Download Topological Dynamics: Minimality, Entropy and Chaos.

Topological Dynamics:
Minimality, Entropy and Chaos.
Sergiy Kolyada
Institute of Mathematics, NAS of Ukraine, Kyiv
Zentrum Mathematik, Technische Universität München,
John-von-Neumann Lecture, 2013
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Throughout this part of the lecture (X , f ) denotes a topological
dynamical system, where X is a (compact) metric (Hausdorff) space with
a metric d and f : X → X is a continuous map.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Throughout this part of the lecture (X , f ) denotes a topological
dynamical system, where X is a (compact) metric (Hausdorff) space with
a metric d and f : X → X is a continuous map.
A subset M of X is a (topologically) minimal set if M is closed,
non-empty and invariant (i.e., f (M) ⊆ M) and if M has no proper subset
with these three properties.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Throughout this part of the lecture (X , f ) denotes a topological
dynamical system, where X is a (compact) metric (Hausdorff) space with
a metric d and f : X → X is a continuous map.
A subset M of X is a (topologically) minimal set if M is closed,
non-empty and invariant (i.e., f (M) ⊆ M) and if M has no proper subset
with these three properties. A dynamical system (X ; f ) is called minimal
if the set X is minimal.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Throughout this part of the lecture (X , f ) denotes a topological
dynamical system, where X is a (compact) metric (Hausdorff) space with
a metric d and f : X → X is a continuous map.
A subset M of X is a (topologically) minimal set if M is closed,
non-empty and invariant (i.e., f (M) ⊆ M) and if M has no proper subset
with these three properties. A dynamical system (X ; f ) is called minimal
if the set X is minimal. In such a case we also say that the map f itself is
minimal.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Throughout this part of the lecture (X , f ) denotes a topological
dynamical system, where X is a (compact) metric (Hausdorff) space with
a metric d and f : X → X is a continuous map.
A subset M of X is a (topologically) minimal set if M is closed,
non-empty and invariant (i.e., f (M) ⊆ M) and if M has no proper subset
with these three properties. A dynamical system (X ; f ) is called minimal
if the set X is minimal. In such a case we also say that the map f itself is
minimal.
Minimal systems are natural generalizations of periodic orbits, and they
are analogues of ergodic measures in topological dynamics. They were
defined by G. D. Birkhoff in 1912.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Given a dynamical system (X , f ) , a set M ⊆ X is called a minimal set if
it is non-empty, closed and invariant and if no proper subset of M has
these three properties. So, M ⊆ X is a minimal set if and only if
(M, f |M ) is a minimal system. A system (X , f ) is minimal if and only if X
is a minimal set in (X , f ) .
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Given a dynamical system (X , f ) , a set M ⊆ X is called a minimal set if
it is non-empty, closed and invariant and if no proper subset of M has
these three properties. So, M ⊆ X is a minimal set if and only if
(M, f |M ) is a minimal system. A system (X , f ) is minimal if and only if X
is a minimal set in (X , f ) .
The basic fact discovered by G. D. Birkhoff is that in any compact
system (X , f ) there are minimal sets. This follows immediately from the
Zorn’s lemma.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Given a dynamical system (X , f ) , a set M ⊆ X is called a minimal set if
it is non-empty, closed and invariant and if no proper subset of M has
these three properties. So, M ⊆ X is a minimal set if and only if
(M, f |M ) is a minimal system. A system (X , f ) is minimal if and only if X
is a minimal set in (X , f ) .
The basic fact discovered by G. D. Birkhoff is that in any compact
system (X , f ) there are minimal sets. This follows immediately from the
Zorn’s lemma.
The following conditions are equivalent:
(X , f ) is minimal,
every orbit is dense in X ,
ωf (x) = X for every x ∈ X .
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Given a dynamical system (X , f ) , a set M ⊆ X is called a minimal set if
it is non-empty, closed and invariant and if no proper subset of M has
these three properties. So, M ⊆ X is a minimal set if and only if
(M, f |M ) is a minimal system. A system (X , f ) is minimal if and only if X
is a minimal set in (X , f ) .
The basic fact discovered by G. D. Birkhoff is that in any compact
system (X , f ) there are minimal sets. This follows immediately from the
Zorn’s lemma.
