Pub Date : 2023-05-10DOI: 10.1080/14689367.2023.2208533
Qi Wang
Let be the collection of homogeneous product Moran sets determined by two sequences of positive integers , and two sequences of positive numbers , . We obtain the maximal and minimal values of the Hausdorff and packing dimensions of elements in .
{"title":"Hausdorff and packing dimensions of homogeneous product Moran sets","authors":"Qi Wang","doi":"10.1080/14689367.2023.2208533","DOIUrl":"https://doi.org/10.1080/14689367.2023.2208533","url":null,"abstract":"Let be the collection of homogeneous product Moran sets determined by two sequences of positive integers , and two sequences of positive numbers , . We obtain the maximal and minimal values of the Hausdorff and packing dimensions of elements in .","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"510 - 524"},"PeriodicalIF":0.5,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45448043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-04DOI: 10.1080/14689367.2023.2194522
Bui Kim My, N. Toan
In this paper, we consider the non-autonomous stochastic FitzHugh–Nagumo system with hereditary memory and a very large class of nonlinearities, which has no restriction on the upper growth of the nonlinearity. The existence of a random pullback attractor is established for this system in all N-dimensional space.
{"title":"Dynamics of stochastic FitzHugh–Nagumo system on unbounded domains with memory","authors":"Bui Kim My, N. Toan","doi":"10.1080/14689367.2023.2194522","DOIUrl":"https://doi.org/10.1080/14689367.2023.2194522","url":null,"abstract":"In this paper, we consider the non-autonomous stochastic FitzHugh–Nagumo system with hereditary memory and a very large class of nonlinearities, which has no restriction on the upper growth of the nonlinearity. The existence of a random pullback attractor is established for this system in all N-dimensional space.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"453 - 476"},"PeriodicalIF":0.5,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41564087","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-26DOI: 10.1080/14689367.2023.2194520
Georgios Lamprinakis
A long-standing question is what invariant subsets can be shared by two maps acting on the same space. A similar question stands for invariant measures. A particular interesting case are expanding Markov maps of the circle. If the two involved maps are commuting the answer is almost complete. However very little is known in the non-commutative case. A first step is to analyse the structure of the invariant subsets of a single map. For a mapping of the circle of class , , we study the topological structure of the set consisting of all compact invariant subsets. Furthermore for a fixed such mapping we examine locally, in the category sense, how big is the set of all maps that have at least one non-trivial joint invariant compact subset. Lastly we show the strong dimensional relation between the maximal invariant subset of a given Markov map contained in a subinterval and the set of all right endpoints of its invariant subsets that are contained in the same subinterval, , as well as the continuous dependence of the dimension on the endpoints of the subinterval .
{"title":"Structure and dimension of invariant subsets of expanding Markov maps and joint invariance","authors":"Georgios Lamprinakis","doi":"10.1080/14689367.2023.2194520","DOIUrl":"https://doi.org/10.1080/14689367.2023.2194520","url":null,"abstract":"A long-standing question is what invariant subsets can be shared by two maps acting on the same space. A similar question stands for invariant measures. A particular interesting case are expanding Markov maps of the circle. If the two involved maps are commuting the answer is almost complete. However very little is known in the non-commutative case. A first step is to analyse the structure of the invariant subsets of a single map. For a mapping of the circle of class , , we study the topological structure of the set consisting of all compact invariant subsets. Furthermore for a fixed such mapping we examine locally, in the category sense, how big is the set of all maps that have at least one non-trivial joint invariant compact subset. Lastly we show the strong dimensional relation between the maximal invariant subset of a given Markov map contained in a subinterval and the set of all right endpoints of its invariant subsets that are contained in the same subinterval, , as well as the continuous dependence of the dimension on the endpoints of the subinterval .","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"405 - 426"},"PeriodicalIF":0.5,"publicationDate":"2023-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41848452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-26DOI: 10.1080/14689367.2023.2194521
Xing Zhou
We consider the non-semisimple 0:1 resonance (i.e. the unperturbed equilibrium has two purely imaginary eigenvalues ( and ) and a non-semisimple double-zero one) Hamiltonian bifurcation with one distinguished parameter, which corresponds to the supercritical Hamiltonian pitchfork bifurcation. Based on BCKV singularity theory established by [H.W. Broer, S. -N. Chow, Y. Kim, and G. Vegter, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys. 44 (3) (1993), pp. 389–432], this bifurcation essentially triggered by the reversible universal unfolding with respect to BCKV-restricted morphisms of the planar non-semisimple singularity (the is regarded as distinguished parameter with respect to the external parameter λ). We first give the plane bifurcation diagram of the integrable Hamiltonian on each level of integral in detail, which is related to the usual supercritical Hamiltonian pitchfork bifurcation. Then, we use the -symmetry generated by the additional pair of imaginary eigenvalues to reconstruct the above plane bifurcation phenomenon caused by the zero eigenvalue pair into the case with two degrees of freedom. Finally, we prove the persistence of typical bifurcation scenarios (e.g. 2-dimensional invariant tori and the symmetric homoclinic orbit) under the small Hamiltonian perturbations, as proposed by [H.W. Broer, S. -N. Chow, Y. Kim, and G. Vegter, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys. 44 (3) (1993), pp. 389–432]. An example system (the coupled Duffing oscillator) with strong linear coupling and weak local nonlinearity is given for this bifurcation.
