PPACKING ENTROPY FOR FIXED-POINT FREE FLOWS
RUIMING LIANG AND HAOYI LEI
Abstract.
Let (
X, φ ) be a compact flow without fixed points. We define the packingtopological entropy h P top ( φ, K ) on subsets of X through considering all the possiblereparametrizations of time. For fixed-point free flows, we prove the following result:for any non-empty compact subset K of X , h P top ( φ, K ) = sup { h µ ( φ ) : µ ( K ) = 1 , µ is a Borel probability measure on X } , where h µ ( φ ) denotes the upper local entropy for a Borel probability measure µ on X . Introduction
Entropy plays a crucial role in the study of the complexity of dynamical systems,which characterizes the growth of geometric or measure-theoretic quantities under longtime dynamical iterations. Among various definitions of entropies, one of frequentlystudied definitions was introduce by Bowen [1] in 1973 in a way resembling the Haus-dorff dimension for discrete dynamical systems, which is now called Bowen topologicalentropy. Bowen topological entropy can be defined on any subset of the state spacewhile the classic topological entropy defined by Adler, Konheim and McAndrew cannot, although they coincide on the whole state space. Similarly, by considering theconcept of packing dimension, Feng and Huang introduced the concept of packingtopological entropy over subsets for discrete systems [4]. In particular, it also coin-cides with Adler-Konheim-McAndrew’s topological entropy on the whole space. Thistwo definitions are of great importance in topological dynamics and dimension theory.Inspired by classical results in geometric-measure theory, Feng and Huang proved thefollowing non-classic variational principles for Bowen entropy and packing entropy:Let (
X, T ) be a topological dynamical system where X is a compact metric space and T : X → X is a continuous map. For any non-empty subset K of X , let h B top ( T, K ) and h P top ( T, K ) denote the Bowen entropy and packing entropy on K , respectively. When K is a compact subset of X , h B top ( T, K ) = sup { h µ ( T ) : µ ( K ) = 1 , µ ∈ M ( X ) } ;(1.1) h P top ( T, K ) = sup { h µ ( T ) : µ ( K ) = 1 , µ ∈ M ( X ) } . (1.2)Here M ( X ) denotes the set of all Borel probability measures on X and h µ ( T ) and h µ ( T ) are the lower and upper local entropies for µ (see [4] for more details). Key words and phrases. measure-theoretic entropy, packing entropy, variational principle, fixed-point free flow, reparametrization. a r X i v : . [ m a t h . D S ] F e b RUIMING LIANG AND HAOYI LEI
Feng and Huang’s variational principles have now been extended to many situationssuch as for more general dynamical systems under countable amenable group actions[13, 3] and for nonautonomous dynamical systems [12]. A basic but nature question ishow the above result works for the case of continuous dynamical systems (flows). Let(
X, d ) be a compact metric space with metric d . Recall that a pair ( X, φ ) is called aflow, if φ : X × R → X is a continuous map satisfying φ t ◦ φ s = φ s + t for all s, t ∈ R and φ t ( · ) = φ ( · , t ) is a homeomorphism on X . A Borel probability measure µ on X iscalled φ − invariant if for any Borel set B , it holds µ ( φ t ( B )) = µ ( B ) for all t ∈ R . Itis called ergodic if any φ − invariant Borel set has measure 0 or 1. For convention, wedenote all φ − invariant Borel probability measures and all ergodic φ − invariant Borelprobability measures on X by M ( X, φ ) and E ( X, φ ) respectively.A nature definition of entropy for flows is to study the time-1 map. This deducesthe entropy theory for flows to discrete case. But for flows there are more depthconsiderations–finding quantitative invariants for flows under orbit equivalents. Thisleads to a series of studies of flows through considering all the possible reparametriza-tions of flows ([9, 10, 7, 8]). Recently Dou etc. in [2] introduced Bowen topologicalentropy for flows via the reparametrization balls and proved the variational principle(1.1) for flows without fixed-points. For further studies on Bowen topological entropyfor fixed-point free flows, see [5, 11]. In the present paper, we will study packingtopological entropy for fixed-point free flows. Especially we will aim on proving thevariational principle (1.2) for flows without fixed-points.The paper is organized as follows. In section 2, we introduce packing topologicalentropy for flows via reparametrization balls and some basic properties are also listedthere. In section 3, we give the exact statement of our variational principle and somepreparatory lemmas for the proof. Finally, in section 4, we give detailed proof of thevariational principle of packing topological entropy for flows without fixed-points. Forthe proof of the lower bound, we need carefully treatments on the reparametrizationballs. For the proof of the upper bound, we will employ Feng and Huang’s methodin [4], the idea of which origins from Joyce and Preiss’s work on packing measures[6]. This idea from geometric measure theory is surprisingly suitable for many cases indynamical systems.2.