The following conditions are equivalent:
(X , f ) is minimal,
every orbit is dense in X ,
ωf (x) = X for every x ∈ X .
A minimal map f is necessarily surjective if X is assumed to be Hausdorff
and compact.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Since any orbit closure is invariant, we get that any compact orbit closure
contains a minimal set.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Since any orbit closure is invariant, we get that any compact orbit closure
contains a minimal set. In particular, this is how compact minimal sets
may appear in non-compact spaces.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Since any orbit closure is invariant, we get that any compact orbit closure
contains a minimal set. In particular, this is how compact minimal sets
may appear in non-compact spaces.
Two minimal sets in (X , f ) either are disjoint or coincide.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Since any orbit closure is invariant, we get that any compact orbit closure
contains a minimal set. In particular, this is how compact minimal sets
may appear in non-compact spaces.
Two minimal sets in (X , f ) either are disjoint or coincide. A minimal set
M is strongly f -invariant, i.e. f (M) = M, provided it is compact
Hausdorff.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Introduction
Since any orbit closure is invariant, we get that any compact orbit closure
contains a minimal set. In particular, this is how compact minimal sets
may appear in non-compact spaces.
Two minimal sets in (X , f ) either are disjoint or coincide. A minimal set
M is strongly f -invariant, i.e. f (M) = M, provided it is compact
Hausdorff.
Some other equivalent definitions of a minimal system:
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Other equivalent definitions of a minimal system
For a compact metric space X and a continuous map f : X → X the
following are equivalent:
(X , f ) is minimal.
f (X ) = X and every backward orbit of every point in X is dense (by
a ’backward orbit’ of x0 ∈ X we mean any set {x0 , x1 , ..., xn , ...} with
f (xi+1 ) = xi for i ≥ 0).
The only closed subsets E of X with f (E ) ⊇ E are ∅ and X .
For every opene set U ⊆ X , there exists N ∈ N such that
S
N
−n
(U) = X .
n=0 f
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
More examples of minimal maps
Example 2.1
Consider a homeomorphism of the 2-torus, S : T → T , of the form
S(x, y ) = (x + α, y + β) , where 1, α, β ∈ R are rationally independent
and + : R/Z × R → R/Z is defined in the obvious way. Then S is
minimal (and ergodic with respect to Lebesgue measure).
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
More examples of minimal maps
Example 2.1
Consider a homeomorphism of the 2-torus, S : T → T , of the form
S(x, y ) = (x + α, y + β) , where 1, α, β ∈ R are rationally independent
and + : R/Z × R → R/Z is defined in the obvious way. Then S is
minimal (and ergodic with respect to Lebesgue measure).
Example 2.2
Let K = (kn )n>0 be a sequence of integers kn ≥ 2 . Let ΣK be the set of
all one-sided infinite sequences (in )n≥1 for which 0 ≤ in ≤ kn − 1 .
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
More examples of minimal maps
Example 2.1
Consider a homeomorphism of the 2-torus, S : T → T , of the form
S(x, y ) = (x + α, y + β) , where 1, α, β ∈ R are rationally independent
and + : R/Z × R → R/Z is defined in the obvious way. Then S is
minimal (and ergodic with respect to Lebesgue measure).
Example 2.2
Let K = (kn )n>0 be a sequence of integers kn ≥ 2 . Let ΣK be the set of
all one-sided infinite sequences (in )n≥1 for which 0 ≤ in ≤ kn − 1 . Think
of these sequences as ’integers’ in multibase notation, the base of the nth
digit in being kn . With the natural (product) topology, ΣK is
homeomorphic to the Cantor set.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
More examples of minimal maps
Example 2.1
Consider a homeomorphism of the 2-torus, S : T → T , of the form
S(x, y ) = (x + α, y + β) , where 1, α, β ∈ R are rationally independent
and + : R/Z × R → R/Z is defined in the obvious way. Then S is
minimal (and ergodic with respect to Lebesgue measure).
