Article The Conjecture

Lvlin Luo 1,2,3,4

1 Arts and Sciences Teaching Department, Shanghai University of Medicine and Health Sciences, Shanghai 201318, China; [email protected] or [email protected] 2 School of Mathematical Sciences, Fudan University, Shanghai 200433, China 3 School of Mathematics, Jilin University, Changchun 130012, China 4 School of Mathematics and Statistics, Xidian University, Xi’an 710071, China

Abstract: For a compact X, let J be the ordered set associated with the set of all

finite open covers of X such that there exists nJ, where nJ is the dimension of X associated with ∂. Therefore, we have Hˇ p(X; Z), where 0 ≤ p ≤ n = nJ. For a continuous self-map f on X, let α ∈ J be f an open cover of X and L f (α) = {L f (U)|U ∈ α}. Then, there exists an open fiber cover L˙ f (α) of X induced by L f (α). In this paper, we define a topological fiber entropy entL( f ) as the supremum of f ent( f , L˙ f (α)) through all finite open covers of X = {L f (U); U ⊂ X}, where L f (U) is the f-fiber of − U, that is the set of images f n(U) and preimages f n(U) for n ∈ N. Then, we prove the conjecture log ρ ≤ entL( f ) for f being a continuous self-map on a given compact Hausdorff space X, where ρ is the maximum absolute eigenvalue of f∗, which is the linear transformation associated with f on the n L Cechˇ homology group Hˇ ∗(X; Z) = Hˇ i(X; Z). i=0

Keywords: algebra equation; Cechˇ homology group; Cechˇ homology germ; eigenvalue; topological fiber entropy

  MSC: Primary 37B40; 55N05; Secondary 28D20

Citation: Luo, L. The Topological Entropy Conjecture. Mathematics 2021, 9, 296. https://doi.org/10.3390/ 1. Introduction math9040296 Recall that the pair (X, f ) is called a topological , which is induced

Academic Editor: Basil Papadopoulos by the iteration: and Marek Lampart n Received: 16 December 2020 f = f ◦ · · · ◦ f , n ∈ N | {z } Accepted: 31 January 2021 n Published: 3 February 2021 and f 0 is denoted the identity self-map on X, where X is a compact Hausdorff space and f is a continuous self-map on X. The preimage of a subset A ⊆ X is denoted by f −1(A). Publisher’s Note: MDPI stays neutral If the preimage of f −(n−1)(A) is defined, then by induction, the preimage of f −(n−1)(A) is with regard to jurisdictional claims in denoted by f −n(A), where n ∈ +. published maps and institutional affil- Z iations. 1.1. Brief History For a topological dynamical system (X, f ), let α and β be the collections of the finite open cover of X, and let:

Copyright: © 2021 by the authors.  α ∨ β = {A ∩ B; A ∈ α, B ∈ β};  Licensee MDPI, Basel, Switzerland.  −1( ) = { −1( ) ∈ }  f α f A ; A α , This article is an open access article f −1(α ∨ β) = f −1(α) ∨ f −1(β); (1) distributed under the terms and  n−1  W −i −1 −(n−1) + conditions of the Creative Commons  f (α) = α ∨ f (α) ∨ · · · ∨ f (α), n ∈ Z . = Attribution (CC BY) license (https:// i 0 creativecommons.org/licenses/by/ 4.0/).

Mathematics 2021, 9, 296. https://doi.org/10.3390/math9040296 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 296 2 of 17

For a finite open cover α of X, let N(α) be the infimum number of the subcover of α. Because X is compact, we get that N(α) is a positive integer. Hence, we define:

H(α) = log N(α) ≥ 0.

Following [1] (p. 81), if α, β are finite open covers of X, then we see:

α < β =⇒ H(α) ≤ H(β).

Definition 1 ([1], p. 89). For any given finite open cover α of X, define:

n−1 1 _ ent( f , α) = lim H( f −i(α)), n→+∞ n i=0

and define the topological entropy of f such that:

ent( f ) = sup{ent( f , α)}, α

where sup is through the all finite open cover of X. α

For a compact manifold M, let Hi(M; Z) be the i-th homology group of integer coeffi- cients, where 0 ≤ i ≤ dim M. In 1974, M. Shub stated the topological entropy conjecture [2], which usually has been called the entropy conjecture [3], that is,

Conjecture 1. The inequality:

log ρ ≤ ent( f ) is valid or not for any C1 self-map f on a compact manifold M, where ent( f ) is the topological entropy of f and ρ is the maximum absolute eigenvalue of f∗, which is the linear transformation associated with f on the homology group:

dim M M H∗(M; Z) = Hi(M; Z). i=0

In the first place, the inequality of Conjecture1 is connected to the work of S. Smale [4–7], M. Shub [8,9], and D. P. Sullivan [10–12]. In 1975, Manning [13] proved that Conjecture1 holds for any of manifolds X for which dim X ≤ 3, Shub and Williams [14] proved Conjecture1 on mani- folds M for no cycle , which are ; also, Ruelle and Sullivan [15] proved Conjecture1 on manifolds M, which have an oriented expanding X ⊂ M. In the same year, Pugh [16] proved that there is a homeomorphism f of some smooth M8 such that Conjecture1 is invalid. In 1977, Misiurewicz et al. [17,18] proved that Conjecture1 holds for any smooth maps + on X = Sn and for any continuous maps on Tn with n ∈ Z . In 1980, Katok [19] proved that if a C1+α (α > 0) f of a compact manifold has a Borel probability continuous (non-atomic) invariant ergodic measure with non-zero Lyapunov exponents, then it has positive topological entropy. In 1986, Katok [20] proved that if the universal covering space of X is homeomorphic to the Eu- clidean space, then Conjecture1 holds for any f ∈ C∞(X); also, he gave a counterexample explaining that the inequality of Conjecture1 is invalid for a continuous map, that is on two-dimensional sphere S2, there is f ∈ C0(S2) such that:

