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1. CJM 2017 (vol 70 pp. 824)

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 Online first

Handelman, David
Nearly approximate transitivity (AT) for circulant matrices
By previous work of Giordano and the author, ergodic actions of $\mathbf Z$ (and other discrete groups) are completely classified measure-theoretically by their dimension space, a construction analogous to the dimension group used in C*-algebras and topological dynamics. Here we investigate how far from AT (approximately transitive) can actions be which derive from circulant (and related) matrices. It turns out not very: although non-AT actions can arise from this method of construction, under very modest additional conditions, ATness arises; in addition, if we drop the positivity requirement in the isomorphism of dimension spaces, then all these ergodic actions satisfy an analogue of AT. Many examples are provided.

Keywords:approximately transitive, ergodic transformation, circulant matrix, hemicirculant matrix, dimension space, matrix-valued random walk
Categories:37A05, 06F25, 28D05, 46B40, 60G50

3. CJM 2014 (vol 67 pp. 795)

Di Nasso, Mauro; Goldbring, Isaac; Jin, Renling; Leth, Steven; Lupini, Martino; Mahlburg, Karl
On a Sumset Conjecture of Erdős
Erdős conjectured that for any set $A\subseteq \mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,C\subseteq \mathbb{N}$ such that $B+C\subseteq A$. We verify Erdős' conjecture in the case that $A$ has Banach density exceeding $\frac{1}{2}$. As a consequence, we prove that, for $A\subseteq \mathbb{N}$ with positive Banach density (a much weaker assumption than positive lower density), we can find infinite $B,C\subseteq \mathbb{N}$ such that $B+C$ is contained in the union of $A$ and a translate of $A$. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős' conjecture for subsets of the natural numbers that are pseudorandom.

Keywords:sumsets of integers, asymptotic density, amenable groups, nonstandard analysis
Categories:11B05, 11B13, 11P70, 28D15, 37A45

4. CJM 2013 (vol 66 pp. 721)

Durand-Cartagena, E.; Ihnatsyeva, L.; Korte, R.; Szumańska, M.
On Whitney-type Characterization of Approximate Differentiability on Metric Measure Spaces
We study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately differentiable functions in this setting. As an application, we prove a Stepanov-type theorem and consider approximate differentiability of Sobolev, $BV$ and maximal functions.

Keywords:approximate differentiability, metric space, strong measurable differentiable structure, Whitney theorem
Categories:26B05, 28A15, 28A75, 46E35

5. CJM 2013 (vol 66 pp. 303)

Elekes, Márton; Steprāns, Juris
Haar Null Sets and the Consistent Reflection of Non-meagreness
A subset $X$ of a Polish group $G$ is called Haar null if there exists a Borel set $B \supset X$ and Borel probability measure $\mu$ on $G$ such that $\mu(gBh)=0$ for every $g,h \in G$. We prove that there exist a set $X \subset \mathbb R$ that is not Lebesgue null and a Borel probability measure $\mu$ such that $\mu(X + t) = 0$ for every $t \in \mathbb R$. This answers a question from David Fremlin's problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.) This result motivates the following Baire category analogue. It is consistent with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor set $C \subset G$ such that for every non-meagre set $X \subset G$ there exists a $t \in G$ such that $C \cap (X + t)$ is relatively non-meagre in $C$. This essentially generalises results of Bartoszyński and Burke-Miller.

Keywords:Haar null, Christensen, non-locally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic real
Categories:28C10, 03E35, 03E17, , , , , 22C05, 28A78

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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.


12. 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.


