7,149 research outputs found
On the Dynamics of Induced Maps on the Space of Probability Measures
For the generic continuous map and for the generic homeomorphism of the
Cantor space, we study the dynamics of the induced map on the space of
probability measures, with emphasis on the notions of Li-Yorke chaos,
topological entropy, equicontinuity, chain continuity, chain mixing, shadowing
and recurrence. We also establish some results concerning induced maps that
hold on arbitrary compact metric spaces.Comment: 23 page
Mean Li-Yorke chaos in Banach spaces
We investigate the notion of mean Li-Yorke chaos for operators on Banach
spaces. We show that it differs from the notion of distributional chaos of type
2, contrary to what happens in the context of topological dynamics on compact
metric spaces. We prove that an operator is mean Li-Yorke chaotic if and only
if it has an absolutely mean irregular vector. As a consequence, absolutely
Ces\`aro bounded operators are never mean Li-Yorke chaotic. Dense mean Li-Yorke
chaos is shown to be equivalent to the existence of a dense (or residual) set
of absolutely mean irregular vectors. As a consequence, every mean Li-Yorke
chaotic operator is densely mean Li-Yorke chaotic on some infinite-dimensional
closed invariant subspace. A (Dense) Mean Li-Yorke Chaos Criterion and a
sufficient condition for the existence of a dense absolutely mean irregular
manifold are also obtained. Moreover, we construct an example of an invertible
hypercyclic operator such that every nonzero vector is absolutely mean
irregular for both and . Several other examples are also presented.
Finally, mean Li-Yorke chaos is also investigated for -semigroups of
operators on Banach spaces.Comment: 26 page
Li-Yorke Chaos for Composition Operators on -Spaces
Li-Yorke chaos is a popular and well-studied notion of chaos. Several simple
and useful characterizations of this notion of chaos in the setting of linear
dynamics were obtained recently. In this note we show that even simpler and
more useful characterizations of Li-Yorke chaos can be given in the special
setting of composition operators on spaces. As a consequence we obtain a
simple characterization of weighted shifts which are Li-Yorke chaotic. We give
numerous examples to show that our results are sharp
A note on the continuity of multilinear mappings in topological modules
In the present note, we obtain a criterion for the equicontinuity of families of multilinear mappings between topological modules. We also give an example which shows that the hypothesis imposed on the neighborhoods of zero is essential for the validity of our theorem
Probing quantum fluctuation theorems in engineered reservoirs
Fluctuation Theorems are central in stochastic thermodynamics, as they allow
for quantifying the irreversibility of single trajectories. Although they have
been experimentally checked in the classical regime, a practical demonstration
in the framework of quantum open systems is still to come. Here we propose a
realistic platform to probe fluctuation theorems in the quantum regime. It is
based on an effective two-level system coupled to an engineered reservoir, that
enables the detection of the photons emitted and absorbed by the system. When
the system is coherently driven, a measurable quantum component in the entropy
production is evidenced. We quantify the error due to photon detection
inefficiency, and show that the missing information can be efficiently
corrected, based solely on the detected events. Our findings provide new
insights into how the quantum character of a physical system impacts its
thermodynamic evolution.Comment: 9 pages, 4 figure
Expansivity and Shadowing in Linear Dynamics
In the early 1970's Eisenberg and Hedlund investigated relationships between
expansivity and spectrum of operators on Banach spaces. In this paper we
establish relationships between notions of expansivity and hypercyclicity,
supercyclicity, Li-Yorke chaos and shadowing. In the case that the Banach space
is or (), we give complete characterizations
of weighted shifts which satisfy various notions of expansivity. We also
establish new relationships between notions of expansivity and spectrum.
Moreover, we study various notions of shadowing for operators on Banach spaces.
In particular, we solve a basic problem in linear dynamics by proving the
existence of nonhyperbolic invertible operators with the shadowing property.
