Different Statistical Future of Dynamical Orbits Yiwei Dong∗ and Xueting Tian† School of Mathematical Sciences, Fudan University Shanghai 200433, People’s Republic of China ∗Email:[email protected]; †Email:[email protected] 7 Abstract 1 0 For expanding or hyperbolic dynamical systems, we use upper and lower natural density and 2 Banachdensitytodividedynamicalorbitsintoseveraldifferentlevelsets. Meanwhile,non-recurrence n a andBirkhoffaveragesareconsideredandweobtainedsimultaneouslevelsetsbymixingthemtogether. J By studyingthe topological entropy via multifractal analysis, we reveal the complexity of each level 8 set. In this process we generalize entropy-densepropertyand saturated property to strong ones. ] S Contents D . h 1 Introduction 2 t a 1.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 m 1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 [ 1.2.1 Symbolic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1 1.2.2 Smooth dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 v 0 1 2 Preliminaries 6 9 2.1 Notions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1 0 2.2 The Space of Borel Probability Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. 2.2.1 Metric Compatible with the Weak∗ Topology . . . . . . . . . . . . . . . . . . . . . 7 0 2.2.2 The Space of Invariant Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7 1 2.3 TopologicalEntropy and Metric Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 : 2.3.1 Topological Entropy for Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . 9 v i 2.3.2 Topological Entropy for Noncompact Set . . . . . . . . . . . . . . . . . . . . . . . 10 X 2.3.3 Metric Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 r a 2.3.4 Characterizing Metric Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Pseudo-orbitTracing Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4.1 Shadowing Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4.2 Limit-shadowing and s-Limit-shadowing . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.3 Specification and Almost Specification Properties . . . . . . . . . . . . . . . . . . . 13 2.5 Uniform Separation Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Density 15 Keywordsandphrases: Minimality;Non-recurrence;Irregularpoints orpointswithhistoricbehavior;ω-limitset. AMSReview: 37D20;37C50;37B20;37B40; 37C45. 1 4 Entropy-dense Properties 19 4.1 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Star-entropy-dense and Strong-entropy-denseProperties . . . . . . . . . . . . . . . . . . . 23 4.3 Basic-entropy-denseand Strong-basic-entropy-denseProperties . . . . . . . . . . . . . . . 25 4.4 Minimal-entropy-dense Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 Saturated Set and Saturated Properties 28 5.1 Star and Locally-star Saturated Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2 Nonrecurrently-star-saturatedProperty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.3 Non-transitive Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6 Multi-fractal Analysis 36 6.1 Useful facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.2 Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 6.3 Auxiliary Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.4 Multi-fractal Analysis for the Irregular Set . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.5 Multifractal analysis for the Level Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7 Proof of Main Theorems 49 7.1 Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 7.2 Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 7.3 Proof of Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 8 Non-expansive case 58 8.1 Construction of Symbolic Factor Subsystem . . . . . . . . . . . . . . . . . . . . . . . . . 59 8.2 Proof of Theorem 8.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 9 Proof of Proposition 4.3 60 9.1 Notions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 9.2 Subshifts of Finite Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 9.3 Basic Lemmas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 9.4 Proof of proposition 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 1 Introduction 1.1 Main Results By a topological dynamical system (TDS for short) (X,f), we mean a continuous map f acting on a compact metric space (X,d). Throughout this paper, we suppose (X,d) has infinitely many points and denote the sets of natural numbers, integer numbers and nonnegative numbers by N,Z,Z+ respectively. For any x∈X, the orbit of x is {fnx}∞ which we denote by orb(x,f). The ω-limit set of x is defined n=0 as ω (x):= {fkx}={y ∈X :∃ n →∞ s.t. fnix→y}. f i n≥1k≥n \ [ It is clear that ω (x) is a nonempty compact f-invariantset. f 2 Definition 1.1. We call x∈ X to be recurrent, if x∈ ω (x). A point x∈ X is called wandering, if there f is a neighborhood U of x such that the sets f−nU, n ≥ 0, are mutually disjoint. Otherwise, x is called non-wandering. A point x ∈ X is almost periodic, if for every open neighborhood U of x, there exists N ∈N such that fk(x)∈U for some k∈[n,n+N] and every n∈N. Denote the set of almost periodic points by AP(f), the set of non-wandering points by Ω(f), the set of recurrent points by Rec(f) and the set of non-recurrent points by NR(f). Now we recall some notions of density to describe different statistical structure. Definition 1.2. Let S ⊆N, define |S∩{0,1,··· ,n−1}| |S∩{0,1,··· ,n−1}| d¯(S):=limsup , d(S):=liminf , n→∞ n n→∞ n where |Y| denotes the cardinality of the set Y. These two concepts are called upper density and lower density of S, respectively. If d¯(S)=d(S)=d, we call S to have density of d. Define |S∩I| |S∩I| B∗(S):=limsup , B (S):=liminf , ∗ |I|→∞ |I| |I|→∞ |I| here I ⊆N is taken from finite continuous integer intervals. These two concepts are calledBanach upper density and Banach lower density of S, respectively. A set S ⊆ N is called syndetic, if there is N ∈ N such that for any n∈N, S∩{n,n+1,··· ,n+N}6=∅. Theseconceptsofdensityarebasicandhaveplayedimportantrolesinthefieldofdynamicalsystems, ergodic theory and number theory, etc. Let U,V ⊆X be two nonempty open subsets and x∈X. Define sets of visiting time