We prove analyticity in time of the particle trajectories associated with the solutions of some transport equations when the initial condition is the characteristic function of a regular bounded domain. These results are obtained from a detailed study of the Beurling transform, that represents a derivative of the velocity field. The precise estimates obtained for the solutions of an equation satisfied by the Lagrangian flow, are a key point in the development.
{"title":"On the analyticity of the trajectories of the particles in the planar patch problem for some active scalar equations","authors":"J. M. Burgués, J. Mateu","doi":"10.3934/dcds.2022005","DOIUrl":"https://doi.org/10.3934/dcds.2022005","url":null,"abstract":"We prove analyticity in time of the particle trajectories associated with the solutions of some transport equations when the initial condition is the characteristic function of a regular bounded domain. These results are obtained from a detailed study of the Beurling transform, that represents a derivative of the velocity field. The precise estimates obtained for the solutions of an equation satisfied by the Lagrangian flow, are a key point in the development.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"98 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85762290","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}
Y. El hadfi, Zhaosheng Feng, A. Ghazdali, A. Laghrib
{"title":"Preface to the special issue \"Partial Differential Equations, Optimization and their Applications\"","authors":"Y. El hadfi, Zhaosheng Feng, A. Ghazdali, A. Laghrib","doi":"10.3934/dcdss.2021163","DOIUrl":"https://doi.org/10.3934/dcdss.2021163","url":null,"abstract":"<jats:p xml:lang=\"fr\" />","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79652297","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}
R. Sakthivel, P. Selvaraj, Yeong-Jae Kim, Dong-Hoon Lee, O. Kwon, R. Sakthivel
This paper investigates the resilient begin{document}$ H_infty $end{document} event-triggered control problem for Takagi-Sugeno fuzzy system with time-varying delay and external disturbance. Contrary to some existing results, the considered event-triggered conditions are verified only at each sampling instant because it is difficult to prove Zeno-freeness for a continuous event-triggered mechanism in the presence of external disturbance. Furthermore, by constructing an appropriate Lyapunov-Krasovskii functional, sufficient conditions are derived in the form of linear matrix inequalities to ensure the asymptotic stability and the begin{document}$ H_infty $end{document} performances of closed-loop systems. More precisely, the proposed control design not only improve robust performance but also save the communication resources. Finally, the obtained theoretical results are verified through numerical simulation, which demonstrate the efficiency and advantages of the proposed method.
This paper investigates the resilient begin{document}$ H_infty $end{document} event-triggered control problem for Takagi-Sugeno fuzzy system with time-varying delay and external disturbance. Contrary to some existing results, the considered event-triggered conditions are verified only at each sampling instant because it is difficult to prove Zeno-freeness for a continuous event-triggered mechanism in the presence of external disturbance. Furthermore, by constructing an appropriate Lyapunov-Krasovskii functional, sufficient conditions are derived in the form of linear matrix inequalities to ensure the asymptotic stability and the begin{document}$ H_infty $end{document} performances of closed-loop systems. More precisely, the proposed control design not only improve robust performance but also save the communication resources. Finally, the obtained theoretical results are verified through numerical simulation, which demonstrate the efficiency and advantages of the proposed method.
