# Degeneration of the Julia set to singular loci of algebraic curves

@article{Saito2013DegenerationOT,
title={Degeneration of the Julia set to singular loci of algebraic curves},
author={Satoru Saito and Noriko Saitoh and Hiromitsu Harada and Tsukasa Yumibayashi and Yuki Wakimoto},
journal={arXiv: Exactly Solvable and Integrable Systems},
year={2013}
}
• S. Saito, +2 authors Y. Wakimoto
• Published 2013
• Mathematics, Physics
• arXiv: Exactly Solvable and Integrable Systems
We show that, when a non-integrable rational map changes to an integrable one continuously, a large part of the Julia set of the map approach indeterminate points (IDP) of the map along algebraic curves. We will see that the IDPs are singular loci of the curves.
1 Citations

#### Figures from this paper

Derivation of higher dimensional periodic recurrence equations by nested structure of complex numbers
We give a procedure for derivation of higher dimensional periodic recurrence equations by nested structure of complex numbers.

#### References

SHOWING 1-10 OF 10 REFERENCES
Invariant Varieties of Periodic Points for Some Higher Dimensional Integrable Maps
• Physics, Mathematics
• 2007
By studying various rational integrable maps on $$\hat{\mathbf{C}}^{d}$$ with p invariants, we show that periodic points form an invariant variety of dimension ≥ p for each period, in contrast to t...
Perturbative Changes of the Nature of Invariant Varieties for Some Higher Dimensional Integrable Maps
• Physics
• 2008
We have shown in our previous paper that the periodic points of some higher dimensional integrable maps form an invariant variety specific for each period when the maps have sufficient number ofExpand
Fate of the Julia set of higher dimensional maps in the integrable limit
• Mathematics
• 2010
By studying higher dimensional rational maps, we have shown, in our previous papers, that periodic points of integrable maps with sufficient number of invariants form invariant varieties of periodicExpand
Invariant Varieties of Periodic Points for the Discrete Euler Top
• Mathematics, Physics
• 2006
The behaviour of periodic points of discrete Euler top is studied. We derive invariant varieties of periodic points explicitly. When the top is axially symmetric they are specified by some particularExpand
On recurrence equations associated with invariant varieties of periodic points
• Mathematics, Physics
• 2007
A recurrence equation is a discrete integrable equation whose solutions are all periodic and the period is fixed. We show that infinitely many recurrence equations can be derived from the informationExpand
Singular points of complex hypersurfaces
The description for this book, Singular Points of Complex Hypersurfaces. (AM-61), will be forthcoming.
Proof of Poincaré’s geometric theorem
In a paper recently published in the Rendiconti del Circolo Matemático di Palermo (vol. 33, 1912, pp. 375-407) and entitled Sur un théorème de Géométrie, Poincaré enunciated a theorem of greatExpand
An Introduction To Chaotic Dynamical Systems
Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic FamilyExpand
An extension of Poincaré's last geometric theorem
The Crowned Memoir b y POINCAR~, ~>Le probl~me de trois corps et les 6quations de la dynamique>>, in volume 13 of the Acta mathematica contained the first great attack upon the non-integrableExpand
Transition to Chaos (Springer-Verlag
• 1992