Continuous Nowhere Differentiable Functions Research Articles

Hausdorff dimension of local level sets of Takagi’s function

Takagi’s function is a continuous nowhere-differentiable function constructed by Takagi in 1903. In this paper, we study the local level sets of Takagi’s function. By using the tools of Moran sets and symbolic spaces, we establish the Hausdorff dimension of local level sets for \(x\in [0,1]\) and obtain the range of the Hausdorff dimension. In addition, we derive a lower bound for the Hausdorff dimension of the set of points \(x\) whose local level set has Hausdorff dimension \(s\).

Hausdorff–Besicovitch dimension of the graph of one continuous nowhere-differentiable function

We investigate fractal properties of the graph of the function $$y = f(x) = \sum\limits_{k - 1}^\infty \frac{{\beta _k }}{{2_k }} \equiv \Delta _{\beta _1 \beta _2 \ldots \beta _{k \ldots } }^2 ,$$ where $$\beta _1 = \left\{ {\begin{array}{lll} 0 & {{\text{if}}} & {{{\alpha }}_{\text{1}} \left( x \right) = 0,} \\ 1 & {{\text{if}}} & {{{\alpha }}_1 \left( x \right) \ne 0,} \\ \end{array} } \right.$$ $$\beta _k = \left\{ {\begin{array}{lllc} {\beta _{k - 1} } \hfill & {{\text{if}}} \hfill & {\alpha _k \left( x \right) = \alpha _{k - 1} \left( x \right),} \hfill & {} \hfill \\ {1 - \beta _{k - 1} } \hfill & {{\text{if}}} \hfill & {\alpha _k \left( x \right) \ne \alpha _{k - 1} \left( x \right),} \hfill & {k > 1,} \hfill \\ \end{array} } \right.$$ and ‎α k (x) is the kth ternary digit of x: In particular, we prove that this graph is a fractal set with Hausdorff–Besicovitch α 0(Г f )=log2(1+2log 3 2 ) dimension and cell dimension α K (Г f )=2-log32.

A "Superfat" Attractor with a Singular-Continuous 2-D Weierstrass Function in a Cross Section

Abstract A 3-D invertible map generating a 4-loop solenoid attractor is presented. The attractor of the "standard" 4-solenoid turns out to be "fat" - that is, fractal dimensionality exceeds topological dimensionality by more than unity. If a fourth variable that is weakly damped is added to this 3-variable map, a new type of attractor is generated. In a cross section it is a continuous nowhere-differentiable function over a 2-D domain which is a 2-D Cantor set. Such an object can be called a "singular-continuous 2-D Weierstrass function". The latter can be "superfat" (dimension gap larger than 2). Realistic systems possessing attractors of this new type are possible.

Continuous Nowhere-Differentiable Functions--an Application of Contraction Mappings

then f is also a continuous nowhere-differentiable function. (See [3, p. 115].) The above examples have concise definitions and establish the existence of continuous nowhere-differentiable functions. However, it is not easy to visualize or guess what their graphs look like, let alone to see intuitively why they work. Our continuous nowhere-differentiable function f: [0, 11 -] R is the uniform limit of a sequence of piecewise linear continuous functions fn: [0, 1] -+ R with steep slopes. In constructing the sequence (f,,>, we will be using a contraction mapping w from the family of all closed subsets of X = [0, 1] x [0, 1] into itself with respect to the Hausdorff metric induced by the Euclidean metric. (A contraction mapping on a metric space (Y, d) is a function g: Y -+ Y for which there is a positive constant k of sets converges to the graph of f in the Hausdorff metric. If A is the diagonal of slope 1 in the square X, then wn(A) is the graph of fn, and hence, a reader can obtain an intuitive idea of the graph of f. This idea first occurred to me when I attended Mr. Gary Church's master's thesis defense at San Jose State University at which he talked about attractors of contraction mappings (see [4]).

Continuous Nowhere-Differentiable Functions — an Application of Contraction Mappings

Continuous Nowhere-Differentiable Functions — an Application of Contraction Mappings