Example 2.2
Let K = (kn )n>0 be a sequence of integers kn ≥ 2 . Let ΣK be the set of
all one-sided infinite sequences (in )n≥1 for which 0 ≤ in ≤ kn − 1 . Think
of these sequences as ’integers’ in multibase notation, the base of the nth
digit in being kn . With the natural (product) topology, ΣK is
homeomorphic to the Cantor set. Define a map αK : ΣK → ΣK which
informally may be described as ’add 1 and carry’ where the addition is
performed at the leftmost term i1 and the carry proceeds to the right in
multibase notation. Then αK is a minimal homeomorphism and is called
a ”generalized adding machine” or an ’odometer’.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
More examples of minimal maps
Example 2.1
Consider a homeomorphism of the 2-torus, S : T → T , of the form
S(x, y ) = (x + α, y + β) , where 1, α, β ∈ R are rationally independent
and + : R/Z × R → R/Z is defined in the obvious way. Then S is
minimal (and ergodic with respect to Lebesgue measure).
Example 2.2
Let K = (kn )n>0 be a sequence of integers kn ≥ 2 . Let ΣK be the set of
all one-sided infinite sequences (in )n≥1 for which 0 ≤ in ≤ kn − 1 . Think
of these sequences as ’integers’ in multibase notation, the base of the nth
digit in being kn . With the natural (product) topology, ΣK is
homeomorphic to the Cantor set. Define a map αK : ΣK → ΣK which
informally may be described as ’add 1 and carry’ where the addition is
performed at the leftmost term i1 and the carry proceeds to the right in
multibase notation. Then αK is a minimal homeomorphism and is called
a ”generalized adding machine” or an ’odometer’.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
On the equivalent formulations of the definition
As a general reference see e.g. T. Downarowicz, Survey of odometers and
Toeplitz flows, Algebraic and topological dynamics, 7-37, Contemp.
Math., 385, Amer. Math. Soc., Providence, RI, 2005.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
A continuous map f : X → Y between topological spaces is called
irreducible if it is surjective and f (A) 6= Y for every proper closed subset
A ⊂ X.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
A continuous map f : X → Y between topological spaces is called
irreducible if it is surjective and f (A) 6= Y for every proper closed subset
A ⊂ X.
A set B ⊆ X is said to be a redundant open set for a map f : X → Y if
B is opene and f (B) ⊆ f (X \ B) (i.e., its removal from the domain of f
does not change the image of f ).
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
A continuous map f : X → Y between topological spaces is called
irreducible if it is surjective and f (A) 6= Y for every proper closed subset
A ⊂ X.
A set B ⊆ X is said to be a redundant open set for a map f : X → Y if
B is opene and f (B) ⊆ f (X \ B) (i.e., its removal from the domain of f
does not change the image of f ).
By taking B = X \ A one can have the following equivalent definition – a
continuous map f : X → Y between topological spaces is irreducible if it
is surjective and has no open redundant sets.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
A continuous map f : X → Y between topological spaces is called
irreducible if it is surjective and f (A) 6= Y for every proper closed subset
A ⊂ X.
A set B ⊆ X is said to be a redundant open set for a map f : X → Y if
B is opene and f (B) ⊆ f (X \ B) (i.e., its removal from the domain of f
does not change the image of f ).
By taking B = X \ A one can have the following equivalent definition – a
continuous map f : X → Y between topological spaces is irreducible if it
is surjective and has no open redundant sets.
A map f : X → Y is called almost open if it sends opene sets to sets
with non-empty interior (the terminology is not unified – instead of
almost open some authors say semi-open, feebly open, somewhat open or
quasi-interior).
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
A continuous map f : X → Y between topological spaces is called
irreducible if it is surjective and f (A) 6= Y for every proper closed subset
A ⊂ X.
A set B ⊆ X is said to be a redundant open set for a map f : X → Y if
B is opene and f (B) ⊆ f (X \ B) (i.e., its removal from the domain of f
does not change the image of f ).
By taking B = X \ A one can have the following equivalent definition – a
continuous map f : X → Y between topological spaces is irreducible if it
is surjective and has no open redundant sets.
A map f : X → Y is called almost open if it sends opene sets to sets
with non-empty interior (the terminology is not unified – instead of
almost open some authors say semi-open, feebly open, somewhat open or
quasi-interior). It is easy to see that a map is almost open if and only if
the inverse image of every dense subset is dense.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Theorem 2.1
Let X be a compact Hausdorff space and f : X → X continuous. Then f
is minimal =⇒ f is irreducible
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Theorem 2.1
Let X be a compact Hausdorff space and f : X → X continuous. Then f
is minimal =⇒ f is irreducible =⇒ f is almost open
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Theorem 2.1
Let X be a compact Hausdorff space and f : X → X continuous. Then f
is minimal =⇒ f is irreducible =⇒ f is almost open and if f is minimal
then the following are equivalent: f is open ⇐⇒ f is injective ⇐⇒ f is a
homeomorphism.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis.