N(U,V):={n≥1|U ∩f−n(V)6=∅} and N(x,U):={n≥1|fn(x)∈U}. Definition 1.3. For x ∈ X and ξ = d, d, B∗, B , a point y ∈ X is called x−ξ−accessible, if for any ∗ ǫ>0, N(x,V (y)) has positive density w.r.t. ξ, where V (x) denotes the ballcenteredatxwith radiusǫ. ǫ ǫ Let X (x):={y ∈X|y is x−ξ−accessible}. ξ For convenience, it is called ξ−ω-limit set of x. Note that X (x)⊆X (x)⊆X (x)⊆X (x)⊆ω (x). (1.1) B∗ d d B∗ f Moreover, they are all compact and invariant (with possibility that some sets are empty). Denote QAP(f) = {x ∈ X : ω (x) is minimal} and call x ∈ QAP((f) a quasi-minimal point. Moreover, f denote WQAP = {x ∈ X : C is minimal} and call x ∈ WQAP(f) a weak-quasi-minimal point. It is x clear that QAP(f)⊆WQAP(f). For a continuous function ϕ on X, define the ϕ−irregular set as n−1 1 I (f) := x∈X : lim ϕ(fix)diverges . ϕ ( n→∞n ) i=0 X Definition 1.4. A topologicaldynamical system (X,f) is called topologically expanding, if it is positively expansiveandsatisfiestheshadowingproperty. Ahomeomorphism(X,f)iscalledtopologicallyhyperbolic, if it is expansive and satisfies the shadowing property. 3 Let Υ := {x ∈ X|f| is transitive} and Υ := {x ∈ X|f| has the shadowing property}. T ωf(x) S ωf(x) Note that for any x∈X, ω (f) is internally chain transitive. If x∈Γ , then ω (x) is transitive. So one f S f has Γ ⊆Γ . Therefore, if we let Υ:=Υ ∩Υ =Γ , X can be written as the disjoint union S T S T S X =Υ⊔Υc ⊔(Υc ∩Υ ). T S T Theorem A. Suppose(X,f)is topologically expanding andtransitive (resp., (X,f)is ahomeomorphism that is topologically hyperbolic and transitive). Let ϕ be a continuous function on X. If I (f)6=∅, then ϕ (1) h ({x∈X|X (x)=X (x)=X (x)=ω (x)}∩Υ∩NR(f)∩I (f))=h (f). top d d B∗ f ϕ top (1’) h ({x∈X|X (x)=X (x)=X (x)(ω (x)}∩Υc ∩NR(f)∩I (f))=h (f). top d d B∗ f T ϕ top (2) h ({x∈X|X (x)=X (x)(X (x)=ω (x)}∩Υ∩NR(f)∩I (f))=h (f). top d d B∗ f ϕ top (2’) h ({x∈X|X (x)=X (x)(X (x)(ω (x)}∩Υc ∩NR(f)∩I (f))=h (f). top d d B∗ f T ϕ top (3) h ({x ∈ X|∅ = X (x) ( X (x) = X (x) = ω (x)} ∩ Υ ∩ NR(f) ∩ I (f)) = h (f); and top d d B∗ f ϕ top h ({x∈X|∅6=X (x)(X (x)=X (x)=ω (x)}∩Υ∩NR(f)∩I (f))=h (f). top d d B∗ f ϕ top (3’) h ({x∈X|∅=X (x)(X (x)=X (x)(ω (x)}∩Υc ∩NR(f)∩I (f))=h (f); and top d d B∗ f T ϕ top h ({x∈X|∅6=X (x)(X (x)=X (x)(ω (x)}∩NR(f)∩I (f))=h (f). top d d B∗ f ϕ top (4) h ({x ∈ X|∅ = X (x) ( X (x) ( X (x) = ω (x)} ∩ Υ ∩ NR(f) ∩ I (f)) = h (f); and top d d B∗ f ϕ top h ({x∈X|∅6=X (x)(X (x)(X (x)=ω (x)}∩Υ∩NR(f)∩I (f))=h (f). top d d B∗ f ϕ top (4’) h ({x∈X|∅=X (x)(X (x)(X (x)(ω (x)}∩Υc ∩NR(f)∩I (f))=h (f); and top d d B∗ f T ϕ top h ({x∈X|∅6=X (x)(X (x)(X (x)(ω (x)}∩Υc ∩NR(f)∩I (f))=h (f). top d d B∗ f T ϕ top (5) h ({x ∈ X|X (x) = X (x) = X (x) = X (x) = ω (x)}∩NR(f)∩I (f)) = h (f). Alterna- top B∗ d d B∗ f ϕ top tively, h (QAP(f)∩NR(f)∩I (f))=h (f). top ϕ top (5’) h ({x ∈X|X (x) =X (x) = X (x) = X (x) ( ω (x)}∩NR(f)∩I (f)) = h (f). Alterna- top B∗ d d B∗ f ϕ top tively, h ((WQAP(f)\QAP(f))∩NR(f)∩I (f))=h (f). top ϕ top For a continuous function ϕ on X, denote L = inf ϕdµ, sup ϕdµ and IntL = inf ϕdµ, sup ϕdµ . ϕ ϕ "µ∈M(f,X)Z µ∈M(f,X)Z # µ∈M(f,X)Z µ∈M(f,X)Z ! For any a∈L , denote ϕ t = sup h : ϕdµ=a a µ µ∈M(f,X)(cid:26) Z (cid:27) and consider the level set n−1 1 R (a) := x∈X : lim ϕ(fix)=a . ϕ ( n→∞n ) i=0 X Theorem B. Suppose(X,f)is topologically expanding and transitive(resp., (X,f)is a homeomorphism that is topologically hyperbolic and transitive).. For a continuous function ϕ on X and a ∈Int(L ), we ϕ have the following conditional variational principle: 4 (1) h ({x∈X|X (x)=X (x)=X (x)=ω (x)}∩Υ∩NR(f)∩R (a))=t . top d d B∗ f ϕ a (1’) h ({x∈X|X (x)=X (x)=X (x)=ω (x)}∩Υc ∩NR(f)∩R (a))=t . top d d B∗ f T ϕ a (2) h ({x∈X|X (x)=X (x)(X (x)=ω (x)}∩Υ∩NR(f)∩R (a))=t . top d d B∗ f ϕ a (2’) h ({x∈X|X (x)=X (x)(X (x)(ω (x)}∩Υc ∩NR(f)∩R (a))=t . top d d B∗ f T ϕ a (3) h ({x∈X|∅=X (x)(X (x)=X (x)=ω (x)}∩Υ∩NR(f)∩R (a))=t ; and top d d B∗ f ϕ a h ({x∈X|∅6=X (x)(X (x)=X (x)=ω (x)}∩Υ∩NR(f)∩R (a))=t . top d d B∗ f ϕ a (3’) h ({x∈X|∅=X (x)(X (x)=X (x)(ω (x)}∩Υc ∩NR(f)∩R (a))=t ; and top d d B∗ f T ϕ a h ({x∈X|∅6=X (x)(X (x)=X (x)(ω (x)}∩Υc ∩NR(f)∩R (a))=t . top d d B∗ f T ϕ a (4) h ({x∈X|∅=X (x)(X (x)(X (x)=ω (x)}∩Υ∩NR(f)∩R (a))=t ; and top d d B∗ f ϕ a h ({x∈X|∅6=X (x)(X (x)(X (x)=ω (x)}∩Υ∩NR(f)∩R (a))=t . top d d B∗ f ϕ a (4’) h ({x∈X|∅=X (x)(X (x)(X (x)(ω (x)}∩Υc ∩NR(f)∩R (a))=t ; and top d d B∗ f T ϕ a h ({x∈X|∅6=X (x)(X (x)(X (x)(ω (x)}∩Υc ∩NR(f)∩R (a))=t . top d d B∗ f T ϕ a (5) For any k ∈N, h (R (a))=h (R (a)∩NR(f)) top ϕ top ϕ = h (R (a)∩NR(f)∩WQAP(f)) top ϕ = h (R (a)∩NR(f)∩QAP(f)) top ϕ = sup h | ϕdµ=a,µ∈M(f,X) µ (cid:26) Z (cid:27) = sup h | ϕdµ=a,µ∈M(f,X) and S 6=X µ µ (cid:26) Z (cid:27) = sup h | ϕdµ=a,µ∈M(f,X), S 6=X, S is minimal and #M(f,S )=k . µ µ µ µ (cid:26) Z (cid:27) 1.2 Examples 1.2.1 Symbolic dynamics Symbol spaces and their shift maps have a long history in dynamics beginning at the latest in 1921with Morse [48]. The definition of subshift of finite type (f.t. subshift for short) is due to Parry[52, 53] in the study of intrinsic Markov chain. It also evolved independently from Smale [68] via Bowen-Lanford [16]. See [76] for proofs and related results. 1.2.2 Smooth dynamics We now suppose that f :M →M is a diffeomorphismof a compactC∞ Riemannian manifold M. Then the derivative of f can be considered a map df : TM → TM where TM = T M is the tangent x∈M x bundle of M and df : T M → T M. A closed subset Λ ⊂ M is hyperbolic if f(Λ) = Λ and each x x f(x) S tangent space T M with x∈Λ can be written as a direct sum x T M =Eu⊕Es x x x of subspaces so that 1. Df(Es)=Es ,Df(Eu)=Eu ; x f(x) x f(x) 5 2. there exist constants c>0 and λ∈(0,1) so that kDfn(v)k≤cλnkvk when v ∈Es,n≥0 x and kDf−n(v)k≤cλnkvk when v ∈Eu,n≥0; x 3. Es,Eu vary continuously with x. x x Furthermore,we say f satisfies Axiom A if Ω(f) is hyperbolic and the periodic points are dense in Ω(f). ItiswellknownthatthebasicsetofAxiomAsystemsisexpansive,transitiveandsatisfiestheshadowing property. 