{"title":"The 0:1 resonance bifurcation associated with the supercritical Hamiltonian pitchfork bifurcation","authors":"Xing Zhou","doi":"10.1080/14689367.2023.2194521","DOIUrl":"https://doi.org/10.1080/14689367.2023.2194521","url":null,"abstract":"We consider the non-semisimple 0:1 resonance (i.e. the unperturbed equilibrium has two purely imaginary eigenvalues ( and ) and a non-semisimple double-zero one) Hamiltonian bifurcation with one distinguished parameter, which corresponds to the supercritical Hamiltonian pitchfork bifurcation. Based on BCKV singularity theory established by [H.W. Broer, S. -N. Chow, Y. Kim, and G. Vegter, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys. 44 (3) (1993), pp. 389–432], this bifurcation essentially triggered by the reversible universal unfolding with respect to BCKV-restricted morphisms of the planar non-semisimple singularity (the is regarded as distinguished parameter with respect to the external parameter λ). We first give the plane bifurcation diagram of the integrable Hamiltonian on each level of integral in detail, which is related to the usual supercritical Hamiltonian pitchfork bifurcation. Then, we use the -symmetry generated by the additional pair of imaginary eigenvalues to reconstruct the above plane bifurcation phenomenon caused by the zero eigenvalue pair into the case with two degrees of freedom. Finally, we prove the persistence of typical bifurcation scenarios (e.g. 2-dimensional invariant tori and the symmetric homoclinic orbit) under the small Hamiltonian perturbations, as proposed by [H.W. Broer, S. -N. Chow, Y. Kim, and G. Vegter, A normally elliptic Hamiltonian bifurcation, Z. Angew. Math. Phys. 44 (3) (1993), pp. 389–432]. An example system (the coupled Duffing oscillator) with strong linear coupling and weak local nonlinearity is given for this bifurcation.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"427 - 452"},"PeriodicalIF":0.5,"publicationDate":"2023-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43798492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-21DOI: 10.1080/14689367.2023.2230160
S. Farhangi
We prove a generalization of van der Corput's difference theorem for sequences of vectors in a Hilbert space. This generalization is obtained by establishing a connection between sequences of vectors in the first Hilbert space with a vector in a new Hilbert space whose spectral type with respect to a certain unitary operator is absolutely continuous with respect to the Lebesgue measure. We use this generalization to obtain applications regarding recurrence and multiple ergodic averages when we have measure preserving automorphisms T and S that do not necessarily commute, but T has a maximal spectral type that is mutually singular with the Lebesgue measure.
