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Search: MSC category 28A80 ( Fractals [See also 37Fxx] )

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1. CJM Online first

Hare, Kathryn; Hare, Kevin; Ng, Michael Ka Shing
Local dimensions of measures of finite type II -- Measures without full support and with non-regular probabilities
Consider a finite sequence of linear contractions $S_{j}(x)=\varrho x+d_{j}$ and probabilities $p_{j}\gt 0$ with $\sum p_{j}=1$. We are interested in the self-similar measure $\mu =\sum p_{j}\mu \circ S_{j}^{-1}$, of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval. Under some mild technical assumptions, we prove that there exists a subset of supp$\mu $ of full $\mu $ and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support and we show that the dimension of the support can be computed using only information about the essential class. To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type, and show that the $kth$ convolution of the associated Cantor measure has local dimension at $x\in (0,1)$ tending to 1 as $k$ tends to infinity. Lastly, we show that within a maximal loop class that is not truly essential, the set of upper local dimensions need not be an interval. This is in contrast to the case for finite type measures with regular probabilities and full interval support.

Keywords:multi-fractal analysis, local dimension, IFS, finite type
Categories:28A80, 28A78, 11R06

2. CJM 2013 (vol 66 pp. 641)

Grigor'yan, Alexander; Hu, Jiaxin
Heat Kernels and Green Functions on Metric Measure Spaces
We prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling propety, the elliptic Harnack inequality and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball, that uses two-sided estimates of a Green function in a ball.

Keywords:Dirichlet form, heat kernel, Green function, capacity
Categories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07

3. CJM 2011 (vol 63 pp. 648)

Ngai, Sze-Man
Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps
We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

Keywords:spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identities
Categories:28A80, , , , 35P20, 35J05, 43A05, 47A75

4. CJM 2010 (vol 63 pp. 153)

Hambly, B. M.
Asymptotics for Functions Associated with Heat Flow on the Sierpinski Carpet
We establish the asymptotic behaviour of the partition function, the heat content, the integrated eigenvalue counting function, and, for certain points, the on-diagonal heat kernel of generalized Sierpinski carpets. For all these functions the leading term is of the form $x^{\gamma}\phi(\log x)$ for a suitable exponent $\gamma$ and $\phi$ a periodic function. We also discuss similar results for the heat content of affine nested fractals.

Categories:35K05, 28A80, 35B40, 60J65

5. CJM 2010 (vol 62 pp. 1182)

Yue, Hong
A Fractal Function Related to the John-Nirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$
A borderline case function $f$ for $ Q_{\alpha}({\mathbb R^n})$ spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter $\beta>0$. On its support $I_0=[0, 1]^n$, $f(x)$ can be expressed by the binary expansions of the coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb R^n})$ if and only if $\alpha<\beta<\frac{n}{2}$, while for $\beta=\alpha$, it was shown by Yue and Dafni that $f$ satisfies a John--Nirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When $\beta\neq 1$, $f$ is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside $I_0$. In addition, it is not monotone along any coordinate direction in any small cube. When the parameter $\beta\in (0, 1)$, $f$ is onto from $I_0$ to $[-\frac{1}{1-2^{-\beta}}, \frac{1}{1-2^{-\beta}}]$, and the graph of $f$ has a non-integer fractal dimension $n+1-\beta$.

Keywords:Haar wavelets, Q spaces, John-Nirenberg inequality, Greedy expansion, self-affine, fractal, Box dimension
Categories:42B35, 42C10, 30D50, 28A80

6. CJM 2010 (vol 62 pp. 543)

Hare, Kevin G.
More Variations on the Sierpiński Sieve
This paper answers a question of Broomhead, Montaldi and Sidorov about the existence of gaskets of a particular type related to the Sierpiński sieve. These gaskets are given by iterated function systems that do not satisfy the open set condition. We use the methods of Ngai and Wang to compute the dimension of these gaskets.

