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Asymptotic measure-expansiveness for generic diffeomorphisms

  • Manseob Lee EMAIL logo
Veröffentlicht/Copyright: 1. Juni 2021
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Open Mathematics
Aus der Zeitschrift Open Mathematics Band 19 Heft 1

Abstract

In this paper, we will assume M to be a compact smooth manifold and f : M → M to be a diffeomorphism. We herein demonstrate that a C 1 generic diffeomorphism f is Axiom A and has no cycles if f is asymptotic measure expansive. Additionally, for a C 1 generic diffeomorphism f , if a homoclinic class H ( p , f ) that contains a hyperbolic periodic point p of f is asymptotic measure-expansive, then H ( p , f ) is hyperbolic of f .

Keywords: expansive; measure expansive; asymptotic measure expansive; generic; Axiom A; homoclinic class; hyperbolic
MSC 2010: 37C20; 37D20

1 Introduction

Throughout this paper, we will assume M to be a compact smooth manifold and d to be the distance on M induced by a Riemannian metric ∥ ⋅ ∥ . We also assume f : M → M to be a diffeomorphism and denote by Diff ( M ) the set of diffeomorphisms of M endowed with C 1 topology. It is to note that expansiveness has been earlier suggested in the study by Utz [1]. A diffeomorphism f is said to be expansive if there exists a positive constant δ > 0 such that for any two points x , y ∈ M if x ≠ y , and if there exists k ∈ Z such that d ( f k ( x ) , f k ( y ) ) > δ . Equivalently, if there is a positive constant δ > 0 such that for any x , y ∈ M if d ( f i ( x ) , f i ( y ) ) ≤ δ , ∀ i ∈ Z , then x = y . Generally speaking, expansiveness means that if any two real orbits are separated by a small distance, the two orbits are identical, and therefore it is appropriate for studying smooth dynamic systems. Expansivities are hence a valuable notion in the investigation of hyperbolic structures (see [2,3, 4,5,6, 7,8,9, 10,11,12, 13,14,15, 16,17], etc.). Mañé [17] proved that a C 1 robustly expansive diffeomorphism f is quasi Anosov, i.e., the set { ∥ D f n ( v ) ∥ : n ∈ Z } is unbounded for all v ∈ T M ⧹ { 0 } .

Morales and Sirvent [18] introduced stochastic perspectives of expansiveness, called measure-expansiveness. Let us assume ℳ ( M ) to be the set of all Borel probability measures on M endowed with the weak ∗ topology and ℳ ∗ ( M ) to be the set of nonatomic measures μ ∈ ℳ ( M ) . It is known that ℳ ∗ ( M ) ⊂ ℳ ( M ) .

For any μ ∈ ℳ ∗ ( M ) , a closed f -invariant set Λ ⊂ M is said to be μ -expansive for f if there is a positive constant δ > 0 such that μ ( Γ ( δ , x ) ) = 0 ∀ x ∈ Λ , where Γ ( δ , x ) = { y ∈ M : d ( f i ( x ) , f i ( y ) ) ≤ δ ∀ i ∈ Z } .

Definition 1.1

A closed f -invariant set Λ ⊂ M is said to be measure expansive for f if Λ is μ -expansive for all μ ∈ ℳ ∗ ( M ) . If Λ = M , then we say that a diffeomorphism f is measure expansive.

Here, δ is called a measure expansive constant of f . Now, we introduce a general notion of measure-expansiveness called the asymptotic measure expansive (see [19, Example 1.1]). The notion was suggested in [19]. Let us assume that μ ∈ ℳ ∗ ( M ) is given. A closed f -invariant set Λ ⊂ M is said to be asymptotic μ -expansive for f if there is δ > 0 such that

lim n → ∞ μ ( f n ( Γ ( δ , x ) ) ) = 0

for any x ∈ Λ .

Definition 1.2

Let us assume that f ∈ Diff ( M ) , and Λ ⊂ M is a closed f -invariant set. We say that Λ is asymptotic measure expansive for f if there is a positive constant δ > 0 such that Λ is asymptotic μ -expansive for f . Moreover, if Λ = M , then we say that f is asymptotic measure expansive.

The following notion is suggested in [20]. A diffeomorphism f on M is said to be continuum-wise expansive if there is a positive constant δ > 0 such that, for any nontrivial compact connected set Λ , there is an integer n ∈ Z such that diam f n ( Λ ) ≥ e , where diam Λ = sup { d ( x , y ) : x , y ∈ Λ } for any subset Λ ⊂ M and λ is nontrivial, which means that Λ is neither one point nor one orbit.

