{"title":"A new example of robustly transitive diffeomorphism","authors":"Pablo D. Carrasco, Davi Obata","doi":"10.4310/MRL.2021.V28.N3.A2","DOIUrl":"https://doi.org/10.4310/MRL.2021.V28.N3.A2","url":null,"abstract":"We present an example of a $mathcal{C}^1$-robustly transitive skew-product with non-trivial, non-hyperbolic action on homology. The example is conservative, ergodic, non-uniformly hyperbolic and its fiber directions cannot be decomposed into two dominated expanded/contracted bundles.","PeriodicalId":407889,"journal":{"name":"arXiv: Dynamical Systems","volume":"8 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127332480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Strong structural controllability under network perturbations","authors":"S. Mousavi, M. Haeri, M. Mesbahi","doi":"10.3929/ETHZ-B-000381690","DOIUrl":"https://doi.org/10.3929/ETHZ-B-000381690","url":null,"abstract":"This paper investigates the robustness of strong structural controllability for linear time-invariant and linear time-varying directed networks with respect to structural perturbations, including edge deletions and additions. In this direction, we introduce a new construct referred to as a perfect graph associated with a network with a given set of control nodes. The tight upper bounds on the number of edges that can be added to, or removed from a network, while ensuring strong structural controllability, are then derived. Moreover, we obtain a characterization of critical edge-sets, the maximal set of edges whose any subset can be respectively added to, or removed from a network, while preserving strong structural controllability. In addition, procedures for combining networks to obtain strongly structurally controllable network-of-networks are proposed. Finally, controllability conditions are proposed for networks whose edge weights, as well as their structures, can vary over time.","PeriodicalId":407889,"journal":{"name":"arXiv: Dynamical Systems","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116489256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Invariance of distributional chaos for backward shifts","authors":"Xinxing Wu, Yang Luo","doi":"10.7153/oam-2020-14-01","DOIUrl":"https://doi.org/10.7153/oam-2020-14-01","url":null,"abstract":"A sufficient and necessary condition ensuring that the backward shift operator on the K\"{o}the sequence space admits an invariant distributionally $varepsilon$-scrambled set for some $varepsilon>0$ is obtained, improving the main results in [F. Mart'{i}nez-Gim'{e}nez, P. Oprocha, A. Peris, J. Math. Anal. Appl., {bf 351} (2009), 607--615].","PeriodicalId":407889,"journal":{"name":"arXiv: Dynamical Systems","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132361760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On different types of stability for linear delay dynamic equations","authors":"Eric P. Braverman, B. Karpuz","doi":"10.4171/zaa/1592","DOIUrl":"https://doi.org/10.4171/zaa/1592","url":null,"abstract":"We provide explicit conditions for uniform stability, global asymptotic stability and uniform exponential stability for dynamic equations with a single delay and a nonnegative coefficient. Some examples on nonstandard time scales are also given to show applicability and sharpness of the new results.","PeriodicalId":407889,"journal":{"name":"arXiv: Dynamical Systems","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122811577","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Max-Min theorems for weak containment, square summable homoclinic points, and completely positive entropy","authors":"Ben Hayes","doi":"10.1512/IUMJ.2021.70.8535","DOIUrl":"https://doi.org/10.1512/IUMJ.2021.70.8535","url":null,"abstract":"We prove a max-min theorem for weak containment in the context of algebraic actions. Namely, we show that given an algebraic action of $G$ on $X,$ there is a maximal, closed $G$-invariant subgroup $Y$ of $X$ so that the action of $G$ on $Y$ is weakly contained in a Bernoulli shift. This subgroup is also the minimal subgroup so that any action weakly contained in a Bernoulli shift is $Gcurvearrowright X/Y$-ergodic \"in the presence of $Gcurvearrowright X$\". We give several applications, including a major simplification of the proof that measure entropy equals topological entropy for principal algebraic actions whose associated convolution operator is injective. We also deduce from our techniques that algebraic actions whose square summable homoclinic group is dense have completely positive entropy when the acting group is sofic.","PeriodicalId":407889,"journal":{"name":"arXiv: Dynamical Systems","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131958243","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"STABILITY OF LYAPUNOV EXPONENTS, WEAK