13. CJM 2009 (vol 61 pp. 656)

McCutcheon, Randall; Quas, Anthony
Generalized Polynomials and Mild Mixing
An unsettled conjecture of V. Bergelson and I. H\aa land proposes that if $(X,\alg,\mu,T)$ is an invertible weak mixing measure preserving system, where $\mu(X)<\infty$, and if $p_1,p_2,\dots ,p_k$ are generalized polynomials (functions built out of regular polynomials via iterated use of the greatest integer or floor function) having the property that no $p_i$, nor any $p_i-p_j$, $i\neq j$, is constant on a set of positive density, then for any measurable sets $A_0,A_1,\dots ,A_k$, there exists a zero-density set $E\subset \z$ such that \[\lim_{\substack{n\to\infty\\ n\not\in E}} \,\mu(A_0\cap T^{p_1(n)}A_1\cap \cdots \cap T^{p_k(n)}A_k)=\prod_{i=0}^k \mu(A_i).\] We formulate and prove a faithful version of this conjecture for mildly mixing systems and partially characterize, in the degree two case, the set of families $\{ p_1,p_2, \dots ,p_k\}$ satisfying the hypotheses of this theorem.

Categories:37A25, 28D05

14. CJM 2009 (vol 61 pp. 124)

Dijkstra, Jan J.; Mill, Jan van
Characterizing Complete Erd\H os Space
The space now known as {\em complete Erd\H os space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the closed subspace of the Hilbert space $\ell^2$ consisting of all vectors such that every coordinate is in the convergent sequence $\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of $\cerdos$. As an application we determine the class of factors of $\cerdos$. In another application we determine precisely which of the spaces that can be constructed in the Banach spaces $\ell^p$ according to the `Erd\H os method' are homeomorphic to $\cerdos$. A novel application states that if $I$ is a Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$, $\Z\times2^\omega$, or $\cerdos$. This last result answers a question that was asked by Stevo Todor{\v{c}}evi{\'c}.

Keywords:Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on $\omega$, $\R$-trees, line-free groups in Banach spaces
Categories:28C10, 46B20, 54F65

15. CJM 2008 (vol 60 pp. 1149)

Petersen, Kathleen L.; Sinclair, Christopher D.
Conjugate Reciprocal Polynomials with All Roots on the Unit Circle
We study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree $N$ is naturally associated to a subset of $\R^{N-1}$. We calculate the volume of this set, prove the set is homeomorphic to the $N-1$ ball and that its isometry group is isomorphic to the dihedral group of order $2N$.

Categories:11C08, 28A75, 15A52, 54H10, 58D19

16. 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

17. CJM 2004 (vol 56 pp. 115)

Kenny, Robert
Estimates of Hausdorff Dimension for the Non-Wandering Set of an Open Planar Billiard
The billiard flow in the plane has a simple geometric definition; the movement along straight lines of points except where elastic reflections are made with the boundary of the billiard domain. We consider a class of open billiards, where the billiard domain is unbounded, and the boundary is that of a finite number of strictly convex obstacles. We estimate the Hausdorff dimension of the nonwandering set $M_0$ of the discrete time billiard ball map, which is known to be a Cantor set and the largest invariant set. Under certain conditions on the obstacles, we use a well-known coding of $M_0$ \cite{Morita} and estimates using convex fronts related to the derivative of the billiard ball map \cite{StAsy} to estimate the Hausdorff dimension of local unstable sets. Consideration of the local product structure then yields the desired estimates, which provide asymptotic bounds on the Hausdorff dimension's convergence to zero as the obstacles are separated.

Categories:37D50, 37C45;, 28A78

18. CJM 2002 (vol 54 pp. 1280)

Skrzypczak, Leszek
Besov Spaces and Hausdorff Dimension For Some Carnot-Carathéodory Metric Spaces
We regard a system of left invariant vector fields $\mathcal{X}=\{X_1,\dots,X_k\}$ satisfying the H\"ormander condition and the related Carnot-Carath\'eodory metric on a unimodular Lie group $G$. We define Besov spaces corresponding to the sub-Laplacian $\Delta=\sum X_i^2$ both with positive and negative smoothness. The atomic decomposition of the spaces is given. In consequence we get the distributional characterization of the Hausdorff dimension of Borel subsets with the Haar measure zero.