This also contrasts with the expected results for nonlinear dynamics on compact
manifolds, illuminating the richness of dynamics of infinite dimensional linear
operators
On the structure of the neritic suprabenthic communities from the Portuguese continental margin
This work presents the investigations made on the neritic suprabenthic communities of the Portuguese margin (continental shelf and upper slope) exposed to seasonal upwelling. These communities were sampled during the AVEIRO-94 cruise at 5 sites located along an E-W bathymetric transect from 21 to 299 m depth using a suprabenthic sled with superposed nets. In the 0 to 100 cm water layer, the total densities ranged from 700.2 to 13591.7 ind. 100m(-2). During daytime, the motile fauna was mainly concentrated within the 0 to 50 cm water layer (76.2 to 97.2% of the total abundance). The night-time sample at the shallower site showed a more even distribution of the fauna in the near-bottom water layers (nocturnal migratory behaviour of some motile species). The Shannon diversity (H') values ranged from 1.84 to 3.54 for the shelf sites and increased at the upper slope site (4.15). Mysids and amphipods were generally dominant except for at the middle part of the shelf where the latter was replaced by euphausiids. The suprabenthic fauna off Aveiro was compared with similar data from the same bathymetric sampling levels off Arcachon (Bay of Biscay). Multivariate analysis showed that differences in faunal composition between the 2 geographic areas were smaller than depth-related variations within geographic areas. The results were discussed in relation to other suprabenthic communities from the northeastern Atlantic.Programa de Cooperação Oceanológica Luso-FrancesaJNICT/Embaixada de FrançaFrench CIRMAT-CNR
On the existence of polynomials with chaotic behaviour
We establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic) polynomials of arbitrary positive degree. Moreover, every complex infinite-dimensional separable Banach space with an unconditional Schauder decomposition and every complex Frèchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree. We also study distributional chaos for polynomials and show that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that have a dense distributionally scrambled linear manifold. © 2013 Nilson C. Bernardes Jr. and Alfredo Peris.The present work was done while the first author was visiting the Departament de Matematica Aplicada at Universitat Politecnica de Valencia (Spain). The first author is very grateful for the hospitality. The first author was supported in part by CAPES: Bolsista, Project no. BEX 4012/11-9. The second author was supported in part by MEC and FEDER, Project MTM2010-14909, and by GVA, Projects PROMETEO/2008/101 and PROMETEOII/2013/013.Bernardes, NC.; Peris Manguillot, A. (2013). On the existence of polynomials with chaotic behaviour. Journal of Function Spaces and Applications. 2013(320961). https://doi.org/10.1155/2013/320961S2013320961Bayart, F., & Matheron, E. (2009). Dynamics of Linear Operators. doi:10.1017/cbo9780511581113Grosse-Erdmann, K.-G., & Peris Manguillot, A. (2011). Linear Chaos. Universitext. doi:10.1007/978-1-4471-2170-1Rolewicz, S. (1969). On orbits of elements. Studia Mathematica, 32(1), 17-22. doi:10.4064/sm-32-1-17-22Herzog, G. (1992). 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Journal of Mathematical Analysis and Applications, 287(2), 487-493. doi:10.1016/s0022-247x(03)00547-xMARTÍNEZ-GIMÉNEZ, F., & PERIS, A. (2010). CHAOTIC POLYNOMIALS ON SEQUENCE AND FUNCTION SPACES. International Journal of Bifurcation and Chaos, 20(09), 2861-2867. doi:10.1142/s0218127410027416Martínez-Giménez, F., & Peris, A. (2009). Existence of hypercyclic polynomials on complex Fréchet spaces. Topology and its Applications, 156(18), 3007-3010. doi:10.1016/j.topol.2009.02.010Bès, J., & Peris, A. (2007). Disjointness in hypercyclicity. Journal of Mathematical Analysis and Applications, 336(1), 297-315. doi:10.1016/j.jmaa.2007.02.043Bernardes, N. C., Bonilla, A., Müller, V., & Peris, A. (2013). Distributional chaos for linear operators. Journal of Functional Analysis, 265(9), 2143-2163. doi:10.1016/j.jfa.2013.06.019Le�n-Saavedra, F., & M�ller, V. (2004). Rotations of Hypercyclic and Supercyclic Operators. Integral Equations and Operator Theory, 50(3), 385-391. doi:10.1007/s00020-003-1299-8Grosse-Erdmann, K.-G., & Peris, A. (2010). Weakly mixing operators on topological vector spaces. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 104(2), 413-426. doi:10.5052/racsam.2010.25Li, T.-Y., & Yorke, J. A. (1975). Period Three Implies Chaos. The American Mathematical Monthly, 82(10), 985. doi:10.2307/2318254Schweizer, B., & Smital, J. (1994). Measures of Chaos and a Spectral Decomposition of Dynamical Systems on the Interval. Transactions of the American Mathematical Society, 344(2), 737. doi:10.2307/2154504Bermúdez, T., Bonilla, A., Martínez-Giménez, F., & Peris, A. (2011). Li–Yorke and distributionally chaotic operators. Journal of Mathematical Analysis and Applications, 373(1), 83-93. doi:10.1016/j.jmaa.2010.06.011Hou, B., Cui, P., & Cao, Y. (2010). Chaos for Cowen-Douglas operators. Proceedings of the American Mathematical Society, 138(03), 929-929. doi:10.1090/s0002-9939-09-10046-1Martínez-Giménez, F., Oprocha, P., & Peris, A. (2009). Distributional chaos for backward shifts. Journal of Mathematical Analysis and Applications, 351(2), 607-615. doi:10.1016/j.jmaa.2008.10.049Schenke, A., & Shkarin, S. (2013). Hypercyclic operators on countably dimensional spaces. Journal of Mathematical Analysis and Applications, 401(1), 209-217. doi:10.1016/j.jmaa.2012.11.013BONET, J., FRERICK, L., PERIS, A., & WENGENROTH, J. (2005). TRANSITIVE AND HYPERCYCLIC OPERATORS ON LOCALLY CONVEX SPACES. Bulletin of the London Mathematical Society, 37(02), 254-264. doi:10.1112/s0024609304003698Shkarin, S. (2012). Hypercyclic operators on topological vector spaces. Journal of the London Mathematical Society, 86(1), 195-213. doi:10.1112/jlms/jdr08
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