (y ) 6= ∅ and
ωf (x) \ Per (f ) 6= ∅.
Będlewo
4/8
π : (X , f ) → (Y , g )
1
1
2
3
4
(n)
Φxy (t) = n1 # i : d(f i (x), f i (y )) < t
(n)
(n)
a pair is DC if
,0 ≤ i < n ,
Φ∗xy (t) = 1, for all t > 0,
2
ω-chaotic pair
1
2
3
3
ωf (x) ∩ ωf (y ) 6= ∅ and
ωf (x) \ Per (f ) 6= ∅.
Będlewo
4/8
How to get π? A cookbook. . .
π : (Λ, f |Λ ) → (Σ+
2 , σ)
1
Fix two disjoint closed sets N0 , N1 .
2
For every sequence α = {a0 , a1 , . . . , an−1 }, ai ∈ {0, 1}
construct pα (periodic ?)
such that f j (pα ) ∈ Naj(mod n) ,
Λ = f j (pα ) : α ∈ {0, 1}+ , j ∈ N ,
3
4
Assign π(f j (pα )) = σ j (αα . . .) and extend continuously on Λ
Będlewo
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π : (Λ, f |Λ ) → (Σ+
2 , σ)
1
2
Λ = f j (pα ) : α ∈ {0, 1}+ , j ∈ N ,
3
4
Będlewo
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π : (Λ, f |Λ ) → (Σ+
2 , σ)
1
2
Λ = f j (pα ) : α ∈ {0, 1}+ , j ∈ N ,
3
4
Będlewo
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π : (Λ, f |Λ ) → (Σ+
2 , σ)
1
2
Λ = f j (pα ) : α ∈ {0, 1}+ , j ∈ N ,
3
4
Będlewo
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One-to-one covering
π : (Λ, f p |Λ ) → (Σ+
2 , σ)
1
z ∈ X is a repelling fixed point for f in U if
U
⊂ f (U)
T∞
−n
(U) = {z}
n=0 f
2
z is a snap-back repeller for f in U if
z is a repelling fixed point
there is y ∈ U \ {z} such that f m (y ) = z for some m > 0 and f m is
open at y .
(Boyarski, Góra, Lioubimov, Nonlinear Analysis, 43 (2001) 591–604)
(Marotto, J. Math. Anal. Appl. 63 (1978) 199–223)
Będlewo
6/8
A little more on π...
π : (X , f ) → (Σ+
2 , σ)
1
if there is y ∈ Σ+
2 such that:
Orb+ (y ) 6= Σ+
2
#Φ−1 ({y }) = 1 (2, more...?)
then f is distributionally chaotic
(joint work with P. Wilczyński)
2
+
+
if there is y ∈ Σ+
2 (Orb (y ) 6= Σ2 )such that:
π −1 ({y }) is at most countable
or
S
Ω = x∈π−1 ({y }) ωf (x) contains at most countably many minimal
subsets
then f is ω-chaotic
(joint work with M. Lampart)
Będlewo
7/8
Problems with thick fibers...
π : (Λ, f |Λ ) → (Σ+
2 , σ)
Λ = f j (pα ) : α ∈ {0, 1}+ , j ∈ N
1
is there x ∈ Λ such that periodic points are dense in Orb+ (x)?
(chaos in the sense of Devaney)
Będlewo
8/8

Chaos and semiconjugacy arguments

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