1. f is not irreducible =⇒ f is not minimal.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis.
1. f is not irreducible =⇒ f is not minimal.
If f is not surjective then f is not minimal. So, let f (X ) = X .
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis.
1. f is not irreducible =⇒ f is not minimal.
If f is not surjective then f is notTminimal. So, let
f (X ) = X . Denote f |A
T∞
∞
by g and consider the set M := k=0 f −k (A) = k=0 g −k (A).
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis.
1. f is not irreducible =⇒ f is not minimal.
If f is not surjective then f is notTminimal. So, let
f (X ) = X . Denote f |A
T∞
∞
by g and consider the set M := k=0 f −k (A) = k=0 g −k (A).
We have X = g (A) ⊇ A.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis.
1. f is not irreducible =⇒ f is not minimal.
If f is not surjective then f is notTminimal. So, let
f (X ) = X . Denote f |A
T∞
∞
by g and consider the set M := k=0 f −k (A) = k=0 g −k (A).
We have X = g (A) ⊇ A. Hence the set M, being the intersection of a
nested sequence of nonempty compact sets, is nonempty.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis.
1. f is not irreducible =⇒ f is not minimal.
If f is not surjective then f is notTminimal. So, let
f (X ) = X . Denote f |A
T∞
∞
by g and consider the set M := k=0 f −k (A) = k=0 g −k (A).
We have X = g (A) ⊇ A. Hence the set M, being the intersection of a
nested sequence of nonempty compact sets, is nonempty.
But the f -trajectory of any point from M does not intersect the
nonempty open set X \ A. Hence f is not minimal.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis (Cont.)
2. f is irreducible =⇒ f is almost open.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis (Cont.)
2. f is irreducible =⇒ f is almost open.
Since f is continuous, for any A ⊆ X we have f (A) ⊆ f (A). This
inclusion is in fact an equality, because X is compact and thus f is
closed.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis (Cont.)
2. f is irreducible =⇒ f is almost open.
Since f is continuous, for any A ⊆ X we have f (A) ⊆ f (A). This
inclusion is in fact an equality, because X is compact and thus f is
closed. Now let D ⊆ X is dense in X .
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis (Cont.)
2. f is irreducible =⇒ f is almost open.
Since f is continuous, for any A ⊆ X we have f (A) ⊆ f (A). This
inclusion is in fact an equality, because X is compact and thus f is
closed. Now let D ⊆ X is dense in X . Since f is surjective, we have
f (f −1 (D)) = D and also
f (f −1 (D)) = f (f −1 (D)) = D = X .
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis (Cont.)
2. f is irreducible =⇒ f is almost open.
Since f is continuous, for any A ⊆ X we have f (A) ⊆ f (A). This
inclusion is in fact an equality, because X is compact and thus f is
closed. Now let D ⊆ X is dense in X . Since f is surjective, we have
f (f −1 (D)) = D and also
f (f −1 (D)) = f (f −1 (D)) = D = X .
Since f is irreducible, this implies f −1 (D) = X .
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis (Cont.)
2. f is irreducible =⇒ f is almost open.
Since f is continuous, for any A ⊆ X we have f (A) ⊆ f (A). This
inclusion is in fact an equality, because X is compact and thus f is
closed. Now let D ⊆ X is dense in X . Since f is surjective, we have
f (f −1 (D)) = D and also
f (f −1 (D)) = f (f −1 (D)) = D = X .
Since f is irreducible, this implies f −1 (D) = X .
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis (Cont.)
3. Let us also show that f is irreducible and open =⇒ f is a
homeomorphism.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis (Cont.)
3. Let us also show that f is irreducible and open =⇒ f is a
homeomorphism.
Suppose f is not a homeomorphism. It means that there are a 6= b with
f (a) = f (b) =: c.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis (Cont.)
3. Let us also show that f is irreducible and open =⇒ f is a
homeomorphism.
