7,760 research outputs found
Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems
We study chaotic properties of uniformly convergent nonautonomous dynamical
systems. We show that, contrary to the autonomous systems on the compact
interval, positivity of topological sequence entropy and occurrence of Li-Yorke
chaos are not equivalent, more precisely, neither of the two possible
implications is true.Comment: 10 pages, 4 figure
The Specification Property and Infinite Entropy for Certain Classes of Linear Operators
We study the specification property and infinite topological entropy for two specific types of linear operators: translation operators on weighted Lebesgue function spaces and weighted backward shift operators on sequence F-spaces.
It is known from the work of Bartoll, MartinĂnez-GimĂ©nez, Murillo-Arcila (2014), and Peris, that for weighted backward shift operators, the existence of a single non-trivial periodic point is sufficient for specification. We show this also holds for translation operators on weighted Lebesgue function spaces. This implies, in particular, that for these operators, the specification property is equivalent to Devaney chaos. We also show that these forms of chaos imply infinite topological entropy, but that the converse does not hold
Two results on entropy, chaos, and independence in symbolic dynamics
We survey the connections between entropy, chaos, and independence in
topological dynamics. We present extensions of two classical results placing
the following notions in the context of symbolic dynamics:
1. Equivalence of positive entropy and the existence of a large (in terms of
asymptotic and Shnirelman densities) set of combinatorial independence for
shift spaces.
2. Existence of a mixing shift space with a dense set of periodic points with
topological entropy zero and without ergodic measure with full support, nor any
distributionally chaotic pair.
Our proofs are new and yield conclusions stronger than what was known before.Comment: Comments are welcome! This preprint contains results from
arXiv:1401.5969v
Measuring complexity in a business cycle
The purpose of this paper is to study the dynamical behavior of a family of two-
dimensional nonlinear maps associated to an economic model. Our objective is to
measure the complexity of the system using techniques of symbolic dynamics in
order to compute the topological entropy. The analysis of the variation of this im-
portant topological invariant with the parameters of the system, allows us to distin-
guish different chaotic scenarios. Finally, we use a another topological invariant to
distinguish isentropic dynamics and we exhibit numerical results about maps with
the same topological entropy. This work provides an illustration of how our under-
standing of higher dimensional economic models can be enhanced by the theory of
dynamical systems
Topological Chaos in Spatially Periodic Mixers
Topologically chaotic fluid advection is examined in two-dimensional flows
with either or both directions spatially periodic. Topological chaos is created
by driving flow with moving stirrers whose trajectories are chosen to form
various braids. For spatially periodic flows, in addition to the usual
stirrer-exchange braiding motions, there are additional
topologically-nontrivial motions corresponding to stirrers traversing the
periodic directions. This leads to a study of the braid group on the cylinder
and the torus. Methods for finding topological entropy lower bounds for such
flows are examined. These bounds are then compared to numerical stirring
simulations of Stokes flow to evaluate their sharpness. The sine flow is also
examined from a topological perspective.Comment: 18 pages, 14 figures. RevTeX4 style with psfrag macros. Final versio
On Almost Automorphic Dynamics in Symbolic Lattices
1991 Mathematics Subject Classification. Primary Primary 37B10, 37A35, 43A60; Secondary
37B20, 54H20.We study the existence, structure, and topological entropy of almost automorphic arrays in symbolic lattice dynamical systems. In particular we show that almost automorphic arrays with arbitrarily large entropy are typical in symbolic lattice dynamical systems. Applications to pattern formation and spatial chaos in infinite dimensional lattice systems are considered,
and the construction of chaotic almost automorphic signals is discussed.The first author was supported by a Max Kade Postdoctoral Fellowship (at Georgia Tech). The second author was partially supported by DFG grant Si 801 and CDSNS, Georgia Tech. The third author was partially supported by NSF Grant DMS-0204119
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