The joint spectral radius of a finite set of real d × d matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the finiteness property if there exists a periodic product which achieves this maximal rate of growth. J. C. Lagarias and Y. Wang conjectured in 1995 that every finite set of real d × d matrices satisfies the finiteness property. However, T. Bousch and J. Mairesse proved in 2002 that counterexamples to the finiteness conjecture exist, showing in particular that there exists a family of pairs of 2 × 2 matrices which contains a counterexample. Similar results were subsequently given by V. D. Blondel, J. Theys and A. A. Vladimirov and by V. S. Kozyakin, but no explicit counterexample to the finiteness conjecture has so far been given. The purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the Lagarias-Wang finiteness conjecture. Namely, for the set

Abstract. We study the multifractal analysis of a class of equicontractive, self-similar measures of finite type, whose support is an interval. Finite type is a property weaker than the open set condition, but stronger than the weak open set condition. Examples include Bernoulli convolutions with contraction factor the inverse of a Pisot number and self-similar measures associated with m-fold sums of Cantor sets with ratio of dissection 1/R for integer R ≤ m.We introduce a combinatorial notion called a loop class and prove that the set of attainable local dimensions of the measure at points in a positive loop class is a closed interval. We prove that the local dimensions at the periodic points in the loop class are dense and give a simple formula for those local dimensions. These self-similar measures have a distinguished positive loop class called the essential class. The set of points in the essential class has full Lebesgue measure in the support of the measure and is often all but the two endpoints of the support. Thus many, but not all, measures of finite type have at most one isolated point in their set of local dimensions.We give examples of Bernoulli convolutions whose sets of attainable local dimensions consist of an interval together with an isolated point. As well, we give an example of a measure of finite type that has exactly two distinct local dimensions.

Consider a sequence of linear contractions S j (x) = ̺x + d j and probabilities p j > 0 with p j = 1. We are interested in the self-similar measure µ = p j µ • S −1 j , of finite type. In this paper we study the multifractal 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 suppµ of full µ 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 ∈ (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.

Let E be a Sidon subset of the integers and suppose X is a Banach space. Then Pisier has shown that E-spectral polynomials with values in X behave like Rademacher sums with respect to L p-norms. We consider the situation when X is a quasi-Banach space. For general quasi-Banach spaces we show that a similar result holds if and only if E is a set of interpolation (I 0-set). However, for certain special classes of quasi-Banach spaces we are able to prove such a result for larger sets. Thus if X is restricted to be "natural" then the result holds for all Sidon sets. We also consider spaces with plurisubharmonic norms and introduce the class of analytic Sidon sets.

We consider the Assouad spectrum, introduced by Fraser and Yu, along with a natural variant that we call the 'upper Assouad spectrum'. These spectra are designed to interpolate between the upper box-counting and Assouad dimensions. It is known that the Assouad spectrum approaches the upper box-counting dimension at the left hand side of its domain, but does not necessarily approach the Assouad dimension on the right. Here we show that it necessarily approaches the quasi-Assouad dimension at the right hand side of its domain. We further show that the upper Assouad spectrum can be expressed in terms of the Assouad spectrum, thus motivating the definition used by Fraser-Yu.We also provide a large family of examples demonstrating new phenomena relating to the form of the Assouad spectrum. For example, we prove that it can be strictly concave, exhibit phase transitions of any order, and need not be piecewise differentiable.Mathematics Subject Classification 2010: primary: 28A80.