Categories:28A80, 28A78, 11R06

7. CJM 2009 (vol 61 pp. 1182)

Strichartz, Robert S.
Periodic and Almost Periodic Functions on Infinite Sierpinski Gaskets
We define periodic functions on infinite blow-ups of the Sierpinski gasket as lifts of functions defined on certain compact fractafolds via covering maps. This is analogous to defining periodic functions on the line as lifts of functions on the circle via covering maps. In our setting there is only a countable set of covering maps. We give two different characterizations of periodic functions in terms of repeating patterns. However, there is no discrete group action that can be used to characterize periodic functions. We also give a Fourier series type description in terms of periodic eigenfunctions of the Laplacian. We define almost periodic functions as uniform limits of periodic functions.


8. CJM 2009 (vol 61 pp. 1151)

Ruan, Huo-Jun; Strichartz, Robert S.
Covering Maps and Periodic Functions on Higher Dimensional Sierpinski Gaskets
We construct covering maps from infinite blowups of the $n$-dimensional Sierpinski gasket $SG_n$ to certain compact fractafolds based on $SG_n$. These maps are fractal analogs of the usual covering maps from the line to the circle. The construction extends work of the second author in the case $n=2$, but a different method of proof is needed, which amounts to solving a Sudoku-type puzzle. We can use the covering maps to define the notion of periodic function on the blowups. We give a characterization of these periodic functions and describe the analog of Fourier series expansions. We study covering maps onto quotient fractalfolds. Finally, we show that such covering maps fail to exist for many other highly symmetric fractals.


9. CJM 2008 (vol 60 pp. 457)

Teplyaev, Alexander
Harmonic Coordinates on Fractals with Finitely Ramified Cell Structure
We define sets with finitely ramified cell structure, which are generalizations of post-crit8cally finite self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically infinite fractals. We prove that if Kigami's resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates.

Keywords:fractals, self-similarity, energy, resistance, Dirichlet forms, diffusions, quantum graphs, generalized Riemannian metric
Categories:28A80, 31C25, 53B99, 58J65, 60J60, 60G18

10. CJM 1999 (vol 51 pp. 1073)

Nielsen, Ole A.
The Hausdorff and Packing Dimensions of Some Sets Related to Sierpi\'nski Carpets
The Sierpi\'nski carpets first considered by C.~McMullen and later studied by Y.~Peres are modified by insisting that the allowed digits in the expansions occur with prescribed frequencies. This paper (i)~~calculates the Hausdorff, box (or Minkowski), and packing dimensions of the modified Sierpi\'nski carpets and (ii)~~shows that for these sets the Hausdorff and packing measures in their dimension are never zero and gives necessary and sufficient conditions for these measures to be infinite.

Categories:28A78, 28A80

11. CJM 1998 (vol 50 pp. 638)

Strichartz, Robert S.
Fractals in the large
A {\it reverse iterated function system} (r.i.f.s.) is defined to be a set of expansive maps $\{T_1,\ldots,T_m\}$ on a discrete metric space $M$. An invariant set $F$ is defined to be a set satisfying $F = \bigcup^m_{j=1} T_jF$, and an invariant measure $\mu$ is defined to be a solution of $\mu = \sum^m_{j=1} p_j\mu\circ T_j^{-1}$ for positive weights $p_j$. The structure and basic properties of such invariant sets and measures is described, and some examples are given. A {\it blowup} $\cal F$ of a self-similar set $F$ in $\Bbb R^n$ is defined to be the union of an increasing sequence of sets, each similar to $F$. We give a general construction of blowups, and show that under certain hypotheses a blowup is the sum set of $F$ with an invariant set for a r.i.f.s. Some examples of blowups of familiar fractals are described. If $\mu$ is an invariant measure on $\Bbb Z^+$ for a linear r.i.f.s., we describe the behavior of its {\it analytic} transform, the power series $\sum^\infty_{n=0} \mu(n)z^n$ on the unit disc.


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