45 research outputs found
Huygens synchronisation of three clocks equidistant from each other
In this paper we study the synchronisation of three identical oscillators,
i.e., clocks, hanging from the same hard support. We consider the case where
each clock interacts with the other two clocks. The synchronisation is attained
through the exchange of small impacts between each pair of oscillators. The
fundamental result of this article is that the final locked state is at phase
difference of ((2{\pi})/3) from successive clocks (clockwise or
counter-clockwise). Moreover, the locked states attract a set whose closure is
the global set of initial conditions. The methodology of our analysis consists
in the construction a model, which is a non-linear discrete dynamical system,
i.e. a non-linear difference equation. The results are extendable to any set of
three oscillators under mutual symmetric interaction, despite the particular
models of the oscillators
On the spectrum of weighted shifts
It is well-known that, in Linear Dynamics, the most studied class of linear
operators is certainly that of weighted shifts, on the separable Banach spaces
and , . Over the last decades, the intensive
study of such operators has produced an incredible number of versatile, deep
and beautiful results, applicable in various areas of Mathematics and the
relationships between various important notions, especially concerning chaos
and hyperbolic properties, and the spectrum of weighted shifts are
investigated. In this paper, we investigate the point spectrum of weighted
shifts and, under some regularity hypotheses on the weight sequence, we deduce
the spectrum
Averaging on n-dimensional rectangles
In this work we investigate families of translation invariant differentiation bases B of rectangles in Rn, for which L log^(n−1) L(R^n) is the largest Orlicz space that B differentiates. In particular, we improve on techniques developed by Stokolos in [11] and [13] (see the attached file)
Shift-like Operators on
In this article we develop a general technique which takes a known
characterization of a property for weighted backward shifts and lifts it up to
a characterization of that property for a large class of operators on .
We call these operators ``shift-like''. The properties of interest include
chaotic properties such as Li-Yorke chaos, hypercyclicity, frequent
hypercyclicity as well as properties related to hyperbolic dynamics such as
shadowing, expansivity and generalized hyperbolicity. Shift-like operators
appear naturally as composition operators on when the underlying space
is a dissipative measure system. In the process of proving the main theorem, we
provide some results concerning when a property is shared by a linear dynamical
system and its factors.Comment: arXiv admin note: text overlap with arXiv:2009.1152
Generalized Hyperbolicity and Shadowing in spaces
It is rather well-known that hyperbolic operators have the shadowing
property. In the setting of finite dimensional Banach spaces, having the
shadowing property is equivalent to being hyperbolic. In 2018, Bernardes et al.
constructed an operator with the shadowing property which is not hyperbolic,
settling an open question. In the process, they introduced a class of operators
which has come to be known as generalized hyperbolic operators. This class of
operators seems to be an important bridge between hyperbolicity and the
shadowing property. In this article, we show that for a large natural class of
operators on the notion of generalized hyperbolicity and the shadowing
property coincide. We do this by giving sufficient and necessary conditions for
a certain class of operators to have the shadowing property. We also introduce
computational tools which allow construction of operators with and without the
shadowing property. Utilizing these tools, we show how some natural probability
distributions, such as the Laplace distribution and the Cauchy distribution,
lead to operators with and without the shadowing property on
Linear Dynamics Induced by Odometers
Weighted shifts are an important concrete class of operators in linear
dynamics. In particular, they are an essential tool in distinguishing variety
dynamical properties. Recently, a systematic study of dynamical properties of
composition operators on spaces has been initiated. This class of
operators includes weighted shifts and also allows flexibility in construction
of other concrete examples. In this article, we study one such concrete class
of operators, namely composition operators induced by measures on odometers. In
particular, we study measures on odometers which induce mixing and transitive
linear operators on spaces.Comment: 15 pages, keywords: linear dynamics, composition operators,
topological mixing, topological transitivity, odometer