Fibonacci fixed point of renormalization

2000 ◽  
Vol 20 (5) ◽  
pp. 1287-1317 ◽  
Author(s):  
XAVIER BUFF
Keyword(s):  
Fixed Point ◽  
Lebesgue Measure ◽  
Julia Set ◽  
Positive Lebesgue Measure ◽  
Quadratic Polynomials ◽  
Polynomial Maps ◽  
Mathematical Sciences

To study the geometry of a Fibonacci map $f$ of even degree $\ell\geq 4$, Lyubich (Dynamics of quadratic polynomials, I–II. Acta Mathematica178 (1997), 185–297) defined a notion of generalized renormalization, so that $f$ is renormalizable infinitely many times. van Strien and Nowicki (Polynomial maps with a Julia set of positive Lebesgue measure: Fibonacci maps. Preprint, Institute for Mathematical Sciences, SUNY at Stony Brook, 1994) proved that the generalized renormalizations ${\cal R}^{\circ n}(f)$ converge to a cycle $\{f_1,f_2\}$ of order two depending only on $\ell$. We will explicitly relate $f_1$ and $f_2$ and show the convergence in shape of Fibonacci puzzle pieces to the Julia set of an appropriate polynomial-like map.

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.

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.

LIMITING DYNAMICS FOR THE COMPLEX STANDARD FAMILY

1995 ◽  
Vol 05 (03) ◽  
pp. 673-699 ◽  
Author(s):  
NÚRIA FAGELLA
Keyword(s):  
Cross Sections ◽  
Julia Set ◽  
Three Dimensional ◽  
Mandelbrot Set ◽  
Dimensional Parameter ◽  
Quadratic Polynomials ◽  
New Family ◽  
Standard Family ◽  
The Family ◽  
Characteristic Features

The complexification of the standard family of circle maps Fαβ(θ)=θ+α+β+β sin(θ) mod (2π) is given by Fαβ(ω)=ωeiαe(β/2)(ω−1/ω) and its lift fαβ(z)=z+a+β sin(z). We investigate the three-dimensional parameter space for Fαβ that results from considering a complex and β real. In particular, we study the two-dimensional cross-sections β=constant as β tends to zero. As the functions tend to the rigid rotation Fα,0, their dynamics tend to the dynamics of the family Gλ(z)=λzez where λ=e−iα. This new family exhibits behavior typical of the exponential family together with characteristic features of quadratic polynomials. For example, we show that the λ-plane contains infinitely many curves for which the Julia set of the corresponding maps is the whole plane. We also prove the existence of infinitely many sets of λ values homeomorphic to the Mandelbrot set.

CRYPTANALYSIS OF AN IMPLEMENTATION SCHEME OF THE TAMED TRANSFORMATION METHOD CRYPTOSYSTEM

2004 ◽  
Vol 03 (03) ◽  
pp. 273-282 ◽  
Author(s):  
JINTAI DING ◽  
TIMOTHY HODGES
Keyword(s):  
Linear Equations ◽  
Transformation Method ◽  
Total Degree ◽  
Quadratic Polynomials ◽  
Polynomial Maps ◽  
Linearly Independent ◽  
Linear Maps ◽  
Implementation Scheme ◽  
Multivariable Polynomial ◽  
Composition Factors

A Tamed Transformation Method (TTM) cryptosystem was proposed by T. T. Moh in 1999. We describe how the first implementation scheme of the TTM system can be defeated. The computational complexity of our attack is 233 computations on the finite field with 28 elements. The cipher of the TTM systems are degree 2 polynomial maps derived from composition of invertible maps of either total degree 2 or linear maps which can be easily calculated and can be easily inverted. To ensure the system to be of degree two, the key construction of the implementation schemes of the TTM systems is a multivariable polynomial Q8(x1,…,xn) and a set of linearly independent quadratic polynomials qi(x1,…,xm), i=1,…,n such that Q8(q1,…,qn) is again a degree 2 polynomials of x1,…,xm. In this paper, we study the first implementation scheme of the TTM systems [6]. We discovered that in this implementation scheme the specific polynomial Q8 can be decomposed further into a factorization in terms of composition. By taking powers of the equality satisfied by the new composition factors, we can actually derive a set of equations, that can produce linear equations satisfied by the plaintext. These linear equations lead us to find a way to defeat this implementation scheme.

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.

scholarly journals On the Lebesgue measure of the Feigenbaum Julia set

2020 ◽  
Vol 221 (1) ◽  
pp. 167-202 ◽  
Author(s):  
Artem Dudko ◽  
Scott Sutherland
Keyword(s):  
Lebesgue Measure ◽  
Julia Set

Parabolic-like mappings

2014 ◽  
Vol 35 (7) ◽  
pp. 2171-2197 ◽  
Author(s):  
LUNA LOMONACO
Keyword(s):  
Fixed Point ◽  
Julia Set ◽  
The Family ◽  
Dynamical Properties ◽  
Image Position ◽  
Parabolic Fixed Point

In this paper we introduce the notion of parabolic-like mapping. Such an object is similar to a polynomial-like mapping, but it has a parabolic external class, i.e. an external map with a parabolic fixed point. We define the notion of parabolic-like mapping and we study the dynamical properties of parabolic-like mappings. We prove a straightening theorem for parabolic-like mappings which states that any parabolic-like mapping of degree two is hybrid conjugate to a member of the family $$\begin{eqnarray}\mathit{Per}_{1}(1)=\left\{[P_{A}]\,\bigg|\,P_{A}(z)=z+\frac{1}{z}+A,~A\in \mathbb{C}\right\}\!,\end{eqnarray}$$ a unique such member if the filled Julia set is connected.

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