0 = ent( f ) < log ρ. Mathematics 2021, 9, 296 3 of 17

For a C∞ mapping, Yomdin [21] in 1987 and Newhouse [22] in 1989 proved Conjecture1 , respectively. + In 1992, for n-dimensional compact Riemannian manifolds with n ∈ Z , Paternain made a relation between the geodesic entropy and topological entropy of the geodesic flow on the unit tangent bundle [23], which is an improvement of Manning’s inequality [24]. In 1994, Ye [25] showed that of Suslinian chainable continua and homeomorphisms of hereditarily decomposable chainable continua induced by square commuting diagrams on inverse systems of intervals have zero topological entropy. + In 1997, for a closed connected C∞ manifold Mn with n ∈ Z , Mañé [26] provided an equality to relate the exponential growth rate of geodesic entropy, as a of T, which is parametrized by the arc length, with the topological entropy of the geodesic flow on the unit tangent bundle. u In 2000, Cogswell gave that µ-a.e. x ∈ X is contained in an open disk Dx ⊂ W (x), which exhibits an exponential volume growth rate greater than or equal to the measure- theoretic entropy of f with respect to µ, where f ∈ C1+1(X) and f is a measure-preserving transformation [27]. In 2002, Knieper et al. [28] showed that every orientable compact surface has a C∞ open and dense set of Riemannian metrics whose geodesic flow has positive topological entropy. In 2005, Bobok et al. [29] proved the inequality of Conjecture1 for a compact man- ifold X and for any continuously differentiable map f : X → X, which is m-fold at all regular values. In 2006, Zhu [30] showed that for Ck-smooth random systems, the volume growth is bounded from above by the topological entropy on compact Riemannian manifolds. In 2008, Marzantowicz et al. [3] proved the inequality of Conjecture1 for all continuous mappings of compact nilmanifolds. In 2010, Saghin et al. [31] proved the inequality of Conjecture1 for partially hyperbolic diffeomorphism with a one-dimensional center bundle. In 2013, Liao et al. [32] proved the inequality of Conjecture1 for diffeomorphism away from ones with homoclinic tangencies. In 2015, Liu et al. [33] proved the inequality of Conjecture1 for diffeomorphism that are partially hyperbolic . In 2016, Cao et al. [34] proved the inequality of Conjecture1 for dominated splittings without mixed behavior. In 2017, Zang et al. [35] proved the inequality of Conjecture1 for controllable domi- nated splitting. In 2019, Lima et al. [36] developed symbolic dynamics for smooth flows with positive topological entropy on three-dimensional closed (compact and boundaryless) Riemannian manifolds. In 2020, Hayashi [37] proved the inequality of Conjecture1 for nonsingular C1 endo- morphisms away from homoclinic tangencies, extending the result of [32]. Lately, for results about random entropy expansiveness and dominated splittings, see [38], and for results about the relations of topological entropy and Lefschetz numbers, see [39–41]. Furthermore, for a variational principle for subadditive preimage topological pressure for continuous bundle random dynamical systems, see [42].

1.2. Motivation and Main Results Conjecture1 is not proven completely. For a compact Hausdorff space X, let J be the ordered set associated with the set of all finite open covers of X such that there exists nJ, where nJ is the dimension of X associated with ∂, which will become clear in Definition3. Therefore, we have Hˇ p(X; Z), where 0 ≤ p ≤ n = nJ. For a continuous self-map f on X, let α ∈ J be an open cover of X and L f (α) = {L f (U)|U ∈ α}. Then, there exists an f open fiber cover L˙ f (α) of X induced by L f (α). In this paper, we define a topological fiber entropy entL( f ) as the supremum of ent( f , L˙ f (α)) through all finite open covers of Mathematics 2021, 9, 296 4 of 17

f n X = {L f (U); U ⊂ X}, where L f (U) is the f-fiber of U, that is the set of images f (U) and − preimages f n(U) for n ∈ N. Then, we prove the inequality

log ρ ≤ entL( f ).

where ρ is the maximum absolute eigenvalue of f∗, which is the linear transformation associated with f on the Cechˇ homology group:

n M Hˇ ∗(X; Z) = Hˇ i(X; Z). i=0 Specifically, in triangulable compact n-dimensional manifold M, we get:

H∗(M; Z) = Hˇ ∗(M; Z). Hence, Conjecture1 is valid for topological fiber entropy. In this paper, we always let Ψ ∈ J be good enough and have enough refinement, i.e., satisfy all the necessary requirements of this paper. Define:

αc = {A; Ac ∈ α, where Ac ∪ A = X and Ac ∩ A = ∅}, and define:   a0 ··· ai−1aˆi, ai+1 ··· ap = a0 ··· ai−1ai+1, ··· ap;  (i)  a ··· a − b a ··· ap = a ··· a − ba ··· ap;  0 i 1 i 0 i 1 i  ··· (i) ··· = ··· ···  a0 ai−1b(k)ai ap ∑ a0 ai−1bmai ap;  m∈(k) ··· (i) ··· = ··· ··· = ··· ···  a0 ai−1b∅ ap ∑ a0 ai−1bmai ap a0 ai−1ai ap;  m∈∅   (k) = {k1, k2, k3, ··· , kn; n = k{a0, ··· , ai−1, bm, ai, ··· , ap}k ≥ 1, m ∈ Z};  d  ( ··· ˆ ··· ) = ··· ··· ∈ ( ) a0 b(k) ap bk1 bki bkn , ki k .

2. Algebra Equation for the Boundary Operator In this paper, let X be a compact Hausdorff space, C0(X) be the set of all continuous self-maps on X, and id be the identity map on X. Let α, β be finite open covers of X, if for any B ∈ β, there is A ∈ α such that B ⊆ A, then we define α ≤ β and say that β is larger than α or β is a refinement of α. For A ∈ α, let A be the closure of A and kAk be the number of elements of A.