Packing topological entropy via reparametrization balls
To introduce the concept of packing topological entropy for subsets of a flow, wefirst need to introduce the definition of a reparametrization ball, which is an analogyof the traditional Bowen ball.
Definition 2.1.
For a closed interval I which contains the origin, a continuous map α : I → R is called a reparametrization if it is a homeomorphism onto its image and α (0) = 0. The set of all such reparametrizations on I is denoted by Rep ( I ). For a flow φ on X , x ∈ X , t ∈ R + and ε >
0, we set B ( x, t, ε, φ ) = { y ∈ X : there exists α ∈ Rep [0 , t ] such that d ( φ α ( s ) x, φ s y ) < ε, for all 0 ≤ s ≤ t } , ACKING ENTROPY FOR FIXED-POINT FREE FLOWS 3 and B ( x, t, ε, φ ) = { y ∈ X : there exists α ∈ Rep [0 , t ] such that d ( φ α ( s ) x, φ s y ) ≤ ε, for all 0 ≤ s ≤ t } , Usually, we call B ( x, t, ε, φ ) a ( t, ε, φ ) − ball or a reparametrization ball in X . Clearly,all the reparametrization balls are open sets.Now, we can give the definition of packing topological entropy defined throughreparametrization balls. Definition 2.2.
Let (
X, φ ) be a flow and Z a subset of X . For s ≥ N ∈ N , and ε >
0, define P sN,ε ( φ, Z ) = sup (cid:88) i exp( − st i ) , where the supremum is taken over all finite or countable families of disjoint closedreparametrization balls { B ( x i , t i , ε, φ ) } such that x i ∈ X , t i ≥ N .The quantity P sN,ε dose not increase as N increases, hence the following limit exists: P sε ( φ, Z ) = lim N →∞ P sN,ε ( φ, Z ) . Define P sε ( Z ) = inf { ∞ (cid:80) i =1 P sε ( φ, Z i ) : Z ⊂ ∞ (cid:83) i =1 Z i } .By definition and simple calculation, we know h P top ( φ, Z, ε ) = inf { s : P sε ( Z ) = 0 } = sup { s : P sε ( Z ) = + ∞} is well-defined.Note that h P top ( φ, Z, ε ) does not decrease as ε decreases. Define the packing topologicalentropy for subset Z of X to be h P top ( φ, Z ) = lim ε → h P top ( φ, Z, ε ) . For packing topological entropy we have the following properties.
Proposition 2.3.
Let
Z, Z (cid:48) and Z , Z , . . . be subsets of X . (1) If Z ⊂ Z (cid:48) , then h P top ( φ, Z ) ≤ h P top ( φ, Z (cid:48) ) . (2) If Z ⊂ ∞ (cid:83) i =1 Z i , then for s ≥ and ε > , we have P sε ( Z ) ≤ ∞ (cid:80) i =1 P sε ( Z i ) and h P top ( φ, Z ) ≤ sup i ≥ h P top ( φ, Z i ) . The proof is simple from the definition and we omit it.Since the Hausdorff dimension and the packing dimension are dual concepts in fractalgeometry, we will compare packing topological entropy with Bowen topological entropyfor a flow. In the following we recall the definition of Bowen topological entropy for aflow (see [2]).
RUIMING LIANG AND HAOYI LEI
Definition 2.4.
