Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Hausdorff dimension of self-affine Sierpinski sponges
184
0
0.0
(
0
)
Authors
Nuno Luzia
Ask ChatGPT about the research
No Arabic abstract
We compute the Hausdorff dimension of limit sets generated by 3-dimensional self-affine mappings with diagonal matrices of the form A_{ijk}=Diag(a_{ijk}, b_{ij}, c_{i}), where 0<a_{ijk}le b_{ij}le c_i<1. By doing so we show that the variational principle for the dimension holds for this class.
rate research
Read More
In this paper we compute the dimension of a class of dynamically defined non-conformal sets. Let $Xsubseteqmathbb{T}^2$ denote a Bedford-McMullen set and $T:Xto X$ the natural expanding toral endomorphism which leaves $X$ invariant. For an open set $Usubset X$ we let X_U={xin X : T^k(x) otin U text{for all}k}. We investigate the box and Hausdorff dimensions of $X_U$ for both a fixed Markov hole and also when $U$ is a shrinking metric ball. We show that the box dimension is controlled by the escape rate of the measure of maximal entropy through $U$, while the Hausdorff dimension depends on the escape rate of the measure of maximal dimension.
In this paper, we study the Hausdorff and the box dimensions of closed invariant subsets of the space of pointed trees, equipped with a pseudogroup action. This pseudogroup dynamical system can be regarded as a generalization of a shift space. We show that the Hausdorff dimension of the space of pointed trees is infinite, and the union of closed invariant subsets with dense orbit and non-equal Hausdorff and box dimensions is dense in the space of pointed trees. We apply our results to the problem of embedding laminations into differentiable foliations of smooth manifolds. To admit such an embedding, a lamination must satisfy at least the following two conditions: first, it must admit a metric and a foliated atlas, such that the generators of the holonomy pseudogroup, associated to the atlas, are bi-Lipschitz maps relative to the metric. Second, it must admit an embedding into a manifold, which is a bi-Lipschitz map. A suspension of the pseudogroup action on the space of pointed graphs gives an example of a lamination where the first condition is satisfied, and the second one is not satisfied, with Hausdorff dimension of the space of pointed trees being the obstruction to the existence of a bi-Lipschitz embedding.
We show that the graph of the classical Weierstrass function $sum_{n=0}^infty lambda^n cos (2pi b^n x)$ has Hausdorff dimension $2+loglambda/log b$, for every integer $bge 2$ and every $lambdain (1/b,1)$. Replacing $cos(2pi x)$ by a general non-constant $C^2$ periodic function, we obtain the same result under a further assumption that $lambda b$ is close to $1$.
A function which is transcendental and meromorphic in the plane has at least two singular values. On one hand, if a meromorphic function has exactly two singular values, it is known that the Hausdorff dimension of the escaping set can only be either $2$ or $1/2$. On the other hand, the Hausdorff dimension of escaping sets of Speiser functions can attain every number in $[0,2]$ (cf. cite{ac1}). In this paper, we show that number of singular values which is needed to attain every Hausdorff dimension of escaping sets is not more than $4$.
We prove that the Hausdorff dimension of the set of three-period orbits in classical billiards is at most one. Moreover, if the set of three-period orbits has Hausdorff dimension one, then it has a tangent line at almost every point.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