Definition 2 ([43], p. 541). Let X be a Hausdorff space, Ψ be a cover of X, and U0, U1, U2, ... , T T T Up ∈ Ψ with p ∈ N. If U0 U1 ··· Up 6= ∅, then we define a p- σp. Hence, we get the p-th chain group Cp, the p-th homology group Hp(Ψ; Z), and the p-th cohomology group Hp(Ψ; Z), where:

∂p+1 ∂p · · · −→ Cp+1(Ψ; Z) −−→ Cp(Ψ; Z) −→ Cp−1(Ψ; Z) −→· · · p i ˆ ∂p(U0 ∩ · · · ∩ Up) = ∑(−) (U0 ∩ · · · ∩ Ui · · · ∩ Up), i

∂p−1 ◦ ∂p = 0, Bp(Ψ; Z) = im ∂p+1, Zp(Ψ; Z) = ker ∂p and Hp(Ψ; Z) = Zp/Bp.

p p p−1 δ Let C (Ψ; Z) = hom(Cp(Ψ; Z), Z). Then, ∂p induces a homomorphism C (Ψ; Z) −→ Cp(Ψ; Z). We obtain that:

+ δp+1 δp − · · · ←− Cp 1(Ψ; Z) ←−− Cp(Ψ; Z) ←− Cp 1(Ψ; Z) ←−· · · ,

+ + δp 1 ◦ δp = 0, Bp(Ψ; Z) = im δp, Zp(Ψ; Z) = ker δp 1 and Hp(Ψ; Z) = Zp/Bp. Mathematics 2021, 9, 296 5 of 17

Lemma 1. Let X be a Hausdorff space and Ψ be a finite open cover of X. Then, we get Cp(Ψ; Z) =∼ i c Cp(Ψ; Z) with p ∈ N. Moreover, let U = Ui . If:

cp = U0 ∩ · · · ∩ Up ∈ Cp(Ψ; Z), then:

cp = U0 ∪ · · · ∪ Up 6= X is an isomorphic representation of the p-simplex of Cp(Ψ; Z).

Proof. Because Z can be treated as a finite generated free ring [44], Cp(Ψ; Z) can be treated as a finite-dimensional Z-module space [45], and Cp(Ψ; Z) can be treated as the dual Z-module space of Cp(Ψ; Z). With the property of the finite-dimensional Z-module space, p ∼ we get C (Ψ; Z) = Cp(Ψ; Z). Because:

p 0 p cp = U0 ∩ · · · ∩ Up 6= ∅ ⇐⇒ c = U ∪ · · · ∪ U 6= X, we get:

p p cp ∈ Cp(Ψ; Z) ⇐⇒ c ∈ C (Ψ; Z). That is, U0 ∪ · · · ∪ Up 6= X is an isomorphic representation of the p-simplex of Cp(Ψ; Z).

Definition 3. Let X be a Hausdorff space, Ψ be a finite open cover of X, and J be the ordered set associated with the refinement of the finite open cover of X. Then, we define the function nΨ = max{n; n ∈ S} on J. Obviously, if α, β ∈ J and α ≤ β, then nα ≤ nβ. If there exists = nJ lim−−→ nΨ, then we say that nJ is the dimension of X associated with ∂, where: Ψ∈J

S = {n; ∂(U0 · · · ∩ Ui ··· Un) 6= ∂(U0 · · · ∩ Ui ··· Un ∩ Un+1), U0, ··· , Un+1 ∈ Ψ}.

Definition 4. Let X be a Hausdorff space, Ψ be a finite open cover of X, and 0 ≤ p ≤ n = nΨ. If for any σp ∈ Cp(Ψ; Z), there exists σn ∈ Cn(Ψ; Z) such that σp = U0 ∪ · · · ∪ Up is the p-th surface of σn and:

d 0 ˆ p k0 kn−p+1 (U ∪ · · · ∪ U(k) · · · ∪ U ) = U ∪ · · · ∪ U .

Then, we say that X is a Poincaré space.

Lemma 2. Let X be a Poincaré space and Ψ be its finite open cover. For 0 ≤ p ≤ n = nΨ, we get p ∼ that H (Ψ; Z) = Hn−p(Ψ; Z).

Proof. By Lemma1, we get the following chains of the mapping:  ∂p+1 ∂p  · · · −→ Cp+1(Ψ; Z) −−→ Cp(Ψ; Z) −→ Cp−1(Ψ; Z) −→· · · + (2) + δp 1 δp −  · · · ←− Cp 1(Ψ; Z) ←−− Cp(Ψ; Z) ←− Cp 1(Ψ; Z) ←−· · · Mathematics 2021, 9, 296 6 of 17

For a fixed p-simplex in Cp(Ψ; Z), we see the algebraic equation:

 p−1 p−1 < ∂cp, c >=< cp, δc >,  p  i  ∂p(U0 · · · ∩ Ui ··· Up) = ∑ (−1) (U0 · · · ∩ Uˆi ··· Up), i=0 (3)  ∂∅ = δ∅ = 0,   < a, ∅ >=< ∅, b >= 0;

and the algebraic equation:

 p  < (− )i( · · · ∩ ˆ ··· ) · · · ∩ ˆ ··· >=< p−1 >  ∑ 1 U0 Ui Up , V0 Vi Vp cp, δc , i=0 p (4)  (−)i( 0 · · · ∪ (i) ··· p) = p( 0 · · · ∪ ˆ ··· p)  ∑ V V(k) V δ V V(k) V . i=0

If (k) = ∅, then we define:

∅ p 0 ˆ p (i) = ∅, (−1) = 0 and δ (U · · · ∪ U(k) ··· U ) = 0.

From (3) and (4), we obtain that:

 p  ( · · · ∩ ··· ) = (− )i( · · · ∩ ˆ ··· )  ∂p U0 Ui Up ∑ 1 U0 Ui Up , i=0 p (5)  p( 0 · · · ∪ ˆ ··· p) = (− )i( 0 · · · ∪ (i) ··· p)  δ U U(k) U ∑ 1 U U(k) U . i=0 That is,

 p  i ˆ  ∂p(U0 · · · ∩ Ui ··· Up) − ∑ (−1) (U0 · · · ∩ Ui ··· Up) = 0,  i=0  p  p( 0 · · · ∪ ˆ ··· p) − (− )i( 0 · · · ∪ (i) ··· p) =  δ U U(k) U ∑ 1 U U(k) U 0, i=0 (6) d  n−p+1 0 ˆ p n−p+1 k1 km kn−p+1  δ ((U · · · ∪ U(k) ··· U ) ) = δ (U · · · ∪ U ··· U ),  p  i (i)  n−p+1( k1 · · · ∪ km ··· kn−p+1 ) = (− ) ( k1 · · · ∪ ··· kn−p+1 )  δ U U U ∑ 1 U U(0,··· ,p) U . i=0

