Topological Entropy and Chaos

Topological Entropy and Li-Yorke Chaos

"Topological entropy of maps on the real line"

Let X be a Hausdorff topological space and let f:XX be a continuous self-map
on X. The pair (X,f) is called a dynamical system.
A subset KX is said to be invariant by f if f(K)K and it is strictly invariant by f if f(K)=K.

We summarize some properties of the topological entropy below.

Theorem 1. Let X and Y be two (metric) compact topological sets and let f:XX and g:YY be two continuous maps. Then the following properties are held:

(a) Let ϕ:XY be continuous such that gϕ=ϕf. Then:
(a1) If the map ϕ is injective, then h(f)h(g).
(a2) If the map ϕ is surjective, then h(f)h(g).
(a3) If the map ϕ is bijective, then h(f)=h(g).

(b) Suppose that X=i=1nXi, where Xi are compact and invariants by f. Then h(f)=max{h(f|xi)}

(c) For any integer n0 it is hold h(fn)=nh(f).

(d) Let f×g:X×YX×Y be defined by (f×g)(x,y)=(f(x),g(y)) for all (x,y) X×Y. Then h(f×g)=h(f)+h(g).

(e) If f is a homeomorphism, then h(f)=h(f1).

(f) Let φ:XY be a continuous surjective map such that φf=gφ. Then max{h(g),sup{h(f,φ1(y)):yY}}h(f)h(g)+sup{h(f,φ1(y)):yY}.

(g) If f:XY and g:YX are continuous, then h(fg)=h(gf).

(h) Let f:XY,g:YX be continuous and let F:X×YX×Y be defined by F(x,y)=(g(y),f(x)) for all (x,y)X×Y. Then

(1)h(F)=h(fg)=h(gf).

(i) If X=n0fn(X) then h(f)=h(f|X).

(j) h(f)=h(fΩ(f)) where xΩ(f) if for all neighborhood U of x there is n0 such that fn(U)U(Ω(f) is called non-wandering set of f).

A dynamical system (X,f) is called minimal if X does not contain any non-empty, proper,  closed $f$-invariant subset. 
In such a case we also say that the map f itself is minimal.
The following conditions are equivalent: 
-$(X,f)$ is minimal,
-every orbit is dense in $X$, 
-$\omega_f(x)=X$ for every $x\in X$. 

Definition of topological entropy on matric space

For continuous maps on a metric space (X,f) the topological entropy of f is defined by

(2)ent(f):=sup{h(f|K):KX,compact and invariant byf}.

By Theorem 1(i) we have

(3)ent(f)=sup{h(f|K):KK(X,f)}

where K(X,f) is the family of all the compact subsets of X which are strictly invariant by f. Notice that this definition makes sense when X is matric or simply a topological space.

Explanation/interpretation (3):

Any compact strictlyf-invariant set K determines uniquely a strictly f-invariant closed set K=n0fn(K)K(X,f) such that h(f|K)=h(f|K).
Conversely, any KK(X,f) we have K=K .So

(4){K=n0fn(K):K compact andfinvariant}={K:KK(X,f)}.

Therefore,

(5)ent(f)=sup{h(f|K):KX,Kcompact andfinvariant}(Th1(i))=sup{h(f|K):KX,Kcompact andfinvariant}=sup{h(f|K):KK(X,f)}

i.e., (3) holds.

Remark.

由上述定义, 为了计算f的拓扑熵,我们需要

  1. 找集合{K:Kcompact andfinvariant},
  2. 再对每一个K 计算h(f|K),
  3. 最后取它们的上确界得到ent(f).

值得注意的是:如果我们认为f|K是当xK, f|K(x)=f(x), 其它地
方, f|K没有定义. 那么有可能f没有紧不变集,如f(x)=μx(x1), μ>4, 定义在全实轴上, 如果K是紧且f-不变的, 那么K={0}. 按照上述定义计算ent(f)=0, 但是这个映射有Li-Yorke混沌. 所以这并不是人们期望的结果. 但在后面我们会看到有f|K的其它定义被给出,克服了这个困难.

We start by studying some properties of the new definition of topological entropy, which is summed up in the next result.

Theorem 2 Let X and Y be two matric spaces and let f:XX and g:YY be continuous.

(a) Let ϕ:XY be continuous map such that ϕf=gϕ. Then
(a1) If the map ϕ is injective then ent(f) ent(g).
(a2) If the map ϕ:XY is surjective and such that the set ϕ1(K) is compact for all compact KX, then ent(f) ent(g).
(a3) If the map ϕ is bijective then ent(f)=ent(g).

(b) ent(fn)=n ent(f).

(c) Let f:XY and g:YX then ent(fg)= ent (gf).

(d) Let F:X×YX×Y be defined by F(x,y)=(f(x),g(y)) then ent(F)= ent(f)+ent(g).

