scholarly journals The M-set of λ exp(z)/z has infinite area

2015 ◽  
Vol 217 ◽  
pp. 133-159 ◽  
Author(s):  
Guoping Zhan ◽  
Liangwen Liao
Keyword(s):  
Hausdorff Dimension ◽  
Fatou Set

AbstractIt is known that the Fatou set of the map exp(z)/z defined on the punctured plane ℂ* is empty. We consider the M-set of λ exp(z)/z consisting of all parameters λ for which the Fatou set of λexp(z)/z is empty. We prove that the M-set of λexp(z)/z has infinite area. In particular, the Hausdorff dimension of the M-set is 2. We also discuss the area of complement of the M-set.

scholarly journals The M-set of λ exp(z)/z has infinite area

2015 ◽  
Vol 217 ◽  
pp. 133-159
Author(s):  
Guoping Zhan ◽  
Liangwen Liao
Keyword(s):  
Hausdorff Dimension ◽  
Fatou Set

AbstractIt is known that the Fatou set of the map exp(z)/zdefined on the punctured plane ℂ*is empty. We consider theM-set of λ exp(z)/zconsisting of all parameters λ for which the Fatou set of λexp(z)/zis empty. We prove that theM-set of λexp(z)/zhas infinite area. In particular, the Hausdorff dimension of theM-set is 2. We also discuss the area of complement of theM-set.

scholarly journals Minimal F2-flows

1985 ◽  
Vol 39 (2) ◽  
pp. 187-193
Author(s):  
Daniel Berend
Keyword(s):  
Hausdorff Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Affine Transformations ◽  
Dimensional Torus ◽  
Minimal Sets ◽  
Necessary And Sufficient

AbstractLet σ be an ergodic endomorphism of the r–dimensional torus and Π a semigroup generated by two affine transformations lying above σ. We show that the flow defined by Π admits minimal sets of positive Hausdorff dimension and we give necessary and sufficient conditions for this flow to be minimal.

Length distortion and the Hausdorff dimension of limit sets

2000 ◽  
Vol 122 (3) ◽  
pp. 465-482 ◽  
Author(s):  
Martin Bridgeman ◽  
Edward C. Taylor
Keyword(s):  
Hausdorff Dimension ◽  
Limit Sets ◽  
Length Distortion

scholarly journals A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1546
Author(s):  
Mohsen Soltanifar
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Virtual World ◽  
Fractal Dimensions ◽  
Definition Of ◽  
The Given

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

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