Suppose f is not a homeomorphism. It means that there are a 6= b with
f (a) = f (b) =: c. Take disjoint open neighbourhoods Ua of a and Ub of
b.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis (Cont.)
3. Let us also show that f is irreducible and open =⇒ f is a
homeomorphism.
Suppose f is not a homeomorphism. It means that there are a 6= b with
f (a) = f (b) =: c. Take disjoint open neighbourhoods Ua of a and Ub of
b. Since f is open, f (Ua ) is open and contains c.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis (Cont.)
3. Let us also show that f is irreducible and open =⇒ f is a
homeomorphism.
Suppose f is not a homeomorphism. It means that there are a 6= b with
f (a) = f (b) =: c. Take disjoint open neighbourhoods Ua of a and Ub of
b. Since f is open, f (Ua ) is open and contains c. Since f is continuous,
there is an open neighbourhood Vb of b such that Vb ⊆ Ub and
f (Vb ) ⊆ f (Ua ).
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis (Cont.)
3. Let us also show that f is irreducible and open =⇒ f is a
homeomorphism.
Suppose f is not a homeomorphism. It means that there are a 6= b with
f (a) = f (b) =: c. Take disjoint open neighbourhoods Ua of a and Ub of
b. Since f is open, f (Ua ) is open and contains c. Since f is continuous,
there is an open neighbourhood Vb of b such that Vb ⊆ Ub and
f (Vb ) ⊆ f (Ua ).
Then f (Vb ) ⊆ f (X \ Vb ).
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Topological properties of minimal maps
Beweis (Cont.)
3. Let us also show that f is irreducible and open =⇒ f is a
homeomorphism.
Suppose f is not a homeomorphism. It means that there are a 6= b with
f (a) = f (b) =: c. Take disjoint open neighbourhoods Ua of a and Ub of
b. Since f is open, f (Ua ) is open and contains c. Since f is continuous,
there is an open neighbourhood Vb of b such that Vb ⊆ Ub and
f (Vb ) ⊆ f (Ua ).
Then f (Vb ) ⊆ f (X \ Vb ). So, B is a redundant open set, hence f is not
irreducible.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Another interesting properties
Another interesting property of minimal maps in compact Hausdorff
spaces is the following one:
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Another interesting properties
Another interesting property of minimal maps in compact Hausdorff
spaces is the following one:
Proposition 2.1
For
non-empty open set U ⊆ X , there exists N ∈ N such that
SN every
n
(U)
=X .
f
n=0
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Another interesting properties
Another interesting property of minimal maps in compact Hausdorff
spaces is the following one:
Proposition 2.1
For
non-empty open set U ⊆ X , there exists N ∈ N such that
SN every
n
(U)
=X .
f
n=0
Though minimal maps need not be invertible, in some aspects they
behave like homeomorphisms.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Another interesting properties
Another interesting property of minimal maps in compact Hausdorff
spaces is the following one:
Proposition 2.1
For
non-empty open set U ⊆ X , there exists N ∈ N such that
SN every
n
(U)
=X .
f
n=0
Though minimal maps need not be invertible, in some aspects they
behave like homeomorphisms. For instance, if f is a minimal map in a
compact Hausdorff space X and A ⊆ X then both f (A) and f −1 (A)
share some topological properties with the set A – namely the ones which
describe how large a set is.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Another interesting properties
Another interesting property of minimal maps in compact Hausdorff
spaces is the following one:
Proposition 2.1
For
non-empty open set U ⊆ X , there exists N ∈ N such that
SN every
n
(U)
=X .
f
n=0
Though minimal maps need not be invertible, in some aspects they
behave like homeomorphisms. For instance, if f is a minimal map in a
compact Hausdorff space X and A ⊆ X then both f (A) and f −1 (A)
share some topological properties with the set A – namely the ones which
describe how large a set is. In fact, the following claims hold.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Another interesting properties
Theorem 2.2
If A is nowhere dense (dense, of 1st category, of 2nd category,
residual) then both f (A) and f −1 (A) are nowhere dense (dense, of
1st category, of 2nd category, residual), respectively.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Another interesting properties
Theorem 2.2
If A is nowhere dense (dense, of 1st category, of 2nd category,
residual) then both f (A) and f −1 (A) are nowhere dense (dense, of
1st category, of 2nd category, residual), respectively.