Let (
X, φ ) be a flow and Z a subset of X . For s ≥ N ∈ N and ε >
0, define M sN,ε ( φ, Z ) = inf (cid:88) i exp( − st i ) , where the infimum is taken over all finite or countable families of reparametrizationballs { B ( x i , t i , ε, φ ) } such that x i ∈ X , t i ≥ N and (cid:83) B ( x i , t i , ε, φ ) ⊃ Z .The quantity M sN,ε dose not decrease as N increases and ε decreases, hence thefollowing limits exist: M sε ( φ, Z ) = lim N →∞ M sN,ε ( φ, Z ) , M s ( φ, Z ) = lim ε → M sε ( φ, Z ) . The
Bowen topological entropy h Btop ( φ, Z ) is defined as a critical value of the parameter s , where M s ( φ, Z ) jumps from ∞ to 0, i.e. h Btop ( φ, Z ) = inf { s : M s ( φ, Z ) = 0 } = sup { s : M s ( φ, Z ) = ∞} . We now list some properties of Bowen topological entropy ([2, Proposition 2]).
Proposition 2.5.
Let
Z, Z (cid:48) and Z , Z , . . . be subsets of X . (1) If Z ⊂ Z (cid:48) , then h B top ( φ, Z ) ≤ h B top φ, Z (cid:48) ) . (2) If Z ⊂ ∞ (cid:83) i =1 Z i , then for s ≥ and ε > , we have M sε ( Z ) ≤ ∞ (cid:80) i =1 M sε ( Z i ) and h B top ( φ, Z ) ≤ sup i ≥ h B top ( φ, Z i ) . Moreover, we have
Proposition 2.6.
For any Z ⊂ X, h B top ( φ, Z ) ≤ h P top ( φ, Z ) .Proof. If h B top ( φ, Z ) = 0 then there is nothing to prove. Assume h B top ( φ, Z ) > < s < h B top ( φ, Z ). For any n ∈ N and ε >
0, let { B ( x i , n, ε, φ ) } Ri =1 be a disjoint familywith x i ∈ Z such that the cardinality R = R n ( Z, ε ) is maximal. Then for any δ > (cid:83) Ri =1 B ( x i , n, ε + δ, φ ) ⊃ Z . Hence M sn, ε + δ ( φ, Z ) ≤ R exp( − sn ) ≤ P sn,ε ( φ, Z ) . Letting n → ∞ , we have M s ε + δ ( φ, Z ) ≤ P sε ( φ, Z ). Thus for any ∪ ∞ i =1 Z i ⊃ Z , M s ε + δ ( φ, Z ) ≤ ∞ (cid:88) i =1 M s ε + δ ( φ, Z i ) ≤ ∞ (cid:88) i =1 P sε ( φ, Z i ) , which implies that M s ε + δ ( φ, Z ) ≤ P sε ( φ, Z ). Note that 0 < s < h B top ( φ, Z ). Hence M s ( φ, Z ) = ∞ and M s ε + δ ( φ, Z ) > ε and δ are sufficiently small. Hence P sε ( φ, Z ) > h P top ( φ, Z, ε ) ≥ s when ε is small. Since h P top ( φ, Z, ε ) does not decreaseas ε decreases, h P top ( φ, Z ) = lim ε → h P top ( φ, Z, ε ) ≥ s . Therefor h P top ( φ, Z ) ≥ h B top ( φ, Z ). (cid:3) ACKING ENTROPY FOR FIXED-POINT FREE FLOWS 5 Statement of main theorem and preparatory lemmas
The measure-theoretic local entropies are defined as follows.
Definition 3.1.
Let µ ∈ M ( X ). The measure-theoretic lower and upper local entropies of µ are defined respectively by h µ ( φ ) = (cid:90) h µ ( φ, x ) dµ, and h µ ( φ ) = (cid:90) h µ ( φ, x ) dµ where h µ ( φ, x ) = lim ε → lim inf t → + ∞ − t log µ ( B ( x, t, ε, φ ))and h µ ( φ, x ) = lim ε → lim sup t → + ∞ − t log µ ( B ( x, t, ε, φ )) . Now we state the main theorem.
