Displaying similar documents to “Maximal distributional chaos of weighted shift operators on Köthe sequence spaces”

On the non-commutative neutrix product ln x + ∘ x + - s

Brian Fisher, Adem Kiliçman, Blagovest Damyanov, J. C. Ault (1996)

Commentationes Mathematicae Universitatis Carolinae

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The non-commutative neutrix product of the distributions ln x + and x + - s is proved to exist for s = 1 , 2 , ... and is evaluated for s = 1 , 2 . The existence of the non-commutative neutrix product of the distributions x + - r and x + - s is then deduced for r , s = 1 , 2 , ... and evaluated for r = s = 1 .

On approximation of functions by certain operators preserving x 2

Lucyna Rempulska, Karolina Tomczak (2008)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we extend the Duman-King idea of approximation of functions by positive linear operators preserving e k ( x ) = x k , k = 0 , 2 . Using a modification of certain operators L n preserving e 0 and e 1 , we introduce operators L n * which preserve e 0 and e 2 and next we define operators L n ; r * for r -times differentiable functions. We show that L n * and L n ; r * have better approximation properties than L n and L n ; r .

A note on the strong maximal operator on ℝⁿ

Jiecheng Chen, Xiangrong Zhu (2004)

Studia Mathematica

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We prove that for f ∈ L ln⁺L(ℝⁿ) with compact support, there is a g ∈ L ln⁺L(ℝⁿ) such that (a) g and f are equidistributed, (b) M S ( g ) ∈ L ¹ ( E ) for any measurable set E of finite measure.

Defining complete and observable chaos

Víctor Jiménez López (1996)

Annales Polonici Mathematici

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For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x,y) ∈ I² for which l i m i n f n → ∞ | f n ( x ) - f n ( y ) | = 0 and l i m s u p n → ∞ | f n ( x ) - f n ( y ) | > 0 . We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ² denotes the Lebesgue measure on the square and Ch(f) is the set of points (x,y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ²(C(f)) > 0 need not imply λ²(Ch(f)) > 0. We use these results to propose some plausible...

Topological size of scrambled sets

François Blanchard, Wen Huang, L'ubomír Snoha (2008)

Colloquium Mathematicae

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A subset S of a topological dynamical system (X,f) containing at least two points is called a scrambled set if for any x,y ∈ S with x ≠ y one has l i m i n f n → ∞ d ( f ⁿ ( x ) , f ⁿ ( y ) ) = 0 and l i m s u p n → ∞ d ( f ⁿ ( x ) , f ⁿ ( y ) ) > 0 , d being the metric on X. The system (X,f) is called Li-Yorke chaotic if it has an uncountable scrambled set. These notions were developed in the context of interval maps, in which the existence of a two-point scrambled set implies Li-Yorke chaos and many other chaotic properties. In the present paper we address several questions about...

Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle

Sophie Grivaux, Maria Roginskaya (2013)

Czechoslovak Mathematical Journal

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We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle 𝕋 . A set of integers is called r -Bohr if it is recurrent for all products of r rotations on 𝕋 , and Bohr if it is recurrent for all products of rotations on 𝕋 . It is a result due to Katznelson that for each r ≥ 1 there exist sets of integers which are r -Bohr but not ( r + 1 ) -Bohr. We present new examples of r -Bohr sets which are not Bohr, thanks to a construction...