2 Preliminaries 2.1 Notions and Notations Consider a metric space (O,d). Let A,B be two nonempty subsets, then the distance from x ∈ X to B is defined as dist(x,A):= inf d(x,y). y∈B Furthermore, the distance from A to B is defined as dist(A,B):= supdist(x,B). x∈A Finally, the Hausdorff distance between A and B is defined as d (A,B):=max{dist(A,B),dist(B,A)}. H Now consider a TDS (X,f). If for every pair of non-empty open sets U,V there is an integer n such that fn(U)∩V 6= ∅ then we call (X,f) topologically transitive. Furthermore, if for every pair of non- empty open sets U,V there exists an integer N such that fn(U)∩V 6= ∅ for every n > N, then we call (X,f) topologically mixing. We say that f is positively expansive if there exists a constant c > 0 such that for any x,y ∈ X, d(fix,fiy) > c for some i ∈ Z+. When f is a homeomorphism, we say that f is expansive if there exists a constant c>0 such that for any x,y ∈X, d(fix,fiy)>c for some i∈Z. We call c the expansive constant. Fix an arbitrary x∈X. Let V (x) denote a ball centered at x with radius ǫ. Then it is easy to check ǫ that x∈AP(f) ⇔ ∀ǫ>0, N(x,V (x)) is syndetic ⇔ ∀ǫ>0, B (N(x,V (x)))>0, ǫ ∗ ǫ x∈Rec(f) ⇔ ∀ǫ>0, N(x,V (x))6=∅, ǫ x∈Ω(f) ⇔ ∀ǫ>0, N(V (x),V (x))6=∅. ǫ ǫ Otherkindofrecurrencesuchasweakalmostperiodic,quasi-weakalmostperiodicandBanachrecurrent have been discussed in [38, 74]. We say (Y,f| ) is a subsystem of (X,f) if Y is a closed f-invariant Y subset of X and f| is the restriction of f on Y. It is not hard to check that Rec(f| ) = Rec(f)∩Y. Y Y Consequently, NR(f| )=NR(f)∩Y. Y 6 A finite sequence C = hx ,··· ,x i,l ∈ N is called a chain. Furthermore, if d(fx ,x ) < ε,1 ≤ i ≤ 1 l i i+1 l−1, we call C an ε-chain. For any m∈N, if there are m chains C =hx ,··· ,x i, l ∈N,1≤i≤m i i,1 i,li i satisfying that x =x ,1≤i≤m−1, then can concatenate C s to constitute a new chain i,li i+1,1 i hx ,··· ,x ,x ,··· ,x ,··· ,x ,··· ,x i 1,1 1,l1 2,2 2,l2 m,2 m,lm which we denote by C C ···C . When m=∞, we can also concatenate C s to obtain a pseudo-orbit 1 2 m i hx ,··· ,x ,x ,··· ,x ,x ,··· ,x ,···i 1,1 1,l1 2,2 2,l2 3,2 3,l3 which we denote by C C ···C ···. 1 2 m Let A⊆X be a nonempty invariant set. We call A internally chain transitive if for any a,b∈A and any ε>0, there is an ε-chain C in A with connecting a and b. ab For any two TDSs (X,f) and (Y,g), if π : (X,f) → (Y,g) is a continuous surjection such that π◦f =g◦π, then we say π is a semiconjugation. We have the following conclusions: π(AP(f))=AP(g) and π(Rec(f))=Rec(g). (2.2) If in addition, π is a homeomorphism, we call π a conjugation. 2.2 The Space of Borel Probability Measures 2.2.1 Metric Compatible with the Weak∗ Topology The space of Borel probability measures on X is denoted by M(X) and the set of continuous functions on X by C(X). We endow ϕ ∈ C(X) the norm kϕk = max{|ϕ(x)| : x ∈ X}. Let {ϕ } be a dense j j∈N subset of C(X), then ∞ | ϕ dξ− ϕ dτ| j j ρ(ξ,τ)= 2jkϕ k j=1 R jR X defines a metric on M(X) for the weak∗ topology [75]. Proposition 2.1. [75] Let µ ,µ∈M(X). Then the following conditions are equivalent: n 1. µ converges weakly to µ. n 2. lim ϕdµ = ϕdµ for all ϕ∈C(X). n→∞ n 3. limsup R µ (C)R≤µ(C) for all closed C ⊆X. n→∞ n 4. liminf µ (U)≥µ(U) for all open U ⊆X. n→∞ n For ν ∈M(X) and r >0, we