Let:

= ( · · · ∩ ··· ) cp ∑ zm U0 Ui Up m. Then, we see that:

d cn−p = z ((U0 · · · ∪ Uˆ ··· Up) ) , where z ∈ . ∑ m (k) m m Z Therefore, we obtain that:

 0 ˆ p 0 ˆ p d  U0 · · · ∩ Ui ··· Up ←→ U · · · ∪ U(k) ··· U ←→ (U · · · ∪ U(k) ··· U ) , n−p n−p+1 cp ∈ ker ∂p ⇐⇒ c ∈ ker δ , (7)  n−p n−p  cp ∈ im ∂p+1 ⇐⇒ c ∈ im δ .

Let: ( ∂ ker (Cp) = Hp(Ψ; Z) = Zp/Bp = ker ∂p/im ∂p+1, im ∗ p p p p p+1 p (8) ∂ ker (C ) = H (Ψ; Z) = Z /B = ker δ /im δ . im Mathematics 2021, 9, 296 7 of 17

n−p+1 Then, ∂p and δ are the dual solutions in the algebraic Equation (6). Similarly, ∗ ∂ ker and ∂ ker are the dual values in the algebraic Equation (8). All the processes of the dual im im maps are linear reversible, i.e., the same style as isomorphisms. Therefore, the p-th value of ∗ n−p ∂ ker on the Cp chain group is isomorphic to the (n − p)-th value of ∂ ker on the C chain im im group, that is,

∼ ∗ n−p ∂ ker (Cp) = ∂ ker (C ). im im For this reason, we see that:

p ∼ H (Ψ; Z) = Hn−p(Ψ; Z).

Like the linear equation in Euclidean space R3, let:

Si : Aix + Biy + Ciz = 0

be a class of lines, or in other words, a class of planes:

∗ Si : Aix + Biy + Ciz = 0.

+ where i ∈ Z and i ≥ 2. The line and plane are a pair of duals. For a fixed space R3, the intrinsic relationships between lines or between planes are never changed. That is, f and g are two good maps such that they are linear, if:

∗ ∗ ∗ ∗ ∗ ∗ fi = f (Si, Si−1), fi = f (Si , Si+1), gi = g( fi) and gi = g( fi ).

∗ then gi and gi is a pair of duals such that there is a natural relationship between gi and ∗ gn−i. For example, that natural relationship may be:

∗ ∗ ∗ gi = gn−i, or gi gn−i = 1, or gi + gn−i = 0, or:

∗ ∗ gi Ak + gn−iBk + Ck = 0 and gn−i Ak + gi Bk + Ck = 0,

and so on. The dual outcomes and the representations of the natural relation between gi ∗ and gn−i only depend on the good maps f and g.

3. Germ and Dual of the Cechˇ Homology Definition 5 ([43], p. 542). Let X be a Hausdorff space and J be the ordered set associated with the T set of all covers of X, U0, U1, U2, ... , Up ∈ Ψ with p ∈ N and Ψ ∈ J. If U0 U1 ∩ · · · ∩ Up 6= ∅, then we define a p-simplex σp. Hence, we get the p-th chain group Cp, the p-th homology group p Hp(Ψ; Z), and the p-th cohomology group H (Ψ; Z). If Ω, Ψ ∈ J and Ω ≤ Ψ, then we get the homomorphisms:

p p fΨΩ : Hp(Ψ; Z) → Hp(Ω; Z), and fΩΨ : H (Ω; Z) → H (Ψ; Z).

Hence, we define the p-th Cechˇ cohomology group:

ˇ p( ) = p( ) H X; Z −−→lim H Ω; Z . Ω∈J

Following Definition5, we have the following definition. Mathematics 2021, 9, 296 8 of 17

Definition 6. Let X be a Hausdorff space and J be the ordered set associated with the set of all finite open covers of X such that there exist nJ. For 0 ≤ p ≤ n = nJ, there exists the p-th Cechˇ homology group:

ˇ ( ) = ( ) Hp X; Z ←−−lim Hp Ω; Z . Ω∈J

Definition 7. Let X be a Poincaré space and J be the ordered set associated with the set of all finite open covers of X such that there exists nJ. For Ω, Ψ ∈ J, let Θ = Ψ ∨ Ω = {α ∩ β; α ∈ Ψ, β ∈ Ω}. Then, we get homomorphisms

fΘΩ : Hp(Θ; Z) → Hp(Ω; Z) and fΘΨ : Hp(Θ; Z) → Hp(Ψ; Z).

Following this, we can define the Cechˇ homology germ Hp(J; Z). Similarly, we define the Cechˇ p ∼ n−p cohomology germ H (J; Z). If there exists Γ ∈ J such that, we get Hp(Ψ; Z) = H (Ψ; Z) for any Ψ ∈ J whenever Γ ≤ Ψ, then we define:

p ∼ H (J; Z) = Hn−p(J; Z),

where n = nJ.

By Lemma2, Definitions5–7, we get the following result.

Lemma 3. Let X be a Poincaré space and J be the ordered set associated with the set of all finite open covers of X such that there exists nJ. For 0 ≤ p ≤ n = nJ, we get that:

p p Hˇ p(X; Z) ∼ Hp(J; Z) and Hˇ (X; Z) ∼ H (J; Z), where ∼ means the different expressions for the same thing.

Definition 8. Let X be a Poincaré space and J be the ordered set associated with the set of all finite open covers of X such that there exists nJ. For n = nJ, if:

∼ n−p Hp(J; Z) = H (J; Z), then we define:

∼ n−p Hˇ p(X; Z) = Hˇ (X; Z).

