5,621 research outputs found
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
Decidability and Universality in Symbolic Dynamical Systems
Many different definitions of computational universality for various types of
dynamical systems have flourished since Turing's work. We propose a general
definition of universality that applies to arbitrary discrete time symbolic
dynamical systems. Universality of a system is defined as undecidability of a
model-checking problem. For Turing machines, counter machines and tag systems,
our definition coincides with the classical one. It yields, however, a new
definition for cellular automata and subshifts. Our definition is robust with
respect to initial condition, which is a desirable feature for physical
realizability.
We derive necessary conditions for undecidability and universality. For
instance, a universal system must have a sensitive point and a proper
subsystem. We conjecture that universal systems have infinite number of
subsystems. We also discuss the thesis according to which computation should
occur at the `edge of chaos' and we exhibit a universal chaotic system.Comment: 23 pages; a shorter version is submitted to conference MCU 2004 v2:
minor orthographic changes v3: section 5.2 (collatz functions) mathematically
improved v4: orthographic corrections, one reference added v5:27 pages.
Important modifications. The formalism is strengthened: temporal logic
replaced by finite automata. New results. Submitte
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
Generic Points for Dynamical Systems with Average Shadowing
It is proved that to every invariant measure of a compact dynamical system
one can associate a certain asymptotic pseudo orbit such that any point
asymptotically tracing in average that pseudo orbit is generic for the measure.
It follows that the asymptotic average shadowing property implies that every
invariant measure has a generic point. The proof is based on the properties of
the Besicovitch pseudometric DB which are of independent interest. It is proved
among the other things that the set of generic points of ergodic measures is a
closed set with respect to DB. It is also showed that the weak specification
property implies the average asymptotic shadowing property thus the theory
presented generalizes most known results on the existence of generic points for
arbitrary invariant measures
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