If A has nonempty interior (has the Baire property) then both f (A)
and f −1 (A) have nonempty interior (have the Baire property),
respectively.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Another interesting properties
Theorem 2.2
If A is nowhere dense (dense, of 1st category, of 2nd category,
residual) then both f (A) and f −1 (A) are nowhere dense (dense, of
1st category, of 2nd category, residual), respectively.
If A has nonempty interior (has the Baire property) then both f (A)
and f −1 (A) have nonempty interior (have the Baire property),
respectively.
If A is open then there is an open set B ⊆ X such that
B ⊆ f (A) ⊆ B (here B may not be unique; the largest of such sets
is always the interior of f (A)).
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Almost one-to-one maps
The fact that in some aspects minimal maps behave like
homeomorphisms will be less surprising in the light of the following result.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Almost one-to-one maps
The fact that in some aspects minimal maps behave like
homeomorphisms will be less surprising in the light of the following result.
Theorem 2.3
Let X be a compact metric space and f : X → X be minimal. Then
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Almost one-to-one maps
The fact that in some aspects minimal maps behave like
homeomorphisms will be less surprising in the light of the following result.
Theorem 2.3
Let X be a compact metric space and f : X → X be minimal. Then
f is almost one-to-one, which means that the set
{x ∈ X : #f −1 (x) = 1} is a Gδ -dense set in X .
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Almost one-to-one maps
The fact that in some aspects minimal maps behave like
homeomorphisms will be less surprising in the light of the following result.
Theorem 2.3
Let X be a compact metric space and f : X → X be minimal. Then
f is almost one-to-one, which means that the set
{x ∈ X : #f −1 (x) = 1} is a Gδ -dense set in X .
there exists a residual set Y ⊆ X such that f (Y ) = Y and f |Y is a
minimal homeomorphism.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Almost one-to-one maps
The fact that in some aspects minimal maps behave like
homeomorphisms will be less surprising in the light of the following result.
Theorem 2.3
Let X be a compact metric space and f : X → X be minimal. Then
f is almost one-to-one, which means that the set
{x ∈ X : #f −1 (x) = 1} is a Gδ -dense set in X .
there exists a residual set Y ⊆ X such that f (Y ) = Y and f |Y is a
minimal homeomorphism. Moreover, (f |Y )−1 is also a minimal
homeomorphism and while f |Y is uniformly continuous, (f |Y )−1 is
uniformly continuous only in the case when f is a homeomorphism
(then one can take Y = X ).
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
Almost one-to-one maps
The fact that in some aspects minimal maps behave like
homeomorphisms will be less surprising in the light of the following result.
Theorem 2.3
Let X be a compact metric space and f : X → X be minimal. Then
f is almost one-to-one, which means that the set
{x ∈ X : #f −1 (x) = 1} is a Gδ -dense set in X .
there exists a residual set Y ⊆ X such that f (Y ) = Y and f |Y is a
minimal homeomorphism. Moreover, (f |Y )−1 is also a minimal
homeomorphism and while f |Y is uniformly continuous, (f |Y )−1 is
uniformly continuous only in the case when f is a homeomorphism
(then one can take Y = X ).
For proofs of these results see – S. Kolyada, . Snoha and S. Trofimchuk,
Noninvertible minimal maps, Fund. Math. 168(2001), 141-163.
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
General references
1. S. Kolyada, . Snoha and S. Trofimchuk, Noninvertible minimal maps,
Fund. Math. 168(2001), 141-163.
2. S. Kolyada and L. Snoha, Minimal dynamical systems, Scholarpedia,
4(11):5803 (2009).
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.
Minimal maps
2. Minimal Maps
HOMEWORK
Exercise 2.1
Let S be the unit circle. It is known (see J. Auslander and J. A. Yorke,
Interval maps, factors of maps, and chaos, Tohoku Math. Journ.
32(1980), 177-188) that any continuous minimal map on S is
topologically conjugate to an irrational rotation. In particular, can not be
noninvertible. Prove it (by showing that any minimal map on the circle S
has no redundant open sets).
Sergiy Kolyada
Topological Dynamics: Minimality, Entropy and Chaos.