scholarly journals An infinitely generated self-similar set with positive Lebesgue measure and empty interior

2019 ◽  
Vol 147 (11) ◽  
pp. 4891-4899
Author(s):  
Simon Baker ◽  
Nikita Sidorov
Keyword(s):  
Lebesgue Measure ◽  
Positive Lebesgue Measure ◽  
Empty Interior ◽  
Self Similar

Self-Affine sets: The Relation Between Positive Lebesgue Measure and Non-Empty Interior

Fractals ◽  
2021 ◽  
Author(s):  
Qi-Rong Deng ◽  
Si-Min Li
Keyword(s):  
Lebesgue Measure ◽  
Positive Lebesgue Measure ◽  
Empty Interior

scholarly journals Do self-similar sets with positive Lebesgue measure contain an interval?

2010 ◽  
Vol 23 (2) ◽  
pp. 207-211 ◽  
Author(s):  
Keqiang Dong ◽  
Pengjian Shang
Keyword(s):  
Lebesgue Measure ◽  
Positive Lebesgue Measure ◽  
Self Similar

Strange Nonchaotic Trajectories on Torus

1997 ◽  
Vol 07 (02) ◽  
pp. 423-429 ◽  
Author(s):  
T. Kapitaniak ◽  
L. O. Chua
Keyword(s):  
Lebesgue Measure ◽  
Parameter Space ◽  
Positive Lebesgue Measure ◽  
Strange Nonchaotic Attractors

In this letter we have shown that aperiodic nonchaotic trajectories characteristic of strange nonchaotic attractors can occur on a two-frequency torus. We found that these trajectories are robust as they exist on a positive Lebesgue measure set in the parameter space.

Jordan Curves of Positive Lebesgue Measure

1994 ◽  
pp. 131-143
Author(s):  
Hans Sagan
Keyword(s):  
Lebesgue Measure ◽  
Positive Lebesgue Measure ◽  
Jordan Curves

scholarly journals Torsion Discriminance for Stability of Linear Time-Invariant Systems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 386
Author(s):  
Yuxin Wang ◽  
Huafei Sun ◽  
Yueqi Cao ◽  
Shiqiang Zhang
Keyword(s):  
Lebesgue Measure ◽  
Linear Time ◽  
Zero Solution ◽  
Positive Lebesgue Measure ◽  
Asymptotically Stable ◽  
Linear Time Invariant ◽  
Measurable Set ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant

This paper extends the former approaches to describe the stability of n-dimensional linear time-invariant systems via the torsion τ ( t ) of the state trajectory. For a system r ˙ ( t ) = A r ( t ) where A is invertible, we show that (1) if there exists a measurable set E 1 with positive Lebesgue measure, such that r ( 0 ) ∈ E 1 implies that lim t → + ∞ τ ( t ) ≠ 0 or lim t → + ∞ τ ( t ) does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set E 2 with positive Lebesgue measure, such that r ( 0 ) ∈ E 2 implies that lim t → + ∞ τ ( t ) = + ∞ , then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the ith curvature ( i = 1 , 2 , ⋯ ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix.

A Sufficient Condition for a Bounded Set of Positive Lebesgue Measure in ℝ2 or ℝ3 to Contain its Centroid

2015 ◽  
Vol 40 (1) ◽  
pp. 179
Author(s):  
Eric A. Hintikka ◽  
Steven G. Krantz
Keyword(s):  
Lebesgue Measure ◽  
Positive Lebesgue Measure ◽  
Sufficient Condition ◽  
Bounded Set

On a Minimization Problem Involving the Critical Sobolev Exponent

2007 ◽  
Vol 7 (4) ◽  
Author(s):  
Francesca Prinari ◽  
Nicola Visciglia
Keyword(s):  
Lebesgue Measure ◽  
Minimization Problem ◽  
Lorentz Space ◽  
Critical Sobolev Exponent ◽  
Positive Lebesgue Measure ◽  
Sobolev Exponent ◽  
Interior Part

AbstractFollowing [3] we study the following minimization problem:in any dimension n ≥ 4 and under suitable assumptions on a(x). Mainly we assume that a(x) belongs to the Lorentz space LN ≡ {x ∈ Ω|a(x) < 0}has positive Lebesgue measure. Notice that this last condition is satisfied when the set N has a nontrivial interior part (in fact this is the typical assumption imposed in the literature on the set N).

Transitive maps which are not ergodic with respect to Lebesgue measure

1996 ◽  
Vol 16 (4) ◽  
pp. 833-848 ◽  
Author(s):  
Sebastian Van Strien
Keyword(s):  
Lebesgue Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Iteration Scheme ◽  
Rational Maps ◽  
Positive Lebesgue Measure ◽  
Rational Map ◽  
Interval Maps ◽  
Open Set ◽  
Totally Disconnected

AbstractIn this paper we shall give examples of rational maps on the Riemann sphere and also of polynomial interval maps which are transitive but not ergodic with respect to Lebesgue measure. In fact, these maps have two disjoint compact attractors whose attractive basins are ‘intermingled’, each having a positive Lebesgue measure in every open set. In addition, we show that there exists a real bimodal polynomial with Fibonacci dynamics (of the type considered by Branner and Hubbard), whose Julia set is totally disconnected and has positive Lebesgue measure. Finally, we show that there exists a rational map associated to the Newton iteration scheme corresponding to a polynomial whose Julia set has positive Lebesgue measure.

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