denote a ball in M(X) centered at ν with radius r by B(ν,r):={ρ(ν,µ)<r:µ∈M(X)}. One notices that ρ(ξ,τ)≤2 for any ξ,τ ∈M(X). (2.3) Itisalsowellknownthatthe naturalimbeddingj :x7→δ iscontinuous. SinceX iscompactandM(X) x isHausdorff,oneseesthatthereisahomeomorphismbetweenX anditsimagej(X). Therefore,without loss of generality we will assume that d(x,y)=ρ(δ ,δ ). (2.4) x y A straight calculation using (2.3) and (2.4) gives 7 Lemma 2.2. Let (X,f) be a dynamical system and let x∈X. 1. Let 0≤k <m<n and x∈X. Then 2 ρ(E (x),E (fk(x)))≤ (n−m+k), m n n 2. Given ε>0 and p∈N, for every y ∈B (x,ε) we have ρ(E (y),E (x))<ε. p p p 3. Given ε > 0 and p,q ∈ N satisfying p ≤ q ≤ (1 + ε/2)p, for every y ∈ B (x,ε) we have p ρ(E (y),E (x))<2ε. q p Definition 2.3. Forµ∈M(X),the setofallx∈X withthepropertythatµ(U)>0forallneighborhood U of x is called the support of µ and denoted by S . Alternatively, S is the (well defined) smallest µ µ closed set C with µ(C)=1. 2.2.2 The Space of Invariant Measures We sayµ∈M(X)is anf-invariantmeasureifforanyBorelmeasurablesetA,onehasµ(A)=µ(f−1A). The set of f-invariant measures are denoted by M(f,X). We remark that if µ ∈ M(f,X), then S is a µ closed f-invariant set. Let (X,f) and (Y,g) be two dynamical systems and π : X → Y is a continuous map. Define π : M(X) → M(Y) as π µ := µ◦π−1 and π∗ : C(Y) → C(X) as π∗ϕ := ϕ◦π. We call π µ the ∗ ∗ ∗ push-forward of µ and π∗ϕ the pull-back of ϕ. It is not hard to see that ϕdπ µ= π∗ϕdµ for any ϕ∈C(Y) and µ∈M(X). (2.5) ∗ Z Z Moreover,one notices that µ∈M(f,X) is equivalent to f µ=µ. ∗ The following are not hard to check. Lemma 2.4. If π :(X,f)→(Y,g) is a conjugation, then (1) there is a metric ρ on M(X) and a metric ρ on M(Y) such that π :(M(X),ρ )→(M(Y),ρ ) X Y ∗ X Y is an isometric isomorphism. In particular, π :M(X)→M(Y) is a homeomorphism. ∗ (2) π (M(f,X))=M(g,Y)andπ | :(M(f,X),ρ )→(M(f,Y),ρ ) is alsoanisometryisomor- ∗ ∗ M(f,X) X Y phism. We say µ ∈ M(X) is an ergodic measure if for any Borel set B with f−1B = B, either µ(B) = 0 or µ(B)=1. We denote the set of ergodic measures on X by M (f,X). It is well known that the ergodic erg measures are exactly the extreme points of M(f,X). We have the following observation. Lemma 2.5. Let Λ,Λ be three closed f-invariant subset of X with Λ ⊆ Λ. Suppose for any x ∈ Λ, 0 0 ω (x)⊆Λ . Then f 0 M(f,Λ)=M(f,Λ ). 0 Proof. Ofcourse,M(f,Λ)⊇M(f,Λ ). WenowprovethatM(f,Λ)⊆M(f,Λ ). Infact,bytheconvexity 0 0 of M(f,Λ) and M(f,Λ ), it is sufficient to prove that for any µ ∈ M (f,Λ), µ ∈ M(f,Λ ). Indeed, 0 erg 0 choose an arbitrary generic point x ∈ Λ of µ. Then S ⊆ ω (x) ⊆ Λ which yields that µ ∈ M(f,Λ ). µ f 0 0 The proof is completed. Moreover,we have the following characterization. 