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Lamination (mathematics)
In topology, a branch of mathematics, a lamination is a : * "topological space partitioned into subsets" * decoration (a structure or property at a point) of a manifold in which some subset of the manifold is partitioned into sheets of some lower dimension, and the sheets are locally parallel. A lamination of a surface is a partition of a closed subset of the surface into smooth curves. It may or may not be possible to fill the gaps in a lamination to make a foliation. Oak Ridge National Laboratory Examples *A geodesic lamination of a 2-dimensional hyperbolic manifold is a closed subset together with a foliation of this closed subset by geodesics. These are used in Thurston's classification of elements of the mapping class group and in his theory of earthquake maps. *Quadratic laminations, which remain invariant under the angle doubling map. These laminations are associated with quadratic maps. It is a closed collection of chords in the unit disc. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbit Portrait
In mathematics, an orbit portrait is a combinatorial tool used in complex dynamics for understanding the behavior of one-complex dimensional quadratic maps. In simple words one can say that it is : * a list of external angles for which rays land on points of that orbit * graph showing above list Definition Given a quadratic map :f_c : z \mapsto z^2 + c. from the complex plane to itself :f_c : \mathbb \to \mathbb and a repelling or parabolic periodic orbit = \ of f, so that f(z_j) = z_ (where subscripts are taken 1 + modulo n), let A_j be the set of angles whose corresponding external rays land at z_j. Then the set = () = \ is called the orbit portrait of the periodic orbit . All of the sets A_j must have the same number of elements, which is called the valence of the portrait. Examples Parabolic or repelling orbit portrait valence 2 = \left \{\left( \frac{1}{3},\frac{2}{3} \right) \right \rbrace {\mathcal P} = \left \{ \left( \frac{3}{7} , \frac{4}{7} \ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Train Track (mathematics)
In the mathematical area of topology, a train track is a family of curves embedded on a surface, meeting the following conditions: #The curves meet at a finite set of vertices called ''switches''. #Away from the switches, the curves are smooth and do not touch each other. #At each switch, three curves meet with the same tangent line, with two curves entering from one direction and one from the other. The main application of train tracks in mathematics is to study laminations of surfaces, that is, partitions of closed subsets of surfaces into unions of smooth curves. Train tracks have also been used in graph drawing. Train tracks and laminations A lamination of a surface is a partition of a closed subset of the surface into smooth curves. The study of train tracks was originally motivated by the following observation: If a generic lamination on a surface is looked at from a distance by a myopic person, it will look like a train track. A switch in a train track models a point w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julia Set
In complex dynamics, the Julia set and the Classification of Fatou components, Fatou set are two complement set, complementary sets (Julia "laces" and Fatou "dusts") defined from a function (mathematics), function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under iterated function, repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small Perturbation theory, perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "chaos theory, chaotic". The Julia set of a function is commonly denoted \operatorname(f), and the Fatou set is denoted \operatorname(f). These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century. Form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mandelbrot Set
The Mandelbrot set () is a two-dimensional set (mathematics), set that is defined in the complex plane as the complex numbers c for which the function f_c(z)=z^2+c does not Stability theory, diverge to infinity when Iteration, iterated starting at z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York. Images of the Mandelbrot set exhibit an infinitely complicated Boundary (topology), boundary that reveals progressively ever-finer Recursion, recursive detail at increasing magnifications; mathematically, the boundary of the Mandelbrot set is a ''fractal curve''. The "style" of this recursive detail depends on the region of the set boundary being ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Quadratic Polynomial
A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable (mathematics), variable are complex numbers. Properties Quadratic polynomials have the following properties, regardless of the form: *It is a unicritical polynomial, i.e. it has one #Critical_points, finite critical point in the complex plane, Dynamical plane consist of maximally 2 basins: the basin of infinity and basin of the finite critical point (if the finite critical point does not escape) *It can be postcritically finite, i.e. the orbit of the critical point can be finite, because the critical point is periodic or preperiodic. * It is a Unimodality#Unimodal function, unimodal function, * It is a rational function, * It is an entire function. Forms When the quadratic polynomial has only one variable (univariate), one can distinguish its four main forms: * The general form: f(x) = a_2 x^2 + a_1 x + a_0 where a_2 \ne 0 * The factored form used for the logistic map: f_r(x) = r x (1-x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubling Map
The dyadic transformation (also known as the dyadic map, bit shift map, 2''x'' mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) : T: , 1) \to [0, 1)^\infty : x \mapsto (x_0, x_1, x_2, \ldots) (where [0, 1)^\infty is the set of sequences from [0, 1)) produced by the rule : x_0 = x : \text n \ge 0,\ x_ = (2 x_n) \bmod 1. Equivalently, the dyadic transformation can also be defined as the iterated function map of the piecewise linear function : T(x)=\begin2x & 0 \le x < \frac \\2x-1 & \frac \le x < 1. \end The name ''bit shift map'' arises because, if the value of an iterate is written in notation, the next iterate is obtained by shifting the binary point one bit to the right, and if the bit to the left of the new binary point is a "one", replacing it with a zero. The dyadic transfo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