{"title":"Robust $ H_infty $ resilient event-triggered control design for T-S fuzzy systems","authors":"R. Sakthivel, P. Selvaraj, Yeong-Jae Kim, Dong-Hoon Lee, O. Kwon, R. Sakthivel","doi":"10.3934/dcdss.2022028","DOIUrl":"https://doi.org/10.3934/dcdss.2022028","url":null,"abstract":"<p style='text-indent:20px;'>This paper investigates the resilient <inline-formula><tex-math id=\"M2\">begin{document}$ H_infty $end{document}</tex-math></inline-formula> event-triggered control problem for Takagi-Sugeno fuzzy system with time-varying delay and external disturbance. Contrary to some existing results, the considered event-triggered conditions are verified only at each sampling instant because it is difficult to prove Zeno-freeness for a continuous event-triggered mechanism in the presence of external disturbance. Furthermore, by constructing an appropriate Lyapunov-Krasovskii functional, sufficient conditions are derived in the form of linear matrix inequalities to ensure the asymptotic stability and the <inline-formula><tex-math id=\"M3\">begin{document}$ H_infty $end{document}</tex-math></inline-formula> performances of closed-loop systems. More precisely, the proposed control design not only improve robust performance but also save the communication resources. Finally, the obtained theoretical results are verified through numerical simulation, which demonstrate the efficiency and advantages of the proposed method.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73454626","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}
Han Zhao, Bangdong Sun, Hui Wang, Shiji Song, Yuli Zhang, Liejun Wang
This paper investigates the optimal operational decisions for the risk-neutral supplier and the risk-averse retailer in the supply chain with a service requirement under the conditional value-at-risk. Specifically, the optimal order and production policies with and without the bidirectional option contract are derived. Further, this paper shows that the optimal conditional value-at-risk of the retailer is non-increasing in the service requirement, while the optimal expected profit of the supplier is non-decreasing in the service requirement. When the service requirement is binding, the optimal conditional value-at-risk of the retailer is increasing in the risk aversion, while the optimal expected profit of the supplier is decreasing in the risk aversion. In addition, it is shown that with the bidirectional option contract, the service level provided by the retailer is equivalent to (higher than) that without them when the service requirement is (not) binding. Finally, this paper demonstrates that the bidirectional option contract can mitigate the effect of risk aversion on the retailer's order quantity, benefit both the retailer and supplier, and improve the performance of the supply chain. Numerical experiments are conducted to further confirm our results.
{"title":"Optimization and coordination in a service-constrained supply chain with the bidirectional option contract under conditional value-at-risk","authors":"Han Zhao, Bangdong Sun, Hui Wang, Shiji Song, Yuli Zhang, Liejun Wang","doi":"10.3934/dcdss.2022021","DOIUrl":"https://doi.org/10.3934/dcdss.2022021","url":null,"abstract":"This paper investigates the optimal operational decisions for the risk-neutral supplier and the risk-averse retailer in the supply chain with a service requirement under the conditional value-at-risk. Specifically, the optimal order and production policies with and without the bidirectional option contract are derived. Further, this paper shows that the optimal conditional value-at-risk of the retailer is non-increasing in the service requirement, while the optimal expected profit of the supplier is non-decreasing in the service requirement. When the service requirement is binding, the optimal conditional value-at-risk of the retailer is increasing in the risk aversion, while the optimal expected profit of the supplier is decreasing in the risk aversion. In addition, it is shown that with the bidirectional option contract, the service level provided by the retailer is equivalent to (higher than) that without them when the service requirement is (not) binding. Finally, this paper demonstrates that the bidirectional option contract can mitigate the effect of risk aversion on the retailer's order quantity, benefit both the retailer and supplier, and improve the performance of the supply chain. Numerical experiments are conducted to further confirm our results.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84726192","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}
In this paper we consider quasi-periodic non-twist mappings with self-intersection property, which depend on a small parameter. Without assuming any twist condition, we prove that for many sufficiently small parameters the mapping admits an invariant curve. As application, we use this result to study Lagrange stability of second order systems.