Theorem 3.2.
Let ( X, φ ) be a compact metric flow without fixed points. If K is anon-empty compact subset of X , then h Ptop ( φ, K ) = sup { h µ ( φ ) : µ ∈ M ( X ) , µ ( K ) = 1 } . We suggest here that there are some further results related to Theorem 3.2 for flowswithout fixed points. Due to Feng and Huang [4], the compact subsets K ’s can beimproved to any analytic subset of X when the classic topological entropy of the flowis finite. (We note that here for the flows without fixed points, the classic topologicalentropy coincides with the topological entropy defined through reparametrization balls[9, 10].)In this section, we first will give some properties about reparametrization balls forflows without fixed points and then give a covering lemma. These lemmas are crutialin proving Theorem 3.2. Lemma 3.3 (Lemma 1.2 of [9]) . Let ( X, φ ) be a compact metric flow without fixedpoints. For any η > , there exists θ > such that for any x, y ∈ X , any interval I containing the origin, and any reparametrization α ∈ Rep ( I ) , if d ( φ α ( s ) ( x ) , φ s ( y )) < θ for all s ∈ I , then it holds that | α ( s ) − s | < (cid:40) η | s | , if | s | > ,η, if | s | ≤ . The following 5 r -lemma for reparametrization balls is proved by Dou etc. [2, Theo-rem 3.5]. This is a variation of the classic 5 r -covering lemma and it will play a crucialrole for the proof of Theorem 3.2. Theorem 3.4 (5 r -lemma for reparametrization balls). Let ( X, φ ) be a compactmetric flow without fixed points. For < η < , let θ > be as in Lemma 3.3. Let B = { B ( x, t, ε, φ ) } ( x,t ) ∈I be a family of reparametrization balls in X with < ε < θ and t > RUIMING LIANG AND HAOYI LEI − η ) . Then there exists a finite or countable subfamily B (cid:48) = { B ( x, t, ε, φ ) } ( x,t ) ∈I (cid:48) ( I (cid:48) ⊂I ) of pairwise disjoint reparametrization balls in B such that (cid:91) B ∈B B ⊆ (cid:91) ( x,t ) ∈I (cid:48) B ( x, ˆ t, ε, φ ) where ˆ t = (1 − η ) t . At the end of this section, we give a frequently used lemma when proving Theorem3.2. It is analogous to [4, Lemma 4.1].
Lemma 3.5.
Let Z ⊂ X and s, ε > . Assume P sε ( Z ) = ∞ . Then for any given finiteinterval ( a, b ) ⊂ R with a ≥ and any N ∈ N , there exists a finite disjoint collection { B ( x i , t i , ε, φ ) } such that x i ∈ Z, t i ≥ N and (cid:80) i e − t i s ∈ ( a, b ) . Proof.
The proof is the same with that of [4, Lemma 4.1] for discrete dynamical systems.For the completeness we give the proof.Let N > N be sufficiently large such that e − N s < b − a . Since P sN,ε ( φ, Z ) does notincrease as N increases and P sε ( φ, Z ) = lim N →∞ P sN,ε ( φ, Z ), we have P sN ,ε ( Z ) ≥ P sε ( Z ) = ∞ . Thus we can choose a finite disjoint collection { B ( x i , t i , ε, φ ) } with x i ∈ Z, t i ≥ N such that (cid:80) i e − t i s > b . Since e − t i s < b − a , we can discard elements in this collectionone by one until we have (cid:80) i e − t i s ∈ ( a, b ) . (cid:3) Proof of Theorem 3.2
With the preparation in Section 3, we can now prove Theorem 3.2. The followinglemma gives the proof of the lower bound.
Lemma 4.1.