4. f-Cechˇ Homology 0 Definition 9. Let X be a Hausdorff space, Ui, V, W ⊆ X and f ∈ C (X), where 0 ≤ i ≤ k and k ∈ Z. Then, we define:  L (U) = (··· , f −n(U), ··· , f −1(U), f 0(U), f 1(U), ··· , f n(U), ··· ),  f  f ◦ L = L ◦ f ,  f f  L (U) ∩ L (V) = L (W) where W = U ∩ V  f f f , , L f (U0) · · · ∩ L f (Ui) ··· L f (Uk) = L f (U0) ∩ (L f (U1) · · · ∩ L f (Ui) ∩ L f (Uk)),  L (U) = (··· , g−n(U) S h−n(U), ··· , g0(U) S h0(U), ··· , gn(U) S hn(U), ··· ),  g+h  L ( ) =  f ∅ ∅,  −1 −1  Lg⊕h(U) = Lg+h(U), when g (U) ∩ h (U) = ∅,

−1 where f (U) is the preimage of U. We say that L f (U) is the f -fiber of U and let f X = {L f (U); U ⊂ X}. Mathematics 2021, 9, 296 9 of 17

−∞ +∞ If X is a , then X+∞ = ∏ X × X × ∏ X is compact as well by the i=−1 i=1 Tychonoff theorem. In fact, in Definition9, L f (U) glues the preimage and image orbit of U. f If X is a discrete Hausdorff space, then we get that X +∞ is the direct limit space X× ∏ X = i 1 f of (X, f ) following [46], but X −∞ is not the inverse limit space of (X, f ). ∏ X×X i=−1

Definition 10. Let X be a Hausdorff space, and let J be the ordered set associated with the set of all 0 finite open covers of X. Let f ∈ C (X), Ψ ∈ J and U0, ··· , Up ∈ Ψ with p ∈ N. If:

f σp = L f (U0) ∩ · · · ∩ L f (Up) 6= ∅,

f then we define an f -Cechˇ p-simplex σp . Hence, we get the f -Cechˇ p-chain group Cp(Ψ, f ; Z), and we get the f -Cechˇ p-th homology group Hp(Ψ, f ; Z), where:

f f ∂p+1 ∂p · · · −→ Cp+1(Ψ, f ; Z) −−→ Cp(Ψ, f ; Z) −→ Cp−1(Ψ, f ; Z) −→· · · , p f i ∂p(L f (U0) · · · ∩ L f (Ui) ··· L f (Up)) = ∑ (−1) (L f (U0) · · · ∩ Lˆ f (Ui) ··· L f (Up)). i=0

f f It is easy to get that ∂p−1 ◦ ∂p = 0, that is,

f f ∂p−1 ◦ ∂p(L f (U0) · · · ∩ L f (Ui) ··· L f (Up)) p i f = ∑ (−1) ∂ (L f (U0) · · · ∩ Lˆ f (Ui) ··· L f (Up)) i p i+j = ∑ ∑ (−1) (L f (U0) · · · ∩ Lˆ f (Uj) · · · ∩ Lˆ f (Ui) ··· L f (Up)) i ji = 0.

Therefore, we see that:

f f Bp(Ψ, f ; Z) = im ∂p+1, Zp(Ψ, f ; Z) = ker ∂p and Hp(Ψ, f ; Z) = Zp(Ψ, f ; G)/Bp(Ψ, f ; Z).

By Lemma3 and Definition9, we easily have the following lemma.

f Lemma 4. A Cechˇ p-chain cp is associated with an f -Cechˇ p-chain cp, that is U0 ∩ U1 ∩ · · · ∩ Up 6= ∅ if and only if L f (U0) ∩ L f (U1) ∩ · · · ∩ L f (Up) 6= ∅. Therefore, the Cechˇ p-chain group is isomorphic to the f -Cechˇ p-chain group.

Definition 11. Let X be a Hausdorff space, f ∈ C0(X), and Ψ be a finite open cover of X. Let J be the ordered set associated with the refinement of the finite open cover of X. Then, we define the function nΨ, f = max{n; n ∈ S} on J. Obviously, if α, β ∈ J and α ≤ β, then nα, f ≤ nβ, f . If there exists: = nJ, f lim−−→ nΨ, f , Ψ∈J

f then we say that nJ, f is the dimension of (X, f ) associated with ∂ , where:

f S = {n; ∂ (L f (U0) · · · ∩ L f (Un)) 6= ∂(L f (U0) · · · ∩ L f (Un) ∩ L f (Un+1)), U0, ··· , Un+1 ∈ Ψ}. Mathematics 2021, 9, 296 10 of 17

Similarly, with Definitions 5–7 and the following Definition 11, we obtain the following definition.

Definition 12. Let X be a Hausdorff space, f ∈ C0(X), and J be the ordered set associated with the refinement of the finite open cover of X such that there exists nJ, f . Let Θ = Ψ ∨ Ω = {α ∩ β; α ∈ Ψ, β ∈ Ω} with Ω, Ψ ∈ J. For 0 ≤ p ≤ n = nJ, f , we get homomorphisms:

fΘΩ : Hp(Θ, f ; Z) → Hp(Ω, f ; Z) and fΘΨ : Hp(Θ, f ; Z) → Hp(Ψ, f ; Z).

Therefore, we get the pth f -Cechˇ homology germ Hp(J, f ; Z) and the pth f -Cechˇ homology group:

ˇ ( ) = ( ) Hp X, f ; Z ←−−lim Hp Ω, f ; Z . Ω∈J

Lemma 5. Let X be a Hausdorff space, f ∈ C0(X), and J be the ordered set associated with the set of all finite open covers of X such that there exist nJ and nJ, f . Then, we have nJ = nJ, f , and we get Hˇ p(X, f ; Z) and Hˇ p(X; Z), where 0 ≤ p ≤ n = nJ. Moreover, for Ψ ∈ J, we get that:

f im ∂p+1 = Bp(Ψ; Z) = Bp(Ψ, f ; Z) = im ∂p+1, f ker ∂p = Zp(Ψ; Z) = Zp(Ψ, f ; Z) = ker ∂p and Zp(Ψ; G)/Bp(Ψ; Z) = Hp(Ψ; Z) = Hp(Ψ, f ; Z) = Zp(Ψ, f ; G)/Bp(Ψ, f ; Z).

Using Lemmas3,5 and Definition 12, we see the following result.