{"title":"A KAM theorem for quasi-periodic non-twist mappings and its application","authors":"Zhi-Hong Ma, Junxiang Xu","doi":"10.3934/dcds.2022013","DOIUrl":"https://doi.org/10.3934/dcds.2022013","url":null,"abstract":"In this paper we consider quasi-periodic non-twist mappings with self-intersection property, which depend on a small parameter. Without assuming any twist condition, we prove that for many sufficiently small parameters the mapping admits an invariant curve. As application, we use this result to study Lagrange stability of second order systems.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83258797","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}
This paper investigates the problem of disturbance-observer-based sliding mode control for stabilization of stochastic systems driven by fractional Brownian motion (fBm). By proposing a novel disturbance observer, an integral-type sliding surface is put forward with the estimated disturbance error confined within a certain value. Meanwhile, by virtue of fractional infinitesimal operator and linear matrix inequality, a sufficient criterion is derived to guarantee the asymptotic stability of obtained sliding mode dynamics. Further, an observer-based sliding mode controller is designed to ensure finite-time reachability of state trajectories onto the predefined sliding surface. Lastly, an illustrative example is utilized to verify the reliability and applicability of the proposed control strategy.
{"title":"Observer-based SMC for stochastic systems with disturbance driven by fractional Brownian motion","authors":"Xin Meng, C. Gao, Baoping Jiang, H. Karimi","doi":"10.3934/dcdss.2022027","DOIUrl":"https://doi.org/10.3934/dcdss.2022027","url":null,"abstract":"This paper investigates the problem of disturbance-observer-based sliding mode control for stabilization of stochastic systems driven by fractional Brownian motion (fBm). By proposing a novel disturbance observer, an integral-type sliding surface is put forward with the estimated disturbance error confined within a certain value. Meanwhile, by virtue of fractional infinitesimal operator and linear matrix inequality, a sufficient criterion is derived to guarantee the asymptotic stability of obtained sliding mode dynamics. Further, an observer-based sliding mode controller is designed to ensure finite-time reachability of state trajectories onto the predefined sliding surface. Lastly, an illustrative example is utilized to verify the reliability and applicability of the proposed control strategy.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89012790","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}
We present a mechanism for the emergence of strange attractors in a one-parameter family of differential equations defined on a 3-dimensional sphere. When the parameter is zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two saddles-foci with different Morse indices. After slightly increasing the parameter, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We will show that, for a set of parameters close enough to zero with positive Lebesgue measure, the dynamics exhibits strange attractors winding around the "ghost'' of a torus and supporting Sinai-Ruelle-Bowen (SRB) measures. We also prove the existence of a sequence of parameter values for which the family exhibits a superstable sink and describe the transition from a Bykov network to a strange attractor.
{"title":"\"Large\" strange attractors in the unfolding of a heteroclinic attractor","authors":"Alexandre A. P. Rodrigues","doi":"10.3934/dcds.2021193","DOIUrl":"https://doi.org/10.3934/dcds.2021193","url":null,"abstract":"We present a mechanism for the emergence of strange attractors in a one-parameter family of differential equations defined on a 3-dimensional sphere. When the parameter is zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two saddles-foci with different Morse indices. After slightly increasing the parameter, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We will show that, for a set of parameters close enough to zero with positive Lebesgue measure, the dynamics exhibits strange attractors winding around the \"ghost'' of a torus and supporting Sinai-Ruelle-Bowen (SRB) measures. We also prove the existence of a sequence of parameter values for which the family exhibits a superstable sink and describe the transition from a Bykov network to a strange attractor.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74568901","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}
Partial differential equations on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schrödinger type equations) or to the analysis of flexible structures (wave type equations). Nevertheless, very few results are available for diffusive models despite an increasing demand arising from life sciences such as neurobiology. This paper analyzes the controllability properties of the heat equation on a compact network under the action of a single input bilinear control.By adapting a recent method due to [F. Alabau-Boussouira, P. Cannarsa, C. Urbani, Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control, arXiv: 1811.08806], an exact controllability result to the eigensolutions of the uncontrolled problem is obtained in this work. A crucial step has been the construction of a suitable biorthogonal family under a non-uniform gap condition of the eigenvalues of the Laplacian on a graph. Application to star graphs and tadpole graphs are included.
