American Institute of Mathematical Sciences

February  2005, 13(2): 271-275. doi: 10.3934/dcds.2005.13.271

On a hypercylicity criterion for strongly continuous semigroups

Received  March 2004 Revised  January 2005 Published  April 2005

In this note we extend the hypercyclicity criterion (HC) to $C_0$- semigroups on separable Banach spaces and characterize semigroups satisfying (HC). Using (HC) we show the hypercyclicity of the translation group on the weighted spaces $L^p_\rho(\mathbb R,\mathbb C)$ or $C_{0,\rho}(\mathbb R,\mathbb C)$ for admissible weight functions $\rho$.
Citation: Samir EL Mourchid. On a hypercylicity criterion for strongly continuous semigroups. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 271-275. doi: 10.3934/dcds.2005.13.271
 [1] Grant Cairns, Barry Jessup, Marcel Nicolau. Topologically transitive homeomorphisms of quotients of tori. Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 291-300. doi: 10.3934/dcds.1999.5.291 [2] Jacek Banasiak, Marcin Moszyński. Hypercyclicity and chaoticity spaces of $C_0$ semigroups. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 577-587. doi: 10.3934/dcds.2008.20.577 [3] Viorel Nitica. Examples of topologically transitive skew-products. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 351-360. doi: 10.3934/dcds.2000.6.351 [4] Jan Kwiatkowski, Artur Siemaszko. Discrete orbits in topologically transitive cylindrical transformations. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 945-961. doi: 10.3934/dcds.2010.27.945 [5] José A. Conejero, Alfredo Peris. Hypercyclic translation $C_0$-semigroups on complex sectors. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1195-1208. doi: 10.3934/dcds.2009.25.1195 [6] George Avalos. Strong stability of PDE semigroups via a generator resolvent criterion. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 207-218. doi: 10.3934/dcdss.2008.1.207 [7] John Banks, Piotr Oprocha, Brett Stanley. Transitive sofic spacing shifts. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4743-4764. doi: 10.3934/dcds.2015.35.4743 [8] Sergiĭ Kolyada, Mykola Matviichuk. On extensions of transitive maps. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 767-777. doi: 10.3934/dcds.2011.30.767 [9] Kesong Yan, Qian Liu, Fanping Zeng. Classification of transitive group actions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5579-5607. doi: 10.3934/dcds.2021089 [10] Andrew D. Barwell, Chris Good, Piotr Oprocha, Brian E. Raines. Characterizations of $\omega$-limit sets in topologically hyperbolic systems. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1819-1833. doi: 10.3934/dcds.2013.33.1819 [11] Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079 [12] Salvador Addas-Zanata, Fábio A. Tal. Homeomorphisms of the annulus with a transitive lift II. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 651-668. doi: 10.3934/dcds.2011.31.651 [13] Shengzhi Zhu, Shaobo Gan, Lan Wen. Indices of singularities of robustly transitive sets. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 945-957. doi: 10.3934/dcds.2008.21.945 [14] Carlos Gutierrez, Simon Lloyd, Vladislav Medvedev, Benito Pires, Evgeny Zhuzhoma. Transitive circle exchange transformations with flips. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 251-263. doi: 10.3934/dcds.2010.26.251 [15] Alastair Fletcher. Quasiregular semigroups with examples. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2157-2172. doi: 10.3934/dcds.2019090 [16] Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447 [17] José A. Conejero, Alfredo Peris. Chaotic translation semigroups. Conference Publications, 2007, 2007 (Special) : 269-276. doi: 10.3934/proc.2007.2007.269 [18] Min He. On continuity in parameters of integrated semigroups. Conference Publications, 2003, 2003 (Special) : 403-412. doi: 10.3934/proc.2003.2003.403 [19] Cheng Cheng, Shaobo Gan, Yi Shi. A robustly transitive diffeomorphism of Kan's type. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 867-888. doi: 10.3934/dcds.2018037 [20] Vladimír Špitalský. Transitive dendrite map with infinite decomposition ideal. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 771-792. doi: 10.3934/dcds.2015.35.771

2021 Impact Factor: 1.588