Lemma 6. Let X be a Hausdorff space, f ∈ C0(X), and J be the ordered set associated with the set of all finite open covers of X such that there exist nJ, f . For 0 ≤ p ≤ n = nJ, f , we obtain:

Hp(J, f ; Z) ∼ Hˇ p(X, f ; Z),

where ∼ means the different expressions for the same thing.

Furthermore, we can define the f -Cechˇ cohomology germ Hp(J, f ; Z), the f -Cechˇ co- p homology group Hˇ (X, f ; Z), and the f -Poincaré space. Obviously, we get that Cp(X; Z) = Cp(X, id; Z). For convenience, let:

n n n L L L Hˇ ∗(X; Z) = Hˇ i(X; Z), C∗(X; Z) = Ci(X; Z), B∗(X; Z) = Bi(X; Z), i=0 i=0 i=0 n n L L Hˇ ∗(X, f ; Z) = Hˇ i(X, f ; Z), C∗(X, f ; Z) = Ci(X, f ; Z) and i=0 i=0 n L B∗(X, f ; Z) = Bi(X, f ; Z). i=0

By Lemmas4 and6, we have the following lemma.

Lemma 7. Let X be a Hausdorff space, f ∈ C0(X), and J be the ordered set associated with the set of all finite open covers of X such that there exist nJ and nJ, f . Then, nJ = nJ, f and for n = nJ = nJ, f . We have Hˇ p(X; Z) and Hˇ p(X, f ; Z), where 0 ≤ p ≤ n. Moreover, there are linear transformations f∗ associated with f on Hˇ ∗(X; Z), on C∗(X; Z), and on Hˇ ∗(X, f ; Z), respectively.

If Ef∗ is the set of all eigenvalues of f∗ and:

kEf∗ k = sup{|a|; a ∈ Ef∗ }, then we obtain the inequalities: Mathematics 2021, 9, 296 11 of 17

kE | k ≤ kE | k ≤ kE | k, f∗ Hˇ ∗(X, f ;Z) f∗ Z∗(X, f ;Z) f∗ C∗(X, f ;Z) k | k ≤ k | k ≤ k | k Ef∗ H∗(X;Z) Ef∗ Z∗(X;Z) Ef∗ C∗(X;Z) and (9) k | k ≤ k | k Ef∗ C∗(X;Z) Ef∗ C∗(X, f ;Z) .

f What is more, we can define the LC0 category that its objects are X and its morphisms are continuous maps, where X is a Hausdorff space and f is a continuous self-map on ˜ ˇ X. Similarly, we can define the LC0 category for which its objects are H∗(X, f ; Z) and 0 0 its morphisms are F∗, where X, Y are Hausdorff spaces, f ∈ C (X), g ∈ C (Y), and F∗ is associated with the continuous map F : X f → Yg. Furthermore, we can define the homotopy and homeomorphism from X f to Xg and research the relations between the ˜ elements of LC0 and LC0 .

Definition 13. Let X, Y be compact Hausdorff spaces, f ∈ C0(X) and g ∈ C0(Y). f g g f (a) If there exist continuous maps F : X −→ Y and D : Y −→ X such that F ◦ D = idYg and f g D ◦ F = idX f , then we say that X and Y are L1-homotopy equivalent. (b) If there exists a continuous map F : X f × [0, 1] −→ Yg such that F(X f , 0) = h(X f ) and f f f g F(X , 1) = r(X ), then we say that h, r : X −→ Y are the L2-homotopy. Hence, h induces a homomorphism:

h∗ : Hˇ ∗(X, f ; Z) −→ Hˇ ∗(Y, g; Z),

and r∗ induced by r.

Let L be the class of objects:

{X f ; X is a compact Hausdorff space, f ∈ C0(X)}.

f g f g f g For each pair X , Y ∈ L, let mors(X , Y ) = L1(X , Y ). By the definition of the L1-homotopy and the composition function ◦, we get the category (L, mors, ◦). f Let L˜ be the class of objects {Hˇ ∗(X, f ; Z); X ∈ L}. Let:

morH(Hˇ ∗(X, f ; Z), Hˇ ∗(Y, g; Z))

be the group homomorphism from Hˇ ∗(X, f ; Z) to Hˇ ∗(Y, g; Z), where Hˇ ∗(X, f ; Z), Hˇ ∗(Y, g; Z) ∈ L˜ . By the induced ∗ homomorphism of the L1-homotopy and the composition function ◦, we get the category (L˜ , morH, ◦). Easily, we get a functor from (L, mors, ◦) to (L˜ , morH, ◦). Then, by diagram chasing, we see the following:

Theorem 2. Let f ∈ C0(X), g ∈ C0(Y), and let X and Y be compact Hausdorff spaces. f g (a)If X and Y are L1-homotopy equivalent, then:

Cp(X, f ; Z) = Cp(X, g; Z) and Hˇ p(X, f ; Z) = Hˇ p(X, g; Z).

f g (b)If h, r : X −→ Y are the L2-homotopy, then h∗ = r∗.

Example 1. Let f ∈ C0(X), g ∈ C0(Y), and let X and Y be compact Hausdorff spaces. If there exists a homeomorphism F from X to Y such that F f = gF, then:

Hˇ p(X, f ; Z) = Hˇ p(X, g; Z). Mathematics 2021, 9, 296 12 of 17

Example 2. Let f ∈ C0(X), g ∈ C0(Y), and let X and Y be compact Hausdorff spaces. If there exists a homeomorphism F from X to Y, then:

Hˇ p(X, f ; Z) = Hˇ p(X, g; Z).

Example 3. Let f ∈ C0(X), g ∈ C0(Y), and let X and Y be compact Hausdorff spaces. If there exists a continuous map F : X × [0, 1] −→ Y such that:

 F(X, 0) = h(X) F(X, 1) = r(X)

that is h and r are homotopies. Then, h∗ = r∗, where:

h∗ : Hˇ ∗(X, f ; Z) −→ Hˇ ∗(Y, g; Z) and r∗ : Hˇ ∗(X, f ; Z) −→ Hˇ ∗(Y, g; Z).