由于网络上的偏微分方程在量子力学(Schrödinger型方程)或柔性结构分析(波动型方程)中的应用，在过去几十年中得到了广泛的研究。然而，尽管神经生物学等生命科学对扩散模型的需求日益增加，但很少有结果可用。本文分析了在单输入双线性控制作用下紧致网络上热方程的可控性。由于[F.]Alabau-Boussouira, P. Cannarsa, C. Urbani，基于双线性控制的抛物型演化方程特征解的精确可控性，[j] . vol . 11: 1811.08806，得到了非控制问题特征解的精确可控性结果。在图上拉普拉斯特征值的非均匀间隙条件下，构造合适的双正交族是关键的一步。应用星图和蝌蚪图包括在内。
{"title":"Exact controllability to eigensolutions of the bilinear heat equation on compact networks","authors":"P. Cannarsa, Alessandro Duca, Cristina Urbani","doi":"10.3934/dcdss.2022011","DOIUrl":"https://doi.org/10.3934/dcdss.2022011","url":null,"abstract":"Partial differential equations on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schrödinger type equations) or to the analysis of flexible structures (wave type equations). Nevertheless, very few results are available for diffusive models despite an increasing demand arising from life sciences such as neurobiology. This paper analyzes the controllability properties of the heat equation on a compact network under the action of a single input bilinear control.By adapting a recent method due to [F. Alabau-Boussouira, P. Cannarsa, C. Urbani, Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control, arXiv: 1811.08806], an exact controllability result to the eigensolutions of the uncontrolled problem is obtained in this work. A crucial step has been the construction of a suitable biorthogonal family under a non-uniform gap condition of the eigenvalues of the Laplacian on a graph. Application to star graphs and tadpole graphs are included.","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82009135","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}
We consider optimal control of fractional in time (subdiffusive, i.e., for begin{document}$ 0) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we begin{document}$mathsf{first;show}$end{document} the existence and regularity of solutions to the forward and the associated begin{document}$mathsf{backward;(adjoint)}$end{document} problems. In the second part, we prove existence of optimal begin{document}$mathsf{controls }$end{document} and characterize the associated begin{document}$mathsf{first;order}$end{document} optimality conditions. Several examples involving fractional in time (and some fractional in space diffusion) equations are described in detail. The most challenging obstacle we overcome is the failure of the semigroup property for the semilinear problem in any scaling of (frequency-domain) Hilbert spaces.
{"title":"A unified framework for optimal control of fractional in time subdiffusive semilinear PDEs","authors":"Harbir Antil, C. Gal, M. Warma","doi":"10.3934/dcdss.2022012","DOIUrl":"https://doi.org/10.3934/dcdss.2022012","url":null,"abstract":"<p style='text-indent:20px;'>We consider optimal control of fractional in time (subdiffusive, i.e., for <inline-formula><tex-math id=\"M1\">begin{document}$ 0<gamma <1 $end{document}</tex-math></inline-formula>) semilinear parabolic PDEs associated with various notions of diffusion operators in an unifying fashion. Under general assumptions on the nonlinearity we <inline-formula><tex-math id=\"M2\">begin{document}$mathsf{first;show}$end{document}</tex-math></inline-formula> the existence and regularity of solutions to the forward and the associated <inline-formula><tex-math id=\"M3\">begin{document}$mathsf{backward;(adjoint)}$end{document}</tex-math></inline-formula> problems. In the second part, we prove existence of optimal <inline-formula><tex-math id=\"M4\">begin{document}$mathsf{controls }$end{document}</tex-math></inline-formula> and characterize the associated <inline-formula><tex-math id=\"M5\">begin{document}$mathsf{first;order}$end{document}</tex-math></inline-formula> optimality conditions. Several examples involving fractional in time (and some fractional in space diffusion) equations are described in detail. The most challenging obstacle we overcome is the failure of the semigroup property for the semilinear problem in any scaling of (frequency-domain) Hilbert spaces.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87914025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about begin{document}$ mathbb{Z} $end{document}-actions extend to this setting:
1. If begin{document}$ (a_n) $end{document} is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.
2. There exists a sequence begin{document}$ (r_n) $end{document} such that every translate is both a rigidity sequence and a set of recurrence.
