Pub Date : 2022-10-04DOI: 10.1080/14689367.2022.2130033
Khadija Ben Rejeb
Let be a continuous flow of homeomorphisms of a connected n-manifold M. The flow G is called periodic if: for some real s>0, . A global section for a flow G is a closed subset K of M such that every orbit under G intersects K in exactly one point. In this paper, we give a topological characterization of periodic flows with global sections for . Next, we consider periodic flows defined on any connected n-manifold M, and we give a similar local characterization.
{"title":"A topological characterization of periodic flows","authors":"Khadija Ben Rejeb","doi":"10.1080/14689367.2022.2130033","DOIUrl":"https://doi.org/10.1080/14689367.2022.2130033","url":null,"abstract":"Let be a continuous flow of homeomorphisms of a connected n-manifold M. The flow G is called periodic if: for some real s>0, . A global section for a flow G is a closed subset K of M such that every orbit under G intersects K in exactly one point. In this paper, we give a topological characterization of periodic flows with global sections for . Next, we consider periodic flows defined on any connected n-manifold M, and we give a similar local characterization.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"20 - 29"},"PeriodicalIF":0.5,"publicationDate":"2022-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41780613","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 : 2022-10-02DOI: 10.1080/14689367.2022.2098091
L. Barreira, Carllos Holanda
ABSTRACT We introduce a version of the nonlinear thermodynamic formalism for flows. Moreover, we discuss the existence, uniqueness, and characterization of equilibrium measures for almost additive families of continuous functions with tempered variation. We also consider with some care the special case of additive families for which it is possible to strengthen some of the results. The proofs are mainly based on multifractal analysis.
{"title":"Nonlinear thermodynamic formalism for flows","authors":"L. Barreira, Carllos Holanda","doi":"10.1080/14689367.2022.2098091","DOIUrl":"https://doi.org/10.1080/14689367.2022.2098091","url":null,"abstract":"ABSTRACT We introduce a version of the nonlinear thermodynamic formalism for flows. Moreover, we discuss the existence, uniqueness, and characterization of equilibrium measures for almost additive families of continuous functions with tempered variation. We also consider with some care the special case of additive families for which it is possible to strengthen some of the results. The proofs are mainly based on multifractal analysis.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"603 - 629"},"PeriodicalIF":0.5,"publicationDate":"2022-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47649738","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 : 2022-09-22DOI: 10.1080/14689367.2023.2229752
K. Burns, Dong Chen
For any rank 1 nonpositively curved surface $M$, it was proved by Burns-Climenhaga-Fisher-Thompson that for any $q<1$, there exists a unique equilibrium state $mu_q$ for $qvarphi^u$, where $varphi^u$ is the geometric potential. We show that as $qto 1-$, the weak$^*$ limit of $mu_q$ is the restriction of the Liouville measure to the regular set.
{"title":"A remark on the phase transition for the geodesic flow of a rank one surface of nonpositive curvature","authors":"K. Burns, Dong Chen","doi":"10.1080/14689367.2023.2229752","DOIUrl":"https://doi.org/10.1080/14689367.2023.2229752","url":null,"abstract":"For any rank 1 nonpositively curved surface $M$, it was proved by Burns-Climenhaga-Fisher-Thompson that for any $q<1$, there exists a unique equilibrium state $mu_q$ for $qvarphi^u$, where $varphi^u$ is the geometric potential. We show that as $qto 1-$, the weak$^*$ limit of $mu_q$ is the restriction of the Liouville measure to the regular set.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45563301","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 : 2022-09-12DOI: 10.1080/14689367.2023.2248013
M. Choudhuri, Gianluca Faraco, Alok Yadav
Any affine map on the (n+1)-dimensional Euclidean space gives rise to a natural map on the n-dimensional sphere whose dynamical aspects are not so well-studied in the literature. We explore the dynamical aspects of these maps by investigating about their distality and expansivity.
{"title":"On the distality and expansivity of certain maps on spheres","authors":"M. Choudhuri, Gianluca Faraco, Alok Yadav","doi":"10.1080/14689367.2023.2248013","DOIUrl":"https://doi.org/10.1080/14689367.2023.2248013","url":null,"abstract":"Any affine map on the (n+1)-dimensional Euclidean space gives rise to a natural map on the n-dimensional sphere whose dynamical aspects are not so well-studied in the literature. We explore the dynamical aspects of these maps by investigating about their distality and expansivity.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42648747","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 : 2022-09-05DOI: 10.1080/14689367.2023.2193677
P. Chocano
We study dynamical systems defined on the combinatorial model of the real line. We prove that using single-valued maps there are no periodic points of period 3, which contrasts with the classical and less restrictive setting. Then, we use Vietoris-like multivalued maps to show that there is more flexibility, at least in terms of periods, in this combinatorial framework than in the usual one because we do not have the conditions about the existence of periods given by the Sharkovski Theorem.
{"title":"On the dynamics of the combinatorial model of the real line","authors":"P. Chocano","doi":"10.1080/14689367.2023.2193677","DOIUrl":"https://doi.org/10.1080/14689367.2023.2193677","url":null,"abstract":"We study dynamical systems defined on the combinatorial model of the real line. We prove that using single-valued maps there are no periodic points of period 3, which contrasts with the classical and less restrictive setting. Then, we use Vietoris-like multivalued maps to show that there is more flexibility, at least in terms of periods, in this combinatorial framework than in the usual one because we do not have the conditions about the existence of periods given by the Sharkovski Theorem.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"395 - 404"},"PeriodicalIF":0.5,"publicationDate":"2022-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47980042","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 : 2022-08-11DOI: 10.1080/14689367.2022.2111295
Lüping Chen
In the present paper, we study the following impulsive pendulum-type equation where , . Under suitable impulses, by Moser's twist theorem, we prove that any solution of the impulsive pendulum-type equation is bounded, and there are infinitely many quasi-periodic solutions for the impulsive pendulum-type equation.