Let ( X, φ ) be a compact metric flow without fixed points. We have h Ptop ( φ, Z ) ≥ sup { h µ ( φ ) : µ ∈ M ( X ) , µ ( Z ) = 1 } for any Borel set Z ⊂ X .Proof. Let µ ∈ M ( X ) with µ ( Z ) = 1 for some Borel set Z ⊂ X . We just need to prove h Ptop ( φ, Z ) ≥ h µ ( φ ). Since the lemma automatically holds when h µ ( φ ) = 0, we mayassume h µ ( φ ) >
0. Let 0 < s < h µ ( φ ), we shall prove h Ptop ( φ, Z ) ≥ s. Choose 0 < η < − η ) < θ > µ ( B ( x, t, ε, φ )) does not increase as ε decreases. Then there exists 0 < ε < θ , δ > A ⊂ Z with µ ( A ) > h µ ( φ, x, ε ) > s + δ, ∀ x ∈ A, where h µ ( φ, x, ε ) = lim t →∞ − t log µ ( B ( x, t, ε, φ )). To see this, we note that h µ ( φ ) = (cid:82) h µ ( φ, x ) dµ > s . And hence there exists δ > A with µ ( A ) > ACKING ENTROPY FOR FIXED-POINT FREE FLOWS 7 that for any x ∈ A , it holds that h µ ( φ, x ) > s + δ . Since h µ ( φ, x ) = lim ε → h µ ( φ, x, ε ),there must exist a sufficiently small ε > P s ε ( Z ) = ∞ , which implies that h P top ( φ, Z ) ≥ h P top ( φ, Z, ε ) ≥ s. Itsuffices to prove that P s ε ( E ) = ∞ for any Borel set E ⊂ A with µ ( E ) >
0. Fix such aset E . Define E t = { x ∈ E : µ ( B ( x, t, ε, φ )) < e − t ( s + δ ) } , t ∈ R + , and (cid:101) E n = { x ∈ E : µ ( B ( x, n, ε, φ )) < e ( − n +1)( s + δ ) ) } , n ∈ N . (4.1)Note that B ( x, (cid:100) t (cid:101) , ε, φ ) ⊂ B ( x, t, ε, φ ). Hence for x ∈ E t , µ ( B ( x, (cid:100) t (cid:101) , ε, φ )) ≤ µ ( B ( x, t, ε, φ )) < e − t ( s + δ ) ≤ e −(cid:100) t (cid:101) ( s + δ ) · e s + δ . This implies x ∈ (cid:101) E n . And thus E t ⊂ (cid:101) E (cid:100) t (cid:101) . By E ⊂ A, we have E = (cid:83) t ≥ N E t ⊂ (cid:83) n ≥ N (cid:101) E n ⊂ E for each N ∈ N . Hence µ ( (cid:83) n ≥ N (cid:101) E n ) ≥ µ ( E ). Then there exists n ≥ N such that µ ( (cid:101) E n ) ≥ n ( n + 1) µ ( E ) . (4.2)Now we consider the family { B ( x, n (1 − η ) , ε , φ ) : x ∈ (cid:101) E n } . By Lemma 3.4, the 5r-lemma for reparametrization balls, there exists a finite or countable union of disjointfamily { B ( x, n (1 − η ) , ε , φ ) } x ∈ Λ (where Λ ⊂ (cid:101) E n is a finite or countable set) such that (cid:101) E n ⊂ (cid:91) x ∈ (cid:101) E n B ( x, n (1 − η ) , ε , φ ) ⊂ (cid:91) x ∈ Λ B ( x, n, ε, φ ) . (4.3)Hence P sN, ε ( E ) ≥ P sN, ε ( (cid:101) E n ) ≥ (cid:88) x ∈ Λ e − ns (since { B ( x, n (1 − η ) , ε , φ ) } x ∈ Λ is a disjoint family)= e nδ · e − ( s + δ ) · (cid:88) x ∈ Λ e ( − n +1)( s + δ ) ≥ e nδ − s − δ · (cid:88) x ∈ Λ µ ( B ( x, n, ε, φ )) (by (4.3)) ≥ e nδ − s − δ µ ( (cid:101) E n ) (by (4.1)) ≥ e nδ − s − δ n ( n + 1) µ ( E ) (by (4.2)) . Let N → ∞ , then we have P s ε ( Z ) = ∞ . This finishes the proof of the lemma. (cid:3) RUIMING LIANG AND HAOYI LEI
In the following we deal with the proof of the upper bound, which is more compli-cated compared with the proof of the lower bound. The core of the proof is to constructa sequence of positive finite measures supported on finite sets satisfying suitable con-ditions on reparametrization balls.