Keywords:Besov spaces, sub-elliptic operators, Carnot-Carathéodory metric, Hausdorff dimension
Categories:46E35, 43A15, 28A78

19. CJM 2000 (vol 52 pp. 332)

del Junco, Andrés; Yassawi, Reem
Multiple Mixing and Rank One Group Actions
Suppose $G$ is a countable, Abelian group with an element of infinite order and let ${\cal X}$ be a mixing rank one action of $G$ on a probability space. Suppose further that the F\o lner sequence $\{F_n\}$ indexing the towers of ${\cal X}$ satisfies a ``bounded intersection property'': there is a constant $p$ such that each $\{F_n\}$ can intersect no more than $p$ disjoint translates of $\{F_n\}$. Then ${\cal X}$ is mixing of all orders. When $G={\bf Z}$, this extends the results of Kalikow and Ryzhikov to a large class of ``funny'' rank one transformations. We follow Ryzhikov's joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of $k$ copies of ${\cal X}$ is necessarily product measure. This method generalizes Ryzhikov's technique.


20. 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

21. CJM 1998 (vol 50 pp. 1163)

Chen, Jingyi; Hsu, Elton P.
Gradient estimates for harmonic Functions on manifolds with Lipschitz metrics
We introduce a distributional Ricci curvature on complete smooth manifolds with Lipschitz continuous metrics. Under an assumption on the volume growth of geodesics balls, we obtain a gradient estimate for weakly harmonic functions if the distributional Ricci curvature is bounded below.

Categories:60D58, 28D05

22. 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.


23. CJM 1997 (vol 49 pp. 1089)

Burke, Maxim R.; Ciesielski, Krzysztof
Sets on which measurable functions are determined by their range
We study sets on which measurable real-valued functions on a measurable space with negligibles are determined by their range.

Keywords:measurable function, measurable space with negligibles, continuous image, set of range uniqueness (SRU)
Categories:28A20, 28A05, 54C05, 26A30, 03E35, 03E50

24. CJM 1997 (vol 49 pp. 3)

Akcoglu, Mustafa A.; Ha, Dzung M.; Jones, Roger L.
Sweeping out properties of operator sequences
Let $L_p=L_p(X,\mu)$, $1\leq p\leq\infty$, be the usual Banach Spaces of real valued functions on a complete non-atomic probability space. Let $(T_1,\ldots,T_{K})$ be $L_2$-contractions. Let $0<\varepsilon < \delta\leq1$. Call a function $f$ a $\delta$-spanning function if $\|f\|_2 = 1$ and if $\|T_kf-Q_{k-1}T_kf\|_2\geq\delta$ for each $k=1,\ldots,K$, where $Q_0=0$ and $Q_k$ is the orthogonal projection on the subspace spanned by $(T_1f,\ldots,T_kf)$. Call a function $h$ a $(\delta,\varepsilon)$-sweeping function if $\|h\|_\infty\leq1$, $\|h\|_1<\varepsilon$, and if $\max_{1\leq k\leq K}|T_kh|>\delta-\varepsilon$ on a set of measure greater than $1-\varepsilon$. The following is the main technical result, which is obtained by elementary estimates. There is an integer $K=K(\varepsilon,\delta)\geq1$ such that if $f$ is a $\delta$-spanning function, and if the joint distribution of $(f,T_1f,\ldots,T_Kf)$ is normal, then $h=\bigl((f\wedge M)\vee(-M)\bigr)/M$ is a $(\delta,\varepsilon)$-sweeping function, for some $M>0$. Furthermore, if $T_k$s are the averages of operators induced by the iterates of a measure preserving ergodic transformation, then a similar result is true without requiring that the joint distribution is normal. This gives the following theorem on a sequence $(T_i)$ of these averages. Assume that for each $K\geq1$ there is a subsequence $(T_{i_1},\ldots,T_{i_K})$ of length $K$, and a $\delta$-spanning function $f_K$ for this subsequence. Then for each $\varepsilon>0$ there is a function $h$, $0\leq h\leq1$, $\|h\|_1<\varepsilon$, such that $\limsup_iT_ih\geq\delta$ a.e.. Another application of the main result gives a refinement of a part of Bourgain's ``Entropy Theorem'', resulting in a different, self contained proof of that theorem.

Keywords:Strong and $\delta$-sweeping out, Gaussian distributions, Bourgain's entropy theorem.
Categories:28D99, 60F99

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