{"title":"Boundedness of solutions for some impulsive pendulum-type equations","authors":"Lüping Chen","doi":"10.1080/14689367.2022.2111295","DOIUrl":"https://doi.org/10.1080/14689367.2022.2111295","url":null,"abstract":"In the present paper, we study the following impulsive pendulum-type equation where , . Under suitable impulses, by Moser's twist theorem, we prove that any solution of the impulsive pendulum-type equation is bounded, and there are infinitely many quasi-periodic solutions for the impulsive pendulum-type equation.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"684 - 698"},"PeriodicalIF":0.5,"publicationDate":"2022-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43965070","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 : 2022-07-01DOI: 10.1080/14689367.2022.2092454
Denis de Carvalho Braga, J. Llibre, Luis Fernando Mello
We present a general mechanism of generation of limit cycles in planar piecewise polynomial differential systems with two zones by means of a transcritical bifurcation at infinity and from a global centre. This study justifies the existence of limit cycles that arise through the intersection of the separation boundary with the one that characterizes the global centre.
{"title":"Transcritical bifurcation at infinity in planar piecewise polynomial differential systems with two zones","authors":"Denis de Carvalho Braga, J. Llibre, Luis Fernando Mello","doi":"10.1080/14689367.2022.2092454","DOIUrl":"https://doi.org/10.1080/14689367.2022.2092454","url":null,"abstract":"We present a general mechanism of generation of limit cycles in planar piecewise polynomial differential systems with two zones by means of a transcritical bifurcation at infinity and from a global centre. This study justifies the existence of limit cycles that arise through the intersection of the separation boundary with the one that characterizes the global centre.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"578 - 602"},"PeriodicalIF":0.5,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47946154","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 : 2022-06-28DOI: 10.1080/14689367.2023.2174000
A. Gorodetski, Alexandro Luna
We study the space of ergodic measures of the map and show that its structure is similar to the graph of Thomae's function.
我们研究了映射遍历测度的空间，证明了它的结构类似于托马函数的图。
{"title":"Thomae's function and the space of ergodic measures","authors":"A. Gorodetski, Alexandro Luna","doi":"10.1080/14689367.2023.2174000","DOIUrl":"https://doi.org/10.1080/14689367.2023.2174000","url":null,"abstract":"We study the space of ergodic measures of the map and show that its structure is similar to the graph of Thomae's function.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"38 1","pages":"268 - 274"},"PeriodicalIF":0.5,"publicationDate":"2022-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47835218","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 : 2022-06-16DOI: 10.1080/14689367.2023.2236036
Pierre-Antoine Guih'eneuf, M. Monge
This paper presents some numerical experiments in relation with the theoretical study of the ergodic short-term behaviour of discretizations of expanding maps done in arXiv:2206.07991 [math.DS]. Our aim is to identify the phenomena driving the evolution of the Cram'er distance between the $t$-th iterate of Lebesgue measure by the dynamics $f$ and the $t$-th iterate of the uniform measure on the grid of order $N$ by the discretization on this grid. Based on numerical simulations we propose some conjectures on the effects of numerical truncation from the ergodic viewpoint.
{"title":"Cramér distance and discretizations of circle expanding maps II: simulations","authors":"Pierre-Antoine Guih'eneuf, M. Monge","doi":"10.1080/14689367.2023.2236036","DOIUrl":"https://doi.org/10.1080/14689367.2023.2236036","url":null,"abstract":"This paper presents some numerical experiments in relation with the theoretical study of the ergodic short-term behaviour of discretizations of expanding maps done in arXiv:2206.07991 [math.DS]. Our aim is to identify the phenomena driving the evolution of the Cram'er distance between the $t$-th iterate of Lebesgue measure by the dynamics $f$ and the $t$-th iterate of the uniform measure on the grid of order $N$ by the discretization on this grid. Based on numerical simulations we propose some conjectures on the effects of numerical truncation from the ergodic viewpoint.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42232744","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 : 2022-06-06DOI: 10.1080/14689367.2022.2086104
Lei Liu, Jinlei Jiao, Xiaoyao Zhou
In this paper, we introduce the Pesin–Pitskel unstable pressure to study dynamical complexity of general subsets in partially hyperbolic systems. We establish some basic results in dimension theory for Pesin–Pitskel unstable pressure, including a pressure distribution principle, and a variational principle for any compact (not necessarily invariant) subset between its Pesin–Pitskel unstable pressure and unstable metric pressure of probability measures supported on this set.
{"title":"Unstable pressure of subsets for partially hyperbolic systems","authors":"Lei Liu, Jinlei Jiao, Xiaoyao Zhou","doi":"10.1080/14689367.2022.2086104","DOIUrl":"https://doi.org/10.1080/14689367.2022.2086104","url":null,"abstract":"In this paper, we introduce the Pesin–Pitskel unstable pressure to study dynamical complexity of general subsets in partially hyperbolic systems. We establish some basic results in dimension theory for Pesin–Pitskel unstable pressure, including a pressure distribution principle, and a variational principle for any compact (not necessarily invariant) subset between its Pesin–Pitskel unstable pressure and unstable metric pressure of probability measures supported on this set.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"564 - 577"},"PeriodicalIF":0.5,"publicationDate":"2022-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41642458","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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