The bifurcation locus for numbers of bounded type (original) (raw)

Self-Similar Fractals in Arithmetic

The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as ‘similarity’ maps. Fractals are subsets of algebraic varieties which can be written as a finite and (almost) disjoint union of ‘similar’ copies. Fractals provide a framework in which one can unite some results and conjectures in Diophantine geometry. We define a well-behaved notion of dimension for fractals. We also prove a fractal version of Roth’s theorem for algebraic points on a variety approximated by elements of a fractal subset. As a consequence, we get a fractal version of Siegel’s and Faltings’ theorems on finiteness of integral points on hyperbolic curves and affine subsets of abelian varieties, respectively.

Bifurcating Continued Fractions II

2000

The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These sequences enable simple representations of roots of cubic equations. In particular, remarkably simple and elegant 'bifurcating continued fraction' representations of Tribonacci and Moore numbers, the cubic variations of the 'golden mean', are obtained. This is further generalized to associate m non-negative integer sequences with a set of m given real numbers so as to provide simple 'bifurcating continued fraction' representation of roots of polynomial equations of degree m+1.

Cantor Set as a Fractal and Its Application in Detecting Chaotic Nature of Piecewise Linear Maps

Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. , 2019

We have investigated the Cantor set from the perspective of fractals and box-counting dimension. Cantor sets can be constructed geometrically by continuous removal of a portion of the closed unit interval [0, 1] infinitely. The set of points remained in the unit interval after this removal process is over is called the Cantor set. The dimension of such a set is not an integer value. In fact, it has a 'fractional' dimension, making it by definition a fractal. The Cantor set is an example of an uncountable set with measure zero and has potential applications in various branches of mathematics such as topology, measure theory, dynamical systems and fractal geometry. In this paper, we have provided three types of generalization of the Cantor set depending on the process of removal. Also, we have discussed some characteristics of the fractal dimensions of these generalized Cantor sets. Further, we have shown its application in detecting chaotic nature of the dynamics produced by iteration of piecewise linear maps.

Dynamics of continued fractions and kneading sequences of unimodal maps, Discrete Contin

2016

In this paper we construct a correspondence between the parameter spaces of two families of one-dimensional dynamical systems, the α-continued fraction transformations Tα and unimodal maps. This correspondence identifies bifurcation parameters in the two families, and allows one to transfer topological and metric properties from one setting to the other. As an application, we recover results about the real slice of the Mandelbrot set, and the set of univoque numbers. 1