Regarding the result of Artigue and Carrasco-Olivera [21], it is observed that a diffeomorphism f is measure-expansive if it is continuum-wise expansive. However, the converse is not true. We already know that a diffeomorphism f is measure-expansive if it is asymptotic measure-expansive. Here, the converse is also untrue. Therefore, we have a question:

What is relation between asymptotic measure expansiveness and continuum-wise expansiveness?

A closed f -invariant set Λ ⊂ M is called hyperbolic if a D f -invariant splitting T Λ M = E s ⊕ E u , there exist constants C > 0 and 0 < λ < 1 such that ∀ x ∈ Λ and n ≥ 0 ,

  1. ∥ D x f n ( v ) ∥ ≤ ∥ v ∥ C λ n for v ∈ E x s ⧹ { 0 } , and

  2. ∥ D x f − n ( u ) ∥ ≤ ∥ u ∥ C λ n for u ∈ E x u ⧹ { 0 } .

If Λ = M , then a diffeomorphism f is said to be Anosov.

It is known that if Λ is hyperbolic for f , then Λ is expansive, thus it is measure-expansive and asymptotic measure-expansive. A point x ∈ M is called periodic if there is n ( x ) > 0 such that f n ( x ) ( x ) = x , and a point x ∈ M is called non-wandering if k > 0 can be found such that f k ( U ) ∩ U ≠ ∅ for any neighborhood U of x . We denote P e r ( f ) as the set of all periodic points of f and Ω ( f ) the set of all non wandering points of f . It is known that P e r ( f ) ⊂ Ω ( f ) . We say that f satisfies Axiom A if the nonwandering set Ω ( f ) = P e r ( f ) ¯ is hyperbolic. According to Aoki [22] and Hayashi [23], f satisfies Axiom A and has no-cycles if f is star.

In this paper, we consider sets of diffeomorphisms that are residual for the Baire category, i.e., sets that contain a countable intersection of dense and open subsets of Diff ( M ) . Regarding C 1 generic diffeomorphisms, it is known that the periodic points are dense in Ω ( f ) by Pugh’s closing lemma [24]. Using the C 1 generic property, Arbieto proved in [25] that f satisfies Axiom A and has no-cycles for a C 1 generic expansive diffeomorphism. Lee [26] proved that a C 1 generic measure expansive diffeomorphism f satisfies Axiom A and has no-cycle. Recently, Lee [27] proved that f satisfies Axiom A and has no-cycles for a C 1 generic continuum-wise expansive diffeomorphism. According to the abovementioned results, we consider general concepts of measure expansiveness. The following is the primary theorem of the paper.

Theorem A

For a C 1 generic f ∈ Diff ( M ) , f satisfies Axiom A and has no-cycles if it is asymptotic measure-expansive.

For any hyperbolic periodic point p , define the following sets W s ( p ) = { x ∈ M : f i ( x ) → p as i → ∞ } and W u ( p ) = { x ∈ M : f i ( x ) → p as i → − ∞ } , where W s ( p ) is called the stable manifold of p and W u ( p ) is called the unstable manifold of p . Denote by dim W s ( p ) = index ( p ) . We say that a hyperbolic p ∈ P e r ( f ) is homoclinically related to q ∈ P e r ( f ) if W s ( p ) ⋔ W u ( p ) ≠ ∅ and W u ( p ) ⋔ W s ( q ) ≠ ∅ . We write p ∼ q . It is clear that index ( p ) = index ( q ) if p ∼ q .

A closed f -invariant set Λ ⊂ M is called transitive if we can take a point x ∈ Λ such that O r b ( x ) ¯ = Λ , where A ¯ is the closure of A . Denote H ( p , f ) = { q ∈ P ( f ) : q ∼ p } ¯ , which is called the homoclinic class. It is known that the set is a closed f -invariant and transitive set. Note that if a diffeomorphism f satisfies Axiom A, then the nonwandering set Ω ( f ) is a disjoint union of transitive invariant closed subsets. In fact, these sets are homoclinic classes that each contain a hyperbolic periodic point. Several researchers are studying these sets and their hyperbolicity (see [4,28,29, 30,31,32, 33,34], etc.). We study whether the homoclinic class is hyperbolic using the asymptotic measure-expansiveness. Yang and Gan [34] proved that a homoclinic class H ( p , f ) is hyperbolic if it is expansive for a C 1 generic diffeomorphism f . Koo et al. [35] proved that a locally maximal homoclinic class H ( p , f ) is hyperbolic if it is measure-expansive for a C 1 generic f . Here, a closed f -invariant set Λ ⊂ M is locally maximal if there exists a neighborhood U of Λ for which Λ = ⋂ n ∈ Z f n ( U ) . Later, Lee proved in [32] that a homoclinic class H ( p , f ) is hyperbolic if it is measure-expansive for a C 1 generic diffeomorphism f . The result is a general version of the proof in [35]. In [31], Lee proved that a homoclinic class H ( p , f ) is hyperbolic if it is continuum-wise expansive for a C 1 generic f . According to the results, we prove the following:

Theorem B

A homoclinic class H ( p , f ) is hyperbolic if it is asymptotic measure-expansive for a C 1 generic f ∈ Diff ( M ) .

