Golden gaskets: variations on the Sierpi\'nski sieve

We consider the iterated function systems (IFSs) that consist of three general similitudes in the plane with centres at three non-collinear points, and with a common contraction factor \la\in(0,1). As is well known, for \la=1/2 the invariant set, \S_\la, is a fractal called the Sierpi\'nski sieve, and for \la<1/2 it is also a fractal. Our goal is to study \S_\la for this IFS for 1/2<\la<2/3, i.e., when there are "overlaps" in \S_\la as well as "holes". In this introductory paper we show that despite the overlaps (i.e., the Open Set Condition breaking down completely), the attractor can still be a totally self-similar fractal, although this happens only for a very special family of algebraic \la's (so-called "multinacci numbers"). We evaluate \dim_H(\S_\la) for these special values by showing that \S_\la is essentially the attractor for an infinite IFS which does satisfy the Open Set Condition. We also show that the set of points in the attractor with a unique ``address'' is self-similar, and compute its dimension. For ``non-multinacci'' values of \la we show that if \la is close to 2/3, then \S_\la has a nonempty interior and that if \la<1/\sqrt{3} then \S_\la$ has zero Lebesgue measure. Finally we discuss higher-dimensional analogues of the model in question.Comment: 27 pages, 10 figure

Dynamical degrees of birational transformations of projective surfaces

The dynamical degree $\lambda(f)$ of a birational transformation $f$ measures the exponential growth rate of the degree of the formulae that define the $n$-th iterate of $f$. We study the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of $\lambda(f)$ and the structure of the conjugacy class of $f$. For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed and well ordered set of algebraic numbers.Comment: 65 page

Multifractal analysis of Bernoulli convolutions associated with Salem numbers

We consider the multifractal structure of the Bernoulli convolution $\nu_{\lambda}$, where $\lambda^{-1}$ is a Salem number in $(1,2)$. Let $\tau(q)$ denote the $L^q$ spectrum of $\nu_\lambda$. We show that if $\alpha \in [\tau'(+\infty), \tau'(0+)]$, then the level set $$E(\alpha):={x\in \R:\; \lim_{r\to 0}\frac{\log \nu_\lambda([x-r, x+r])}{\log r}=\alpha}$$ is non-empty and $\dim_HE(\alpha)=\tau^*(\alpha)$, where $\tau^*$ denotes the Legendre transform of $\tau$. This result extends to all self-conformal measures satisfying the asymptotically weak separation condition. We point out that the interval $[\tau'(+\infty), \tau'(0+)]$ is not a singleton when $\lambda^{-1}$ is the largest real root of the polynomial $x^{n}-x^{n-1}-... -x+1$, $n\geq 4$. An example is constructed to show that absolutely continuous self-similar measures may also have rich multifractal structures.Comment: 26 pages. Accepted by Adv. Mat

A classification of all 1-Salem graphs

Abstract. One way to study certain classes of polynomials is by considering examples that are attached to combinatorial objects. Any graph G has an asso-ciated reciprocal polynomial RG, and with two particular classes of reciprocal polynomials in mind one can ask the questions: (a) when is RG a product of cy-clotomic polynomials (giving the cyclotomic graphs)? (b) when does RG have the minimal polynomial of a Salem number as its only non-cyclotomic factor (the non-trival Salem graphs)? Cyclotomic graphs were classified by Smith in 1970; the maximal connected ones are known as Smith graphs. Salem graphs are ‘spectrally close ’ to being cyclotomic, in that nearly all their eigenvalues are in the critical interval [−2, 2]. On the other hand Salem graphs do not need to be ‘combinatorially close ’ to being cyclotomic: the largest cyclotomic induced subgraph might be comparatively tiny. We define an m-Salem graph to be a connected Salem graph G for which m is minimal such that there exists an induced cyclotomic subgraph of G that has m fewer vertices than G. The 1-Salem subgraphs are both spectrally close and combinatorially close to being cyclotomic. Moreover, every Salem graph contains a 1-Salem graph as an induced subgraph, so these 1-Salem graphs provide some necessary substructure of all Salem graphs. The main result of this paper is a complete combinatorial description of all 1-Salem graphs: in the non-bipartite case there are 25 infinite families and 383 sporadic examples