scholarly journals Bifurcation loci of exponential maps and quadratic polynomials: Local connectivity, triviality of fibers, and density of hyperbolicity

2008 ◽  
pp. 177-196 ◽  
Author(s):  
Lasse Rempe ◽  
Dierk Schleicher
Keyword(s):  
Local Connectivity ◽  
Quadratic Polynomials ◽  
Exponential Maps

scholarly journals On (non-)local connectivity of some Julia sets

2014 ◽  
Author(s):  
Alexandre Dezotti ◽  
Pascale Roesch
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Complete Characterization ◽  
Rational Maps ◽  
Local Connectivity ◽  
Rational Map ◽  
Critical Set ◽  
Quadratic Polynomials ◽  
Non Local

This chapter deals with the question of local connectivity of the Julia set of polynomials and rational maps. It discusses when the Julia set of a rational map is considered connected but not locally connected. The question of the local connectivity of the Julia set has been studied extensively for quadratic polynomials, but there is still no complete characterization of when a quadratic polynomial has a connected and locally connected Julia set. This chapter thus proposes some conjectures and develops a model of non-locally connected Julia sets in the case of infinitely renormalizable quadratic polynomials. This model presents the structure of what the post-critical set in that setting should be.

scholarly journals Distribution of polynomials with cycles of a given multiplier

2011 ◽  
Vol 201 ◽  
pp. 23-43 ◽  
Author(s):  
Giovanni Bassanelli ◽  
François Berteloot
Keyword(s):  
Quadratic Polynomials ◽  
Bifurcation Locus

AbstractIn the space of degreedpolynomials, the hypersurfaces defined by the existence of a cycle of periodnand multipliereiθare known to be contained in the bifurcation locus. We prove that these hypersurfaces equidistribute the bifurcation current. This is a new result, even for the space of quadratic polynomials.

scholarly journals On the dimension of points which escape to infinity at given rate under exponential iteration

2021 ◽  
pp. 1-33
Author(s):  
KRZYSZTOF BARAŃSKI ◽  
BOGUSŁAWA KARPIŃSKA
Keyword(s):  
Packing Dimension ◽  
Exponential Maps

Abstract We prove a number of results concerning the Hausdorff and packing dimension of sets of points which escape (at least in average) to infinity at a given rate under non-autonomous iteration of exponential maps. In particular, we generalize the results proved by Sixsmith in 2016 and answer his question on annular itineraries for exponential maps.

A Generalized Mandelbrot Set of Polynomials of Type 𝐸𝑑 for Bicomplex Numbers

2008 ◽  
Vol 15 (1) ◽  
pp. 189-194
Author(s):  
Ahmad Zireh
Keyword(s):  
Mandelbrot Set ◽  
Complex Numbers ◽  
Quadratic Polynomials ◽  
Bicomplex Numbers

Abstract We use a commutative generalization of complex numbers called bicomplex numbers to introduce the bicomplex dynamics of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑. Rochon [Fractals 8: 355–368, 2000] proved that the Mandelbrot set of quadratic polynomials in bicomplex numbers of the form 𝑤2 + 𝑐 is connected. We prove that our generalized Mandelbrot set of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑, is connected.

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