2 Proof of Theorem A

Theorem A will be proven in this section, which requires some notions to be taken into account. A point p ∈ P e r ( f ) is weak hyperbolic if there is g C 1 close to f such that the derivative map D p g π ( p ) has an eigenvalue λ with ∣ λ ∣ = 1 . For any ε > 0 , we consider a closed curve η to be ε simply periodic if η satisfies the following conditions:

  1. there is k > 0 such that f k ( η ) = η ,

  2. 0 < l ( f i ( η ) ) ≤ ε for 0 ≤ i ≤ k , and

  3. η is normally hyperbolic (see [34]).

If a p ∈ P e r ( f ) is hyperbolic, then there are a C 1 neighborhood U ( f ) of f and a locally maximal neighborhood U of p such that there exists the hyperbolic periodic p g = ⋂ n ∈ Z g n ( U ) for any g ∈ U ( f ) . Here, p g is called a continuation.

The following is called Franks’ lemma [36], which is a useful notion for a C 1 robust property.

Lemma 2.1

U ( f ) be any given C 1 neighborhood of f . Then there exist ε > 0 and a C 1 neighborhood U 0 ( f ) ⊂ U ( f ) of f such that, if a set A = { x 1 , x 2 , … , x k } , a neighborhood U of A , and linear maps L i : T x i M → T g ( x i ) M satisfy ∥ L i − D x i g ∥ ≤ ε ∀ x i ∈ A , then for any g ∈ U 0 ( f ) , there exists g ^ ∈ U ( f ) for which g ^ ( x ) = g ( x ) if x ∈ A ∪ ( M ⧹ U ) and D x i g ^ = L i for all 1 ≤ i ≤ k .

Lemma 2.2

If a diffeomorphism f has a weak hyperbolic periodic point, then for any neighborhood U ( f ) of f and any ε > 0 , there are g ∈ U ( f ) and a small curve J with the following property:

  1. J is g periodic, i.e., there is n ∈ Z such that g n ( J ) = J ;

  2. the length of g i ( J ) is less than ε for all i ∈ Z ;

  3. the endpoints of the curve J are hyperbolic;

  4. J is normally hyperbolic with respect to g (see [37]).

Proof

Let us assume p to be a weak hyperbolic periodic point of f and U ( f ) to be a C 1 neighborhood of f . For simplicity, we may assume that f ( p ) = p . According to Lemma 2.1, there is g ∈ U ( f ) such that g ( p ) = p and the derivative map D p g has an eigenvalue λ with ∣ λ ∣ = 1 , i.e., g has a non hyperbolic periodic point p . As given in the proof of [26, Lemma 2.2], h C 1 can be found close to g (also, h ∈ U ( f ) ) such that

  1. h k ( J ) = J for some k ∈ Z , and

  2. h k ∣ J : J → J is the identity map.

In items (i) and (ii), k = 1 and k = 2 if the eigenvalue λ is a positive or negative real number, respectively. If the eigenvalue λ is a complex number, then one can take l > 0 such that k = l . As in the proof of [26, Lemma 2.2], it is clear that J is normally hyperbolic and the length of J is less than ε . Therefore, the small closed curve J satisfies items (a), (b), and (d).

Finally, we show item (c). Let us assume that q and r are the endpoints of the closed curve J . For simplification, we assume that h k = h . It is observed that the eigenvalue of the derivative maps D q h = 1 and D r h = 1 . Again, using Lemma 2.1, there is h 1 C 1 close to h (also, h 1 ∈ U ( f ) ) such that h 1 ( q ) = q , h 1 ( r ) = r , and the norm of every eigenvalue of the derivative map D q h 1 and D r h 1 are not one. Therefore, we have a small curve J that satisfies items (a), (b), (c), and (d). This completes the proof.□

A diffeomorphism f is star if we can take a C 1 neighborhood U ( f ) of f for which every periodic point of g is hyperbolic for g ∈ U ( f ) .

