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 nonregular 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
selfsimilar measure $\mu =\sum p_{j}\mu \circ S_{j}^{1}$, of
finite type.
In this paper we study the multifractal analysis of such measures,
extending the theory to measures arising from nonregular 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:multifractal 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 twosided subGaussian 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 twosided 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, SzeMan

Spectral Asymptotics of Laplacians Associated with Onedimensional Iterated Function Systems with Overlaps
We set up a framework for computing the spectral dimension of a class of onedimensional
selfsimilar measures that are defined by iterated function systems
with overlaps and satisfy a family of secondorder selfsimilar
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 Cantortype measures.
The main novelty of our result is that the iterated function systems
we consider are not postcritically finite and do not satisfy the
wellknown open set condition.
Keywords:spectral dimension, fractal, Laplacian, selfsimilar measure, iterated function system with overlaps, secondorder selfsimilar 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 ondiagonal 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 JohnNirenberg 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
JohnNirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When
$\beta\neq 1$, $f$ is a selfaffine 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 $[\frac{1}{12^{\beta}}, \frac{1}{12^{\beta}}]$, and the
graph of $f$ has a noninteger fractal dimension $n+1\beta$.
Keywords:Haar wavelets, Q spaces, JohnNirenberg inequality, Greedy expansion, selfaffine, fractal, Box dimension Categories:42B35, 42C10, 30D50, 28A80 

6. CJM 2010 (vol 62 pp. 543)
 Hare, Kevin G.

More Variations on the SierpiÅski Sieve
This paper answers a question of Broomhead, Montaldi and Sidorov about the existence of gaskets of a particular type related to the SierpiÅski 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 blowups 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.
Category:28A80 

8. CJM 2009 (vol 61 pp. 1151)
 Ruan, HuoJun; 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
Sudokutype 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.
Category:28A80 

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 postcrit8cally finite selfsimilar
sets introduced by Kigami and of fractafolds introduced by Strichartz. In general,
we do not assume even local selfsimilarity, and allow countably many cells
connected at each junction point.
In particular, we consider postcritically 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, selfsimilarity, 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 selfsimilar 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.
Category:28A80 