(e) Let f:XY and g:YX then F(x,y)=(g(y),f(x)):X×YX×Y satisfies ent(F)=ent(fg)=ent (gf)

(f) The conclusion that if X=i=1nXi with Xi compact f-invariant then ent(f)=max{h|Xi:i=1n} no holds in general.

(g) Let ϕ:XY be continuous, surjective map and ϕ1(K) is compact for any K compact in Y, then max{ent(g),sup{h(f,ϕ1(y)):yY}}h(f)ent(g)+sup{h(f,ϕ1(y)):yY}.

(h) Let f:XX be a homeomorphism then ent(f)=ent(f1).

(i) Let X=n0fn(X). Then ent(f)=ent(f|X).

(j) ent(f)=ent(f|Ω(f)), where xΩ(f) if for any neighborhood U of x there exist positive integer n such that Ufn(U).

Proof.

(a)

Since ϕf=gϕ, f(K)=K implies g(ϕ(K))=ϕ(K), i.e., {ϕ(K):KK(X,f)}K(Y,g).

(a1) If ϕ is injective then

(6)ent(f)=sup{h(f|K):KK(X,f)},(by Th1(a1))sup{h(g|ϕ(K)):KK(X,f)}{h(g|L):LK(Y,g)}=ent(g).

(a2) ϕf=gϕ implies ϕ1(K)K(X,f) for KK(Y,g).
If ϕ is surjective then

(7)ent(g)=sup{h(g|K):KK(X,g)},(by Th1(a2))sup{h(f|ϕ1(K)):KK(X,g)}sup{h(f|L):LK(X,f)}=ent(f).

(a3) It is immediate consequence of (a1) and (a2).

(b)

We prove ent(fn)=n ent (f). Since K(X,f)K(X,fn) and
h((f|K)n)=h(fn|K),
{(f|K)n=f|Kf|Kf|Kf|K=f|fn1(K)f|f(K)f|K=fn|K}

(8)nent(f)=sup{nh(f|K):KK(X,f)}=sup{h((f|K)n):KK(X,f)}=sup{h(fn|K):KK(X,f)}sup{h((f|K)n):KK(X,fn)}=ent(fn).

On the other hand, for any K(X,fn), we have Kn=i=0n1fi(K)K(X,f).
Hence,
{Kn:KK(X,fn)}K(X,f)
and

(9)nent(f)=nsup{h(f|K):KK(X,f)}nsup{h(f|Kn):KK(X,fn)}sup{h(fn|Kn=i=0n1fi(K)):KK(X,fn)}(byTh1(b))sup{h(fn|K):KK(X,fn)}=ent(fn).

(c)

We prove that if f:XY and g:YX are continuous then ent (fg)= ent (gf).

fg(K)=Kgf(g(K))=g(K)g(K)K(X,gf)

{g(K):KK(Y,fg)}K(X,gf)

(10)ent(fg)=sup{h(fg|K):KK(Y,fg)}(byTh1(g))=sup{h(gf|g(K)):KK(Y,fg)}sup{h(gf|L):LK(Y,fg)}=ent(gf).

By a symmetrical reasoning, we prove that ent(fg) ent(gf).

(d)

We prove that if F:X×YX×Y be defined by F(x,y)=(f(x),g(y)) then ent(F) = ent(f)+ent(g).

Let Π1(x,y)=x,Π2(x,y)=y. For any K, we have KK1×K2, where K1=Π1(K),K2=Π2(K).

alt text

For K1K(X,f),K2K(Y,g), we have K1×K2K(X×Y,F).
Therefore,

{K1×K2:K1K(X,f),K2K(Y,g)K(X×Y,F)}
and

(11)ent(F)=sup{h(F|K):KK(X×Y,F)}sup{h(F|K1×K2):K1K(X,f),K2K(Y,g)}(byTh1(d))=sup{h(f|K1)+h(g|K2):K1K(X,f),K2K(Y,g)}=ent(f)+ent(g).

On the other hand, let i:KK1×K2 is inclusion, i.e., i(x)=x, an injective.

F|K1×K2:K1×K2,F|K:KK,i:KK1×K2,injective,

and

F|K1×K2i=iF|K.

By Th1(a2)

h(F|K)h(F|K1×K2).

Then

(12)ent(F)=sup{h(F|K):KK(X×Y,F)}sup{h(F|K1×K2):K1K(X,f),K2K(X,g)}(byTh1(d))=sup{h(f|K1)+h(f|K2):K1K(X,f),K2K(X,g)}sup{h(f|K1):K1K(X,f)}+sup{h(f|K2):K2K(Y,g)}=ent(f)+ent(g).