Lemma 2.3

There is a residual set ℛ subset in Diff ( M ) such that, for given f ∈ ℛ , we have:

  1. either f is star or

  2. f has a simple periodic curve J with hyperbolic endpoints.

Proof

The proof is similar to [26, Lemma 2.4].□

Proof of Theorem A

For f ∈ ℛ , we assume that f is asymptotic measure-expansive. According to Aoki [22] and Hayashi [23], it is sufficient to show that f is a star. Suppose, by contradiction, that f is not a star. If f is not a star, f has a simple periodic curve J with hyperbolic endpoints according to Lemma 2.3. That is, there is k ∈ Z such that f k ( J ) = J , the length of f i ( J ) is less than ε ( 0 ≤ i ≤ k ) , the endpoints of the curve J are hyperbolic, and J is normally hyperbolic.

Let us now assume ν J to be a normalized Lebesgue measure on J . μ ∈ ℳ ( M ) is defined as

μ ( B ) = 1 k ∑ i = 0 k − 1 ν J ( f − i ( B ) ∩ f i ( J ) )

for any Borel set B ⊂ M . It is clear that μ ∈ ℳ ∗ ( M ) and μ is invariant. For any small δ > 0 and x ∈ J , we define Φ ( δ , x ) = { y ∈ J : d ( f i ( x ) , f i ( y ) ) ≤ δ for all i ∈ Z } ⊂ J . It is also assumed that Γ ( δ , x ) = { y ∈ M : d ( f i ( x ) , f i ( y ) ) ≤ δ for all i ∈ Z } . It can therefore be seen that Φ ( δ , x ) ⊂ Γ ( δ , x ) . Because Φ ( δ , x ) ⊂ J , it is observed that f k n ( Φ ( δ , x ) ) = Φ ( δ , x ) for all n ∈ Z . Therefore, we know that

lim n → ∞ μ ( f n ( Φ ( δ , x ) ) ) ↛ 0 .

μ ( f n ( Γ ( δ , x ) ) ) → 0 as n → ∞ because f is asymptotic measure-expansive. This is a contradiction because μ ( f n ( Φ ( δ , x ) ) ) ↛ 0 ( n → ∞ ) . Therefore, for C 1 generic f , f satisfies both Axiom A and the no-cycle condition if f is asymptotic measure-expansive.□

3 Proof of Theorem B

Theorem B will be proven in this section using various results of a C 1 generic property.

For any δ > 0 , we consider a point p to be a δ weak hyperbolic periodic point if

( 1 − λ ) π ( p ) ≤ ∣ λ ∣ ≤ ( 1 + δ ) π ( p ) ,

where λ is the eigenvalue λ of D p f , and π ( p ) is the period of p .

We consider f to be Kupka-Smale if every periodic point is hyperbolic and its stable and unstable manifolds are transversal intersections. It is well known that a diffeomorphism f is a residual subset of Diff ( M ) if it is Kupka-Smale (see [38]).

For any ε > 0 , a sequence of points { x i } i = a b is a ε -pseudo orbit of f if d ( f ( x i ) , x i + 1 ) < ε for all − ∞ ≤ a ≤ i < b ≤ ∞ . A point x ∈ M is called chain recurrent if there is a finite ε -pseudo orbit { x i } i = 0 n such that x 0 = x and x n = x for any ε > 0 . Let us assume C ℛ ( f ) to be the set of all chain recurrence sets of f . We define a relation ⇌ on C ℛ ( f ) by x ⇌ y if there exists a finite ε -pseudo orbit { w i } i = 0 n for any ε > 0 such that w 0 = x , w n = y , and another ε -pseudo orbit { z n } i = 0 n such that z 0 = y and y n = x . It is clear that ⇌ is an equivalence relation on C ℛ ( f ) . The equivalence classes are called the chain recurrent classes of f .

For any hyperbolic periodic point p , we denote C ( p , f ) = { x ∈ M : x ⇌ p } . It is clear that C ( p , f ) is a closed f -invariant set and H ( p , f ) ⊂ C ( p , f ) .

Lemma 3.1

There is a residual subset G 1 in Diff ( M ) such that, for given f ∈ G 1 , we have the following:

  1. f is Kupka-Smale (see [38]);

  2. for any δ > 0 and any p ∈ P e r ( f ) , there exists g ∈ U ( f ) for any C 1 neighborhood U ( f ) of f such that g has a δ simply periodic curve ℐ , where the two endpoints of ℐ are homoclinically related to p g . Therefore, f has a δ simply periodic curve J such that the two endpoints of J are homoclinically related to p (see [34]);

  3. H ( p , f ) = C ( p , f ) (see [39]).

Lemma 3.2

Let us assume q ∈ H ( p , f ) ∩ P e r ( f ) and δ > 0 to be given. If q is a δ weak hyperbolic periodic point for f , there exists g C 1 close to f such that g has a δ simply periodic curve ℐ with endpoints that are homoclinically related to p g .