5. Topological Fiber Entropy In this section, X is a compact Hausdorff space and J is the set of all finite open covers of X such that there exists nJ. For n = nJ, we have Hˇ p(X; Z), where 0 ≤ p ≤ n. Let α be an open cover of X and L f (α) = {L f (U)|U ∈ α}. Then, there exists an open f fiber cover L˙ f (α) of X induced by L f (α).

f Definition 14. For a fixed open fiber cover L˙ f (α) of X , define:

 −1 ˙ f (L f (α)) −1 ˙ ˙  ˙ = max k{ f L f (U) ∩ L f (U)}k;  L f (α) U∈α  ˙  f (L f (α)) ˙ ˙ ˙ = max k{ f L f (U) ∩ L f (U)}k; L f (α) U∈α −1 ˙ ˙  f (L f (α)) f (L f (α))  Ld = max{ ˙ , ˙ };  L f (α) L f (α)   ent( f , L˙ f (α)) = ent( f , α) + log Ld.

and define the topological fiber entropy of f by:

entL( f ) = sup {ent( f , L˙ f (α))}, L˙ f (α)

where sup is through all finite open covers of X f . L˙ f (α)

Lemma 8 ([1], p. 102). If f is the shift operator on a k-symbolic space, then ent( f ) = log k.

Corollary 1. If f is the shift operator on a k-symbolic space, then:

entL( f ) = ent( f ) + log k = 2 log k.

 {1} → {1, 2, ··· , k},   {2} → {1, 2, ··· , k}, Example 4. Let {1, 2, ··· , k} = X and f : . . . . Then:  . . .   {k} → {1, 2, ··· , k}

ent( f ) = 0, entL( f ) = 0.

Example 5. Let {1, 2, ··· , k} = X and f : {1, 2, ··· , k} → {1} . Then:

ent( f ) = 0, entL( f ) = 0. Mathematics 2021, 9, 296 13 of 17

Example 6. Let [0, 1] = X and f (x) = kx, 0 < k < 1. Then:

ent( f ) = 0, entL( f ) = 0.

Lemma 9. For m ∈ Z and m > 2, there are p, q ∈ Z such that p 6= q and m = p + q, where 1 ≤ p, 1 ≤ q.

0 Let f ∈ C (X) and f∗ be the linear transformation on Hˇ ∗(X, f ; Z) associated with f . We say that a Cechˇ eigenvalue chain is the chain belonging to an eigenvalue of f∗. Then, any Cechˇ eigenchain can be associated with an open cover of X f .

Lemma 10. Let X be a compact Hausdorff space and J be the ordered set associated with the set of all finite open covers of X such that there exist nJ and nJ, f . Then, nJ = nJ, f , and for n = nJ = nJ, f , we have Hˇ p(X; Z) and Hˇ p(X, f ; Z), where 0 ≤ p ≤ n. Let α ∈ J be an open cover of X. If L f (α) is a Cechˇ eigenchain belonging to the eigenvalue m, then L f (α) has a factor conjugating with a shift operator on m-symbolic space or Ld = m, where m ∈ N.

k Proof. By Lemma 6, for an eigenchain L f = ∑ aiσˇi belonging to the eigenvalue m, there i=0 exists the f -Cechˇ homology germ Hp(J, f ; Z) such that:

Hp(J, f ; Z) ∼ Hˇ p(X, f ; Z), 0 ≤ p ≤ nJ.

where σˇi ∈ Hˇ ∗(X, f ; Z) and m, ai ∈ Z. Hence, there exists Φ ∈ J such that L f ∈ H∗(Φ, f ; G) and:

f∗(L f ) = m(L f ).

That can be extended to an equation on C∗(Φ, f ; G), and we get the equation:

f](σˇi) = m(σˇi), i ∈ {0, ··· , k},

where σˇi ∈ C∗(Φ, f ; G) and m ∈ Z. Just thinking of f] on C∗(Φ, f ; G), let U0, ··· , Uj be open subsets of X and:

σˇi = L f (U0) ∩ · · · ∩ L f (Uj).

Then, we see:

−n −1 0 1 n L f (Uη) = (··· , f (Uη), ··· , f (Uη), f (Uη), f (Uη) ··· , f (Uη), ··· ),

where η ∈ {0, ··· , j}. Therefore, T T T T f](σˇi) = f](L f (U0) ··· L f (Uj)) = L f ( f (U0)) ··· L f ( f (Uj)) T T = m(L f (U0) ··· L f (Uj)).

That is,

j T −n −1 0 n m( (··· , f (Uη), ··· , f (Uη), f (Uη), ··· , f (Uη), ··· )) η=0 j T −n −1 0 n = (··· , f ( f (Uη)), ··· , f ( f (Uη)), f ( f (Uη)), ··· , f ( f (Uη)), ··· ) η=0 j T −(n−1) −1 2 n+1 = (··· , f ( f (Uη)), ··· , f ( f (Uη)), f (Uη), f (Uη), ··· , f (Uη), ··· ). η=0 Mathematics 2021, 9, 296 14 of 17

Therefore, we see that:

j j \ \ m( L f (Uη)) = ( L f ( f (Uη))). η=0 η=0

Without loss of generality, let j = 0. Then:

L f ( f (U0)) = m(L f (U0)).

If L f (U0) is torsion, then the conclusion is trivial. Next, we only prove the conclusion for L f (U0), which is torsion free. Now, let L f (U0) be a torsion free element. (i) m = 0, 1; the conclusion is trivial. −1 (ii) If m = 2, then there exists U ⊆ f ( f (U0)) such that U * U0 and U0 * U, where U0, U are non-empty open subsets of X. −1 If f ( f (U0)) = U0, then:

L f ( f (U0)) = (L f (U0)) = 2(L f (U0));

this is a contradiction for the property that Z is a free group. Because of U * U0 and U0 * U, with the property of the Hausdorff space, there exist points x, y such that x ∈ U0, but x ∈/ U, and y ∈ U, but y ∈/ U0. Then, there exist open neighborhoods O(x) of x and O(y) of y, respectively, such that:

x ∈ O(x) ⊆ U0 but O(x) * U and y ∈ O(y) ⊆ U but O(y) * U0.