Lemma 4.2.
Let ( X, φ ) be a compact metric flow without fixed points. For any compact K ⊂ X with h Ptop ( φ, K ) > and any < s < h Ptop ( φ, K ) , there exists µ ∈ M ( K ) suchthat h µ ( φ ) ≥ s. Proof.
Let ε > < s < h
Ptop ( φ, K, ε ) and let s < t
For each x ∈ K , we will construct as in Step 1, a finite set E ( x ) ⊂ K ∩ B ( x, γ m : E ( x ) → [max { m ( y ) : y ∈ K } , ∞ )such that the elements in { B ( y, m ( y ) , ε, φ ) } y ∈ E ( x ) are disjoint, and µ ( { x } ) < (cid:88) y ∈ E ( x ) e − m ( y ) s < (1 + 2 − ) µ ( { x } ) . Fix x ∈ K and set F = K ∩ B ( x, γ ). Let H x = (cid:91) { G ⊂ X : G is open and P tε ( F ∩ G ) = 0 } . Let F (cid:48) = F \ H x . Then as in Step 1, we can show that P tε ( F (cid:48) ) = P tε ( F ) >
0. Fur-thermore, P tε ( F (cid:48) ∩ G ) > G with G ∩ F (cid:48) (cid:54) = ∅ . Since s < t , P sε ( F (cid:48) ) = ∞ . Hence by Lemma 3.5, we can find a finite set E ( x ) ⊂ F (cid:48) and a map m : E ( x ) → [max { m ( y ) : y ∈ K } , ∞ ) such that(1) the elements in { B ( y, m ( y ) , ε, φ ) } y ∈ E ( x ) are disjoint;(2) µ ( { x } ) < (cid:80) y ∈ E ( x ) e − m ( y ) s < (1 + 2 − ) µ ( { x } ) . Since the family { B ( x, γ ) } x ∈ K is disjoint, E ( x ) ∩ E ( x (cid:48) ) = ∅ for different x, x (cid:48) ∈ K .Define K = (cid:91) x ∈ K E ( x ) and µ = (cid:88) y ∈ K e − m ( y ) s δ y . The elements in { B ( y, m ( y ) , ε, φ ) } y ∈ K are pairwise disjoint because the elements inevery { B ( y, m ( y ) , ε, φ ) } y ∈ E ( x ) are disjoint and elements from two different E ( x )’s arealso disjoint. By similar argument as in Step 1, we can take 0 < γ < γ small enoughsuch that for any function z : K → X with d ( x, z ( x )) < γ for each x ∈ K , we have (cid:18) B ( z ( x ) , γ ) ∪ B ( z ( x ) , m ( x ) , ε, φ ) (cid:19) (cid:92)(cid:18) (cid:91) y ∈ K \{ x } B ( z ( y ) , γ ) ∪ B ( z ( y ) , m ( y ) , ε, φ ) (cid:19) = ∅ for each x ∈ K .Step 3. Assume that K i , µ i , m i ( · ) and γ i have been constructed for i = 1 , , ..., p .In particular, assume that for any function z : K p → X with d ( x, z ( x )) < γ p for each x ∈ K p , we have (cid:18) B ( z ( x ) , γ p ) ∪ B ( z ( x ) , m p ( x ) , ε, φ ) (cid:19) (cid:92)(cid:18) (cid:91) y ∈ K p \{ x } B ( z ( y ) , γ p ) ∪ B ( z ( y ) , m p ( y ) , ε, φ ) (cid:19) = ∅ for each x ∈ K p . We will construct K i , µ i , m i ( · ) and γ i for i = p + 1 in a way similar toStep 2.Note that the elements in { B ( x, γ p ) } x ∈ K p are pairwise disjoint. For each x ∈ K p ,since P tε ( K ∩ B ( x, γ p )) > , we can construct as in Step 2, a finite set E p +1 ⊂ K ∩ B ( x, γ p m p +1 : E p +1 ( x ) → [max { m p ( y ) : y ∈ K p } , ∞ )such that the