Proof

See the proof of [33,34].□

Lemma 3.3

For a C 1 generic f , every periodic point in a homoclinic class H ( p , f ) is not weak hyperbolic if H ( p , f ) is asymptotic measure-expansive for f .

Proof

For f ∈ G 1 , assume that f is asymptotic measure-expansive. We shall derive a contradiction. Suppose that there is q ∈ H ( p , f ) ∩ P e r ( f ) such that q is weak hyperbolic. Therefore, there is g C 1 close to f such that g has a δ simply periodic curve ℐ with endpoints that are homoclinically related to p g according to Lemma 3.2. Because f ∈ G 1 , f has a δ simply periodic curve J such that the two endpoints of J are homoclinically related to p by Lemma 3.1.

Assume that the period of J is L > 0 . Let us assume that μ J be a normalized Lebesgue measure on J . χ ∈ ℳ ( M ) is defined by

χ ( B ) = 1 L ∑ i = 0 L − 1 μ J ( f − 1 ( B ) ∩ f i ( J ) )

for any Borel set B ⊂ M . It is clear that χ is invariant and χ ∈ ℳ ∗ ( M ) . For any δ > 0 and x ∈ J , we define Θ ( δ , x ) = { y ∈ J : d ( f i ( x ) , f i ( y ) ) ≤ δ for all i ∈ Z } ⊂ J . Let us assume that Γ ( δ , x ) = { y ∈ M : d ( f i ( x ) , f i ( y ) ) ≤ δ for all i ∈ Z } . It is clear that Θ ( δ , x ) ⊂ Γ ( δ , x ) . It can be observed that f L n ( Θ ( δ , x ) ) = Θ ( δ , x ) for all n ∈ Z because Θ ( δ , x ) ⊂ J . Therefore, we know that

(1) lim n → ∞ χ ( f n ( Θ ( δ , x ) ) ) ↛ 0 .

Because H ( p , f ) is asymptotic measure-expansive, χ ( f n ( Γ ( δ , x ) ) ) = 0 as n → ∞ . This is a contradiction by (1) because Θ ( δ , x ) ⊂ Γ ( δ , x ) .□

We assume p to be a hyperbolic periodic point of f having a period of π ( p ) . Therefore, if μ 1 , μ 2 , … , μ n are the eigenvalues of D p f , then

λ i = 1 π ( p ) log ∣ μ i ∣ ,

for i = 1 , 2 , … , d are called the Lyapunov exponents of p . The following was proven by Wang [40]. In fact, Wang proved that there is a q ∈ H ( p , f ) ∩ P e r ( f ) such that q has a Lyapunov exponent arbitrarily closed to 0 for a C 1 generic diffeomorphism f if a homoclinic class H ( p , f ) is not hyperbolic. It can be observed that for a C 1 generic f , if a periodic point q ∈ H ( p , f ) , then q ∼ p . We modified the statement as follows:

Lemma 3.4

There is a residual subset G 2 in Diff ( M ) such that there exists a weak hyperbolic periodic point q ∈ H ( p , f ) for any f ∈ G 2 , if a homoclinic class H ( p , f ) is not hyperbolic for f .

Proof of Theorem B

Let us assume that f ∈ G 1 ∩ G 2 be asymptotic measure-expansive. We shall derive a contradiction. It is assumed that H ( p , f ) is not hyperbolic. According to Lemma 3.4, there is q ∈ H ( p , f ) ∩ P e r ( f ) such that q is a weak hyperbolic periodic point. This is a contradiction by Lemma 3.3 because H ( p , f ) is asymptotic measure-expansive. Therefore, H ( p , f ) is hyperbolic for C 1 generic diffeomorphism f if a homoclinic class H ( p , f ) is asymptotic measure-expansive for f .□

  1. Funding information: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MIST) (NRF-2020R1F1A1A01051370).

  2. Conflict of interest: Author states no conflict of interest.

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Received: 2020-11-29
Accepted: 2021-02-28
Published Online: 2021-06-01

© 2021 Manseob Lee, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0037
Schlagwörter für diesen Artikel
expansive; measure expansive; asymptotic measure expansive; generic; Axiom A; homoclinic class; hyperbolic
Creative Commons
BY 4.0

Artikel in diesem Heft

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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