13,929 research outputs found
Chaos among self-maps of the Cantor space
No full textAbstractGlasner and Weiss have shown that a generic homeomorphism of the Cantor space has zero topological entropy. Hochman has shown that a generic transitive homeomorphism of the Cantor space has the property that it is topologically conjugate to the universal odometer and hence far from being chaotic in any sense. We show that a generic self-map of the Cantor space has zero topological entropy. Moreover, we show that a generic self-map of the Cantor space has no periodic points and hence is not Devaney chaotic nor Devaney chaotic on any subsystem. However, we exhibit a dense subset of the space of all self-maps of the Cantor space each element of which has infinite topological entropy and is Devaney chaotic on some subsystem
Cn Functions, Hausdorff Measures, and Analytic Sets
No full textAbstractWe characterize in terms of Hausdorff measures and descriptive complexity subsets M⊆R which are (1)the image under some Cn function f of the set of points where derivatives of first n orders are zero, (2)the set of points where the level sets of some Cn function are perfect, and (3)the set of points where the level sets of some Cn function are uncountable
Complexity of curves
No full textWe show that each of the classes of hereditarily locally connected, finitely Suslinian, and Suslinian continua is \Pi^1_1-complete, while the class of regular continua is \Pi^0_4-complete
Shift-like Operators on
No full textIn 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
Lineability of non-differentiable Pettis primitives
No full textLet X be an infinite-dimensional Banach space. In 1995, settling a long outstanding problem of Pettis, Dilworth and Girardi constructed an X-valued Pettis integrable function on [0,1] whose primitive is nowhere weakly differentiable. Using their technique and some new ideas we show that ND, the set of strongly measurable Pettis integrable functions with nowhere weakly differentiable primitives, is lineable, i.e., there is an infinite dimensional vector space whose nonzero vectors belong to ND
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