INTEGRAL SEPARATION AND NONUNIFORM DICHOTOMY SPECTRUM","authors":"Hailong Zhu, Zhaoxiang Li","doi":"10.22541/au.163257147.79616595/v1","DOIUrl":"https://doi.org/10.22541/au.163257147.79616595/v1","url":null,"abstract":"In this paper, a necessary and sufficient condition for the stability of\u0000Lyapunov exponents of linear differential system is proved in the sense\u0000that the equations satisfy the weaker form of integral separation\u0000instead of its classical one. Furthermore, the existence of full\u0000nonuniform exponential dichotomy spectrum under the condition of weak\u0000integral separateness is also presented.","PeriodicalId":407889,"journal":{"name":"arXiv: Dynamical Systems","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116797866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Geometry and entropies in a fixed conformal class on surfaces","authors":"Thomas Barthelm'e, A. Erchenko","doi":"10.5802/aif.3410","DOIUrl":"https://doi.org/10.5802/aif.3410","url":null,"abstract":"We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a thick-thin decomposition, and precompactness for the considered class of metrics. Also, we extend some of the results to metrics of fixed total area in a fixed conformal class with no focal points and with some integral bounds on the positive part of the Gaussian curvature.","PeriodicalId":407889,"journal":{"name":"arXiv: Dynamical Systems","volume":"89 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132691482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Une invitation aux surfaces de dilatation","authors":"Selim Ghazouani","doi":"10.5802/tsg.364","DOIUrl":"https://doi.org/10.5802/tsg.364","url":null,"abstract":"This text is an introduction to dilation surfaces. We attempt to expose some geometric and dynamical aspects of the subject: moduli spaces, directional foliations and the Teichm\"uller flow.","PeriodicalId":407889,"journal":{"name":"arXiv: Dynamical Systems","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116879523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Beta Transformation","authors":"L. Vepstas","doi":"10.13140/RG.2.2.17132.26248","DOIUrl":"https://doi.org/10.13140/RG.2.2.17132.26248","url":null,"abstract":"The beta transformation is the iterated map $beta xmod1$. The special case of $beta=2$ is known as the Bernoulli map, and is exactly solvable. The Bernoulli map provides a model for pure, unrestrained chaotic (ergodic) behavior: it is the full invariant shift on the Cantor space ${0,1}^{omega}$. The beta transformation defines a subshift: iterated on the unit interval, it singles out a subspace of the Cantor space, in such a way that it is invariant under the action of the left-shift operator. That is, lopping off one bit at a time gives back the same subspace. The beta transform seems to capture something basic about the multiplication of two real numbers: $beta$ and $x$. It offers a window into understanding the nature of multiplication. Iterating on multiplication, one would get $beta^{n}x$ - that is, exponentiation; although the mod 1 of the beta transform contorts this in interesting ways. The work presented here is a research diary: a pastiche of observations and some shallow insights. One is that chaos seems to be rooted in how the carry bit behaves during multiplication. Another is that one can surgically insert \"islands of stability\" into chaotic (ergodic) systems, and have some fair amount of control over how those islands of stability behave. In particular, one can have islands with, or without a period-doubling \"route to chaos\". The eigenvalues of the transfer operator seem to lie on a circle of radius $1/beta$ in the complex plane. Given that the transfer operator is purely real, the appearance of such a quasi-unitary spectrum seems surprising. The spectrum appears to be the limit of a dense set of quasi-cyclotomic polynomials, the positive real roots of which include the Golden and silver ratios, the Pisot numbers, the n-bonnaci (tribonacci, tetranacci, etc.) numbers.","PeriodicalId":407889,"journal":{"name":"arXiv: Dynamical Systems","volume":"159 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127292737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Circular slider graphs: de Bruijn, Kautz, Rauzy, lamplighters and spiders","authors":"V. Kaimanovich","doi":"10.1090/CONM/719/14472","DOIUrl":"https://doi.org/10.1090/CONM/719/14472","url":null,"abstract":"We suggest a new point of view on de Bruijn graphs and their subgraphs based on using circular words rather than linear ones.","PeriodicalId":407889,"journal":{"name":"arXiv: Dynamical Systems","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127726186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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