我们证明了Hilbert空间中向量序列的van der Corput差分定理的一个推广。这种推广是通过在第一希尔伯特空间中的向量序列与新希尔伯特空间的向量之间建立连接来获得的，该新希尔伯特空的谱类型相对于某个酉算子相对于勒贝格测度是绝对连续的。当我们有保测度的自同构T和S时，我们使用这种推广来获得关于递推和多重遍历平均的应用，它们不一定是可交换的，但T具有与Lebesgue测度互奇异的最大谱类型。
{"title":"A generalization of van der Corput's difference theorem with applications to recurrence and multiple ergodic averages","authors":"S. Farhangi","doi":"10.1080/14689367.2023.2230160","DOIUrl":"https://doi.org/10.1080/14689367.2023.2230160","url":null,"abstract":"We prove a generalization of van der Corput's difference theorem for sequences of vectors in a Hilbert space. This generalization is obtained by establishing a connection between sequences of vectors in the first Hilbert space with a vector in a new Hilbert space whose spectral type with respect to a certain unitary operator is absolutely continuous with respect to the Lebesgue measure. We use this generalization to obtain applications regarding recurrence and multiple ergodic averages when we have measure preserving automorphisms T and S that do not necessarily commute, but T has a maximal spectral type that is mutually singular with the Lebesgue measure.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44136068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-24DOI: 10.1080/14689367.2023.2184328
Na Yuan, Bing Li
Let T be an expanding Markov map with a repeller E defined on . This paper concerns the Hausdorff dimension of the sets and where is a sequence of Lipschitz functions with a uniform Lipschitz constant, , f is a positive continuous function on , is the sum and ψ is a positive function defined on . The results can be applied to cookie-cutter dynamical systems and continued fraction dynamical systems, etc.
{"title":"Hausdorff dimensions of recurrent and shrinking target sets under Lipschitz functions for expanding Markov maps","authors":"Na Yuan, Bing Li","doi":"10.1080/14689367.2023.2184328","DOIUrl":"https://doi.org/10.1080/14689367.2023.2184328","url":null,"abstract":"Let T be an expanding Markov map with a repeller E defined on . This paper concerns the Hausdorff dimension of the sets and where is a sequence of Lipschitz functions with a uniform Lipschitz constant, , f is a positive continuous function on , is the sum and ψ is a positive function defined on . The results can be applied to cookie-cutter dynamical systems and continued fraction dynamical systems, etc.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"365 - 394"},"PeriodicalIF":0.5,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42646541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-23DOI: 10.1080/14689367.2023.2182183
Lei Liu, Cao Zhao
Let be a compact metric space and be a sequence of continuous maps from X to itself. Denote by the sequence and by the induced nonautonomous dynamical system. In this paper, we give the definitions of upper capacity topological pressure and Pesin-Pitskel topological pressure on a noncompact subset for nonautonomous dynamical systems from the dimension theory. Moreover, we propose the equivalent definition of Pesin-Pitskel topological pressure and investigate some properties of topological pressure. In contrast to Bowen's inequality, we discuss a relation for two topological pressures and establish an inequality formula for two topological pressures with a factor map of nonautonomous dynamical systems.
{"title":"Topological pressure of a factor map for nonautonomous dynamical systems","authors":"Lei Liu, Cao Zhao","doi":"10.1080/14689367.2023.2182183","DOIUrl":"https://doi.org/10.1080/14689367.2023.2182183","url":null,"abstract":"Let be a compact metric space and be a sequence of continuous maps from X to itself. Denote by the sequence and by the induced nonautonomous dynamical system. In this paper, we give the definitions of upper capacity topological pressure and Pesin-Pitskel topological pressure on a noncompact subset for nonautonomous dynamical systems from the dimension theory. Moreover, we propose the equivalent definition of Pesin-Pitskel topological pressure and investigate some properties of topological pressure. In contrast to Bowen's inequality, we discuss a relation for two topological pressures and establish an inequality formula for two topological pressures with a factor map of nonautonomous dynamical systems.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"353 - 364"},"PeriodicalIF":0.5,"publicationDate":"2023-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48472157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-23DOI: 10.1080/14689367.2023.2182182
M. Freitas, A. Ramos, M. Aouadi, D. S. Almeida Júnior
In this paper, we study the long-time dynamics of Bresse system under mixed homogeneous Dirichlet–Neumann boundary conditions. The heat conduction is given by Cattaneo's law. Only the shear angle displacement is damped via the dissipation from the Cattaneo's law, and the vertical displacement and the longitudinal displacement are free. Under quite general assumptions on the source term and based on the semigroup theory, we establish the global well-posedness and the existence of global attractors with finite fractal dimension in natural space energy. Finally, we prove the upper semicontinuous with respect to the relaxation time τ as it converges to zero.