8 Lemma 2.6. If µ∈M (f,X), then for any n∈N, there is a ν ∈M (fn,X) and an m∈N with m|n erg erg such that µ can decompose as 1 µ= (ν+f ν+···+fm−1ν). m ∗ ∗ Moreover, there is a X ⊆ X with ν(X ) = 1 and fmX = X such that X has a mod 0 measurable 0 0 0 0 partition X = m−1fiX . i=0 0 For any x∈FX, we define the measure center of x as C := S . x µ µ∈M([f,ωf(x)) Furthermore, we define the measure center of Λ as C := S . Λ µ µ∈M[(f,Λ) Lemma 2.7. We have the following relations: (1) S = S . µ∈Merg(f,Λ) µ µ∈M(f,Λ) µ T T (2) C = S . Λ µ∈Merg(f,Λ) µ Proof. (1).SItisclearthat S ⊇ S . Soweonlyneedtoprovethat S ⊆ µ∈Merg(f,Λ) µ µ∈M(f,Λ) µ µ∈Merg(f,Λ) µ S . Indeed, for any x∈ S and ε>0, one has µ∈M(f,Λ) µ T µ∈Merg(fT,Λ) µ T T T µ(B(x,ε))>0 for any µ∈M (f,Λ). (2.6) erg Ifν(B(x,ε))=0forsomeν ∈M(f,Λ),thenbytheergodicdecompositiontheorem[75],thereisaunique measure τ on the Borel subsets of the compact metrisable space M(f,Λ) such that τ(M(f,Λ))=1 and 0=ν(B(x,ε))= µ(B(x,ε))dτ(µ). ZM(f,Λ) Therefore, for τ-a.e. µ ∈ M (f,Λ), µ(B(x,ε)) = 0, contradicting (2.6). Thus x∈ S which erg µ∈M(f,Λ) µ implies that S ⊆ S . µ∈Merg(f,Λ) µ µ∈M(f,Λ) µ T (2). Itisclearthat S ⊆ S . Soitissufficienttoprovethat S ⊇ T µ∈Merg(fT,Λ) µ µ∈M(f,Λ) µ µ∈Merg(f,Λ) µ S . Indeed, for any µ∈M(f,Λ) and any x∈S , one has that µ∈M(f,Λ) µ S S µ S S µ(B(x,ε))>0 for any ε>0. By the ergodic decomposition theorem, there is a µ ∈ M (f,Λ) with µ (B(x,ε)) > 0. This implies ε erg ε that B(x,ε)∩S 6= ∅. Since ε > 0 is arbitrary, x ∈ , which yields that S ⊇ µε µ∈Merg(f,Λ) µ∈Merg(f,Λ) µ S . µ∈M(f,Λ) µ S S S 2.3 Topological Entropy and Metric Entropy 2.3.1 Topological Entropy for Compact Sets The classical topological entropy for (X,f) was introduced by Adler et al [1] using open covers. Later on, Bowen gave a equivalent definition of topological entropy for (X,f) using separated and spanning sets [14] which we shall recall now. For x,y ∈X and n∈N, the Bowen distance between x,y is defined as d (x,y):=max{d(fix,fiy):i=0,1,··· ,n−1} n 9 and the Bowen ball centered at x with radius ε>0 is defined as B (x,ε):={y ∈X :d (x,y)<ε}. n n Let Z ⊂ X. A set S is (n,ε)-separated for Z if S ⊂ Z and d (x,y) > ε for any x,y ∈ S and x 6= y. n A set S ⊂Z if (n,ε)-spanning for Z if for any x∈Z, there exists y ∈S such that d (x,y)≤ε. n Define s (Z,ε) = sup {|S|:S is (n,ε)−separated for Z}, n r (Z,ε) = inf {|S|:S is (n,ε)−spanning for Z}, n where |S| denotes the cardinality of S. It is not hard to see that [75] r (Z,ε)≤s (Z,ε)≤r (Z,ε/2). (2.7) n n n Definition 2.8. The topological entropy for a compact set K ⊂X is defined (by Bowen) as 1 1 h (f,K)=limsup logs (K,ε)=limsup logr (K,ε) d n n n n n→∞ n→∞ and for a general set E ⊂X as h (f,E)=sup{h (f,K): K ⊂E compact}. d d 2.3.2 Topological Entropy for Noncompact Set As for noncompactsets , Bowen also developed a satisfying definition via dimension language [13] which we now illustrate. Let E ⊆ X, and G (E,σ) be the collection of all finite or countable covers of E by sets of the form n B (x,σ) with u≥n. We set u C(E;t,n,σ,f):= inf e−tu C∈Gn(E,σ) Bu(Xx,σ)∈C and C(E;t,σ,f):= lim C(E;t,n,σ,f). n→∞ Then we define h (E;σ,f):=inf{t:C(E;t,σ,f)=0}=sup{t:C(E;t,σ,f)=∞} top The Bowen topological entropy of E is h (f,E):= lim h (E;σ,f). (2.8) top top σ→0 We have the following lemma from Bowen [?]. Lemma 2.1. Let f :X →X be a continuous map on a compact metric space. Set QR(t)={x∈X | ∃τ ∈V (x) with h (T)≤t}. f τ Then h (f,QR(t))≤t. top 10