elements in { B ( y, m p +1 ( y ) , ε, φ ) } y ∈ E p +1 ( x ) are disjoint, and µ p ( { x } ) < (cid:88) y ∈ E p +1 ( x ) e − m p +1 ( y ) s < (1 + 2 − p − ) µ p ( { x } ) . Clearly E p +1 ( x ) ∩ E p +1 ( x (cid:48) ) = ∅ for different x, x (cid:48) ∈ K p . Define K p +1 = (cid:91) x ∈ K p E p +1 ( x ) and µ p +1 = (cid:88) y ∈ K p +1 e − m p +1 ( y ) s δ y . Since the elements in { B ( y, m p ( y ) , ε, φ ) } y ∈ K p are pairwise disjoint, we can take 0 <γ p +1 < γ p small enough such that for any function z : K p +1 → X with d ( x, z ( x )) < γ p +1 for x ∈ K p +1 , we have for each x ∈ K p +1 , (cid:18) B ( z ( x ) , γ p +1 ) ∪ B ( z ( x ) , m p +1 ( x ) , ε, φ ) (cid:19) (cid:92)(cid:18) (cid:91) y ∈ K p +1 \{ x } B ( z ( y ) , γ p +1 ) ∪ B ( z ( y ) , m p +1 ( y ) , ε, φ ) (cid:19) = ∅ . As in the above steps, we can construct inductively the sequences { K i } , { µ i } , { m i ( · ) } and { γ i } . We list some of their properties here:(a) For any i ∈ N , the elements in the family F i = { B ( x, γ i ) : x ∈ K i } are disjoint dueto the choice of γ i . Each element in F i +1 is a subset of B ( x, γ i ) for some x ∈ K i . To seethis, first recall that K i +1 = (cid:83) x ∈ K i E i +1 ( x ), E i +1 ⊂ K ∩ B ( x, γ i ) and 0 < γ i +1 < γ i . Thenfor every x ∈ K i +1 , there exists a point x ∈ K i such that x ∈ E i +1 ( x ) ⊂ K ∩ B ( x, γ i ).And hence B ( x , γ i +1 ) ⊂ B ( x , γ i ) ⊂ B ( x, γ i ) . (b) For each x ∈ K i and z ∈ B ( x, γ i ), B ( z, m i ( x ) , ε, φ ) ∩ (cid:91) y ∈ K i \{ x } B ( y, γ i ) = ∅ (4.4)and µ i ( B ( x, γ i )) = e − m i ( x ) s ≤ (cid:88) y ∈ E i +1 ( x ) e − m i +1 ( y ) s ≤ (1 + 2 − i − ) µ i ( B ( x, γ i )) , (4.5)where E i +1 ( x ) = B ( x, γ i ) ∩ K i +1 . ACKING ENTROPY FOR FIXED-POINT FREE FLOWS 11
By (a), for every j ≥ i, K j ⊂ (cid:83) x ∈ K i B ( x, γ i ) which implies µ i ( F i ) ≤ µ j ( F i ) , ∀ F i ∈ F i . By (4.4), µ i ( F i ) ≤ µ i +1 ( F i ) = (cid:88) F ∈F i +1 : F ⊂ F i µ i +1 ( F ) ≤ (1 + 2 − i − ) µ i ( F i ) , F i ∈ F i . Applying the above inequalities repeatedly, it holds that for any j > i,µ i ( F i ) ≤ µ j ( F i ) ≤ j (cid:89) n = i +1 (1 + 2 − n ) µ i ( F i ) ≤ Cµ i ( F i ) , ∀ F i ∈ F i . (4.6)where C = ∞ (cid:81) n =1 (1 + 2 − n ) < ∞ . Let ˜ µ be a limit point of { µ i } in the weak-star topology and ˜ K = ∞ (cid:84) n =1 (cid:83) i ≥ n K i . It’seasy to check that ˜ µ is supported on ˜ K . Moreover, ˜ K is a compact subset of K becauseeach (cid:83) i ≥ n K i is a closed subset of the compact set K .By (4.6), for each x ∈ K i , we have e − m i ( x ) s = µ i ( B ( x, γ i )) ≤ (cid:101) µ ( B ( x, γ i )) ≤ Cµ i (cid:0) B ( x, γ i ) (cid:1) = Ce − m