(e)

We prove that if f:XY and g:YX are continuous then F(x,y)=(g(y),f(x)):X×YX×Y satisfies

ent(F)=ent(fg)=ent(gf)

(13)2ent(F)=ent(F2)=(gf(x),fg(y))(by what just proved (d))=ent(gf)+ent(fg)=2ent(gf)=2ent(fg)

Thus
ent(F)=ent(gf)=ent(fg).

(f)

We prove that the conclusion that if X=i=1nXi with Xi compact f-invariant then ent
(f)=max{h|Xi:i=1n} no holds in general.

Let f:XX be a minimal homeomorphism defined on a compact set with positive topological entropy.

For example:

alt text

Let xX. X1=FullOrbf(x)={fn(x):xZ} and X2=XX1.
It is clear that both sets are invariant by f and that they do not have compact invariant subsets (in case they have compact invariant subsets, the map f is not minimal).
Therefore, ent(f|Xi)=0, for i=1,2, while ent(f)=h(f)>0.

(g)

Let ϕ:XY be continuous, surjective map and ϕ1(K) is compact for any K
compact in Y, then

max{ent(g),sup{h(f,ϕ1(y)):yY}}h(f)ent(g)+sup{h(f,ϕ1(y)):yY}.

(h)

We prove that if f:XX be a homeomorphism then

ent(f)=ent(f1).

Since f(K)=K implies f1(K)=K,

(14)ent(f)=sup{h(f|K):KK(X,f)}(byTh1(e))=sup{h(f1|K):KK(X,f)}(byTh1(e))=sup{h(f1|K):KK(X,f1)}(byTh1(e))=ent(f1).

(i)

We prove that if let X=n0fn(X). Then

ent(f)=ent(f|X).

For any KK(X,f), we have KX. Therefore,

f(K)=Kf|X(K)=K

ent(f)=sup{h(f|K):KK(X,f)}=sup{h(fX|K):KK(X,f|X)}=ent(f|X)

(j)

We prove that ent(f)=ent(f|Ω(f)),
where xΩ(f) if for any neighborhood U of x
there exist positive integer n such that Ufn(U).

f|Ω(f):Ω(f)Ω(f),f:XX,i:Ω(f)X(inclusion map,injective)

satisfy the exchange diagram

alt text

Since i is injective, by (a1) in the theorem,

ent(f|Ω(f))ent(f).

We prove the converse inequality. For any KK(X,f), KΩ(f) is fΩ(f)-invariant.
So

{KΩ(f):KK(X,f)}{K:KisfΩ(f)invariant}.

Moreover,

(15)ent(f)=sup{h(f|K):KK(X,f)}(byTh1(j))=sup{h((f|K)|Ω(f)):KK(X,f)}=sup{h(f|I):I=KΩ(f)(f|Ω(f)invariant),KK(X,f)}

({I=KΩ(f) is f|Ω(f)invariant, KK(X,f)}{JΩ(f):J is f|Ω(f)invariant})

(16)(15)sup{h(f|J):JΩ(f) is f|Ω(f)invariant}=sup{h((f|Ω(f))|J):JΩ(f) is f|Ω(f)invariant}=ent(f|Ω(f)).

参考文献:

 R.L. Adler, A.G. Konheim, M.H. McAndrew, Topological entropy, *Trans. Amer. Math. Soc.* **114** (1965) 309–319.

 - R. Bowen, Entropy for group endomorphism and homogeneous spaces, *Trans. Amer. Math. Soc.* **153** (1971) 401–414.


    - J. S. Cánovas, J. M. Rodríguez, Topological entropy of maps on the real line, *Topology Appl.*, **153**(2005), 735--746.


    - T-Y. Li,  J. A. Yorke, Period three implies chaos, *Amer. Math. Monthly*, **82**(1975), 985--992.


 - J. Milnor, W. Thurston, On iterated maps of the interval, *Dynamical Systems*,  Lecture Notes in Mathematics, vol. **1342**, ed. A. Dold and B. Eckmann, Springer, Berlin, 1988: 465--563.


 - M. Rees, A minimal positive entropy homeomorphism of the 2-torus, *J. London Math. Soc.*, **23** (1981) 537–550.

 (https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms/s2-23.3.537)

 - X. Ye, D-function of a minimal set and an extension of Sharkovskiĭ's theorem to minimal sets, *Ergodic Theory Dynam. Systems,*  **12**(1992), 365-376.

 (https://www.cambridge.org/core/services/aop-cambridge-core/content/view/7737952BD34F742FC1118C8353DB3CE0/S0143385700006817a.pdf/d-function-of-a-minimal-set-and-an-extension-of-sharkovskiis-theorem-to-minimal-sets.pdf)


 - http://www.scholarpedia.org/article/Minimal_dynamical_systems#Minimality_of_a_map_and_its_iterates (minimal system)
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