−1 T That is, O(x), O(y) ⊆ f ( f (U0)) and O(x) O(y) = ∅. Hence, Ld = 2, and for m = 2, the conclusion is true. (iii) m ≥ 3; from the mathematical induction, let the conclusion be right for m = n − 1. Then, we see the conclusion for m = n. Using Lemma9, we get m = p + q, p 6= q and:

L f ( f (U0)) = p(L f (U0)) + q(L f (U0)).

Therefore, there exists f |U0 = h + g such that:

Lh( f (U0)) = p(L f (U0)) and Lg( f (U0)) = q(L f (U0)).

(1) If L f 6= Lh⊕g, then using (ii) with the same computing, we get:

Ld = m.

(2) If L f = Lh⊕g, then we get:

L f ( f (U0)) = Lh( f (U0)) ⊕ Lg( f (U0));

else, we get: −1 \ −1 h ( f (U0)) g ( f (U0)) = W 6= ∅.

That is, we get p(L f (W)) = q(L f (W)), and it is a contradiction of the property that Z is a free group. For m = p + q, we get that p, q ≤ n − 1, and by mathematical induction, we obtain:

 −1 T h ( f (U0)) ⊇ U0i, U0j, U0i U0j = ∅, 1 ≤ i, j ≤ p −1 T g ( f (U0)) ⊇ U1k, U1l U1k U1l = ∅, 1 ≤ k, l ≤ q

where U0i, U0j, U1k, and U1l are non-empty open subsets. Mathematics 2021, 9, 296 15 of 17

With the decomposition:

L f ( f (U0)) = Lh( f (U0)) ⊕ Lg( f (U0)),

−1 T we get that Ui, Uj ⊆ f ( f (U0)), Ui Uj = ∅, and Ui, Uj are non-empty open subsets of X, where 1 ≤ i, j ≤ m. Therefore, Ld = m or there exists an m-symbolic space Sm conjugating with a shift operator on Sm, that is L f (U0) has a factor conjugating with a shift operator on Sm. Therefore, for m = n, the conclusion is right, and by mathematical induction, the conclusion is right for any eigenvalue m, where m ∈ N. Now, we give the following definition.

Definition 15. For two topological dynamic systems (X1, f ) and (X2, g), if there exists a homeo- morphism H from X1 to X2 such that H ◦ f = g ◦ H, then we say that H is a from (X1, f ) to (X2, g) or just say that (X1, f ) is topologically conjugate to (X2, g); moreover, if X = X1 = X2, then we say that f is topologically conjugate to g on X.

From the proof of Lemma 10, it is easy to see that Ld(·) is invariant for topological conjugacy. Furthermore, we know that the topological entropy ent(·) is invariant for topological conjugacy. Hence, we obtain that:

Proposition 1. The topological fiber entropy is invariant for topological conjugacy.

Theorem 3. Let X be a compact Hausdorff space and J be the ordered set associated with the set of all finite open covers of X such that there exists nJ. For n = nJ, we have Hˇ p(X; Z), where 0 ≤ p ≤ n. For f ∈ C0(X), we get:

log kE k ≤ ent ( f ), f∗ Hˇ ∗(X;Z) L

Moreover, for 0 ≤ p ≤ n, we get:

log kE k ≤ ent ( f ). f∗ Hˇ ∗(X, f ;Z) L

Proof. It is easy to obtain that

ent ( f ) ≥ ent( f , L˙ (α)) ≥ log kE | k ≥ log kE | k L f f∗ C∗(X, f ;Z) f∗ Hˇ ∗(X;Z) and: ent ( f ) ≥ ent( f , L˙ (α)) ≥ log kE | k ≥ log kE | k. L f f∗ C∗(X, f ;Z) f∗ Hˇ ∗(X, f ;Z)

By simple computing, we get the following results.

Proposition 2. entL( f ) ≥ ent( f ); the inequality can be strict.

Proposition 3. entL(id) = ent(id) = 0, where id is the identical map.

Corollary 2. Let X be a compact Poincaré space and J be the ordered set associated with the set of all finite open covers of X such that there exists nJ. For n = nJ, we have Hˇ p(X; Z), where 0 ≤ p ≤ n. The topological entropy conjecture is valid for the topological fiber entropy and Cechˇ cohomology. Moreover, the topological entropy conjecture is valid for the topological fiber entropy and the f -Cechˇ homology. Mathematics 2021, 9, 296 16 of 17

Corollary 3. In triangulable compact n-dimensional manifold M, the topological entropy conjec- ture is valid for the topological fiber entropy and homology group:

n M H∗(M; Z) = Hi(M; Z), i=0

where Hi(M; Z) is the i-th integer coefficients’ homology group of M.

6. Conclusions If we replace Z with any free abelian group G that is finite generated, then the conclu- sion is also valid. Because the counterexample of A. B. Katok [20] is on a two-dimension sphere S2 and f ∈ C0(S2), with Corollary 3, we get that the inequality of the topological entropy conjecture is valid again with our definition, that is,

log ρ ≤ entL( f ).

Others may be more interested in what the topological fiber entropy entL( f ) measures. From the definition:

entL( f ) = sup {ent( f , α) + log Ld}, L˙ f (α)

we get that the topological fiber entropy ent( fL) is sup on the sum: L˙ f (α)

ent( f , α) + log Ld.

The first part ent( f , α) is the usually one. The second part log Ld is likely some fiber ratio or fiber degree of the dynamics (X, f ); it is likely the “reference system” or “initial value” of the first part ent( f , α).

Funding: The author is partially supported by the National Nature Science Foundation of China (Grant No. 11801428). Acknowledgments: The author is partially supported by the National Nature Science Foundation of China (Grant No. 11801428). I would like to thank the referee for his/her careful reading of the paper and helpful comments and suggestions. Furthermore, I extend my thanks to all those who have offered their help to me, and I would like to show my deepest gratitude to Prof. Bingzhe Hou for his helpful suggestions for Definition 3 in the first version of this paper in 2012. Lastly, I sincerely appreciate the support and cultivation of Fudan University, Jilin University, and Xidian University. Conflicts of Interest: The author declares no conflict of interest.

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