The first of these results was shown for begin{document}$ mathbb{Z} $end{document}-actions by Adams [1], Fayad and Thouvenot [20], and Badea and Grivaux [2]. The latter was established in begin{document}$ mathbb{Z} $end{document} by Griesmer [23]. While techniques for handling begin{document}$ mathbb{Z} $end{document}-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.
As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in begin{document}$ mathbb{Z} $end{document}, while others exhibit new phenomena.
The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about begin{document}$ mathbb{Z} $end{document}-actions extend to this setting:1. If begin{document}$ (a_n) $end{document} is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.2. There exists a sequence begin{document}$ (r_n) $end{document} such that every translate is both a rigidity sequence and a set of recurrence.The first of these results was shown for begin{document}$ mathbb{Z} $end{document}-actions by Adams [1], Fayad and Thouvenot [20], and Badea and Grivaux [2]. The latter was established in begin{document}$ mathbb{Z} $end{document} by Griesmer [23]. While techniques for handling begin{document}$ mathbb{Z} $end{document}-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in begin{document}$ mathbb{Z} $end{document}, while others exhibit new phenomena.
{"title":"Rigidity, weak mixing, and recurrence in abelian groups","authors":"Ethan Ackelsberg","doi":"10.3934/dcds.2021168","DOIUrl":"https://doi.org/10.3934/dcds.2021168","url":null,"abstract":"<p style='text-indent:20px;'>The focus of this paper is the phenomenon of rigidity for measure-preserving actions of countable discrete abelian groups and its interactions with weak mixing and recurrence. We prove that results about <inline-formula><tex-math id=\"M1\">begin{document}$ mathbb{Z} $end{document}</tex-math></inline-formula>-actions extend to this setting:</p><p style='text-indent:20px;'>1. If <inline-formula><tex-math id=\"M2\">begin{document}$ (a_n) $end{document}</tex-math></inline-formula> is a rigidity sequence for an ergodic measure-preserving system, then it is a rigidity sequence for some weakly mixing system.</p><p style='text-indent:20px;'>2. There exists a sequence <inline-formula><tex-math id=\"M3\">begin{document}$ (r_n) $end{document}</tex-math></inline-formula> such that every translate is both a rigidity sequence and a set of recurrence.</p><p style='text-indent:20px;'>The first of these results was shown for <inline-formula><tex-math id=\"M4\">begin{document}$ mathbb{Z} $end{document}</tex-math></inline-formula>-actions by Adams [<xref ref-type=\"bibr\" rid=\"b1\">1</xref>], Fayad and Thouvenot [<xref ref-type=\"bibr\" rid=\"b20\">20</xref>], and Badea and Grivaux [<xref ref-type=\"bibr\" rid=\"b2\">2</xref>]. The latter was established in <inline-formula><tex-math id=\"M5\">begin{document}$ mathbb{Z} $end{document}</tex-math></inline-formula> by Griesmer [<xref ref-type=\"bibr\" rid=\"b23\">23</xref>]. While techniques for handling <inline-formula><tex-math id=\"M6\">begin{document}$ mathbb{Z} $end{document}</tex-math></inline-formula>-actions play a key role in our proofs, additional ideas must be introduced for dealing with groups with multiple generators.</p><p style='text-indent:20px;'>As an application of our results, we give several new constructions of rigidity sequences in torsion groups. Some of these are parallel to examples of rigidity sequences in <inline-formula><tex-math id=\"M7\">begin{document}$ mathbb{Z} $end{document}</tex-math></inline-formula>, while others exhibit new phenomena.</p>","PeriodicalId":11254,"journal":{"name":"Discrete & Continuous Dynamical Systems - S","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80737913","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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