i ( x ) s . Particularly, we have1 < (cid:88) x ∈ K e − m ( x ) s = (cid:88) x ∈ K µ ( B ( x, γ )) ≤ µ ( ˜ K ) ≤ ˜ µ ( ˜ K ) ≤ (cid:88) x ∈ K Cµ ( B ( x, γ )) ≤ C. Note that ˜ K ⊂ (cid:83) x ∈ K i B ( x, γ i ). By (4.4), for each x ∈ K i and z ∈ B ( x, γ i ), we have˜ µ ( B ( z, m i ( x ) , ε, φ )) ≤ ˜ µ ( B ( x, γ i ≤ Ce − m i ( x ) s . For each z ∈ ˜ K and i ∈ N , z ∈ B ( x, γ i ) for some x ∈ K i . Hence˜ µ ( B ( z, m i ( x ) , ε, φ )) ≤ Ce − m i ( x ) s . Define µ = ˜ µ/ ˜ µ ( ˜ K ). Then µ ∈ M ( ˜ K ) ⊂ M ( K ) ⊂ M ( X ). For each z ∈ ˜ K, thereexists a sequence t i (cid:37) ∞ such that µ ( B ( z, t i , ε, φ )) ≤ Ce − t i s / ˜ µ ( ˜ K ). It follows that h µ ( φ ) = (cid:90) lim ε → lim sup t → + ∞ − t log µ ( B ( x, t, ε, φ )) dµ ≥ s. (cid:3) Combining Lemma 4.1 and Lemma 4.2, we complete the proof of Theorem 3.2.
Acknowledgements
This work is supported by the National University InnovationContest Grant of Nanjing University under the guidance of Professor Dou Dou. Theauthors thank Professor Fei Yang for his encouragement.
References [1] R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 12–136.[2] D. Dou, M. Fan, H. Qiu,Topological entropy on subsets for fixed-point free flows, Discrete Contin.Dyn. Syst. 37 (2017) 6319–6331.[3] D. Dou, D. Zheng, X. Zhou, Packing topological entropy for amenable group actions,arXiv:2010.14719v1.[4] D. F. Feng and W. Huang, Variational principles for topological entropies of subsets, J. Funct.Anal. 263(8)(2012),2228–2254.[5] Y. Ji, E. Chen, Y. Wang, C. Zhao, Bowen entropy for fixed-point free flows, Discrete Contin.Dyn. Syst., 39 (2019) 6231–6239.[6] H. Joyce, D. Preiss, On the existence of subsets of finite positive packing measure, Mathematika,42 (1995), 15–24.[7] Wenxiang Sun, Measure-theoretic entropy for flows, Science in China Series A: Mathematics,40(7)(1997), 725–731.[8] Wenxiang Sun and Edson Vargas, Entropy of flows, revisited, Bol. Soc. Brasil. Mat. (N.S.),30(1999), 315–333.[9] Romeo F. Thomas, Entropy of expansive flows, Ergod. Th. Dynam. Sys., 7(1987), 611–625.[10] Romeo F. Thomas, Topological entropy of fixed-point free flows, Trans. Amer. Math. Soc.,319(1990), 601–618.[11] Y. Wang, E. Chen, Z. Lin, T. Wu, Bowen entropy of sets of generic points for fixed-point freeflows, J. Differential Equations, 269 (2020), no. 11, 9846–9867.[12] L. Xu, X. Zhou, Variational principles for entropies of nonautonomous dynamical systems, J.Dynam. Differential Equations, 30 (2018), no. 3, 1053–1062.[13] D. Zheng, E. Chen, Bowen entropy for actions of amenable groups, Israel Journal of Mathematics,212 (2016), 895–911.
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