{"title":"Upper semicontinuity of the global attractor for Bresse system with second sound","authors":"M. Freitas, A. Ramos, M. Aouadi, D. S. Almeida Júnior","doi":"10.1080/14689367.2023.2182182","DOIUrl":"https://doi.org/10.1080/14689367.2023.2182182","url":null,"abstract":"In this paper, we study the long-time dynamics of Bresse system under mixed homogeneous Dirichlet–Neumann boundary conditions. The heat conduction is given by Cattaneo's law. Only the shear angle displacement is damped via the dissipation from the Cattaneo's law, and the vertical displacement and the longitudinal displacement are free. Under quite general assumptions on the source term and based on the semigroup theory, we establish the global well-posedness and the existence of global attractors with finite fractal dimension in natural space energy. Finally, we prove the upper semicontinuous with respect to the relaxation time τ as it converges to zero.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"321 - 352"},"PeriodicalIF":0.5,"publicationDate":"2023-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43335676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-19DOI: 10.1080/14689367.2023.2170775
Drew D. Ash, A. Dykstra, M. LeMasurier
ABSTRACT A bounded topological speedup of a Cantor minimal system is a minimal system , where for some bounded function , or any system topologically conjugate to such an . Assuming the system is represented by a properly ordered Bratteli diagram , we provide a method for constructing a new, perfectly ordered Bratteli diagram that represents the sped-up system . The diagram relates back to in a manner that enables us to see how certain dynamical properties are preserved under speedup. As an application, in the case that is a substitution minimal system, we show how to use to write an explicit substitution rule that generates the sped-up system , answering an open question from [L. Alvin, D.D. Ash, and N.S. Ormes, Bounded topological speedups, Dyn. Syst. 33(2) (2018), pp. 303–331.].
摘要:Cantor最小系统的有界拓扑加速是这样的最小系统，其中对于某个有界函数，或与这样的函数拓扑共轭的任何系统。假设系统由一个适当有序的Bratteli图表示，我们提供了一种构造一个新的、完全有序的Bratteli图来表示加速系统的方法。该图以一种使我们能够看到某些动态特性在加速下是如何保持的方式联系起来。作为一个应用程序，在替换最小系统的情况下，我们展示了如何使用编写显式替换规则来生成加速系统，回答了来自[L]的开放问题。Alvin, D.D. Ash和N.S. Ormes，有界拓扑加速，Dyn. system . 33(2) (2018)， pp. 303-331。
{"title":"Bratteli diagrams for bounded topological speedups","authors":"Drew D. Ash, A. Dykstra, M. LeMasurier","doi":"10.1080/14689367.2023.2170775","DOIUrl":"https://doi.org/10.1080/14689367.2023.2170775","url":null,"abstract":"ABSTRACT A bounded topological speedup of a Cantor minimal system is a minimal system , where for some bounded function , or any system topologically conjugate to such an . Assuming the system is represented by a properly ordered Bratteli diagram , we provide a method for constructing a new, perfectly ordered Bratteli diagram that represents the sped-up system . The diagram relates back to in a manner that enables us to see how certain dynamical properties are preserved under speedup. As an application, in the case that is a substitution minimal system, we show how to use to write an explicit substitution rule that generates the sped-up system , answering an open question from [L. Alvin, D.D. Ash, and N.S. Ormes, Bounded topological speedups, Dyn. Syst. 33(2) (2018), pp. 303–331.].","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"249 - 267"},"PeriodicalIF":0.5,"publicationDate":"2023-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43041553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-05DOI: 10.1080/14689367.2022.2164708
Xiang Zhang
We show that if a closed smooth n-manifold M admits a u-dimension one special Anosov endomorphism f, then M is homeomorphic to and f is topologically conjugate to a hyperbolic toral endomorphism. Moreover, any u-dimension one special Anosov endomorphism is transitive and the stable foliation is minimal.
{"title":"On codimension one special Anosov endomorphisms","authors":"Xiang Zhang","doi":"10.1080/14689367.2022.2164708","DOIUrl":"https://doi.org/10.1080/14689367.2022.2164708","url":null,"abstract":"We show that if a closed smooth n-manifold M admits a u-dimension one special Anosov endomorphism f, then M is homeomorphic to and f is topologically conjugate to a hyperbolic toral endomorphism. Moreover, any u-dimension one special Anosov endomorphism is transitive and the stable foliation is minimal.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"197 - 216"},"PeriodicalIF":0.5,"publicationDate":"2023-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43265856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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