Beta-expansions, natural extensions and multiple tilings associated with Pisot units

From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different transformations generating expansions in base $\beta$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $\beta$-expansions. Remarkably, the symmetric $\beta$-transformation does not satisfy this condition when $\beta$ is the smallest Pisot number or the Tribonacci number. This means that the Pisot conjecture on tilings cannot be extended to the symmetric $\beta$-transformation. Closely related to these (multiple) tilings are natural extensions of the transformations, which have many nice properties: they are invariant under the Lebesgue measure; under certain conditions, they provide Markov partitions of the torus; they characterize the numbers with purely periodic expansion, and they allow determining any digit in an expansion without knowing the other digits

Invariant measures, matching and the frequency of 0 for signed binary expansions

We introduce a parametrised family of maps $\{S_{\eta}\}_{\eta \in [1,2]}$, called symmetric doubling maps, defined on $[-1,1]$ by $S_\eta (x)=2x-d\eta$, where $d\in \{-1,0,1 \}$. Each map $S_\eta$ generates binary expansions with digits $-1$, 0 and 1. We study the frequency of the digit 0 in typical expansions as a function of the parameter $\eta$. The transformations $S_\eta$ have a natural ergodic invariant measure $\mu_\eta$ that is absolutely continuous with respect to Lebesgue measure. The frequency of the digit 0 is related to the measure $\mu_{\eta}([-\frac12,\frac12])$ by the Ergodic Theorem. We show that the density of $\mu_\eta$ is piecewise smooth except for a set of parameters of zero Lebesgue measure and full Hausdorff dimension and give a full description of the structure of the maximal parameter intervals on which the density is piecewise smooth. We give an explicit formula for the frequency of the digit 0 in typical signed binary expansions on each of these parameter intervals and show that this frequency depends continuously on the parameter $\eta$. Moreover, it takes the value $\frac23$ only on the interval $\big[ \frac65, \frac32\big]$ and it is strictly less than $\frac23$ on the remainder of the parameter space.Comment: 30 pages, 4 figure

Decay of correlations for critically intermittent systems

For a family of random intermittent dynamical systems with a superattracting fixed point we prove that a phase transition occurs between the existence of an absolutely continuous invariant probability measure and infinite measure depending on the randomness parameters and the orders of the maps at the superattracting fixed point. In case the systems have an absolutely continuous invariant probability measure, we show that the systems are mixing and that the correlations decay polynomially even though some of the deterministic maps present in the system have exponential decay. This contrasts other known results, where random systems adopt the best decay rate of the deterministic maps in the systems.Comment: 32 pages, 2 figure

Besicovitch-Eggleston sets for finite GLS number systems with redundancy

In this article we study Besicovitch-Eggleston sets for finite GLS number systems with redundancy. These number systems produce number expansions reminiscent of Cantor base expansions. The redundancy refers to the fact that each number $x \in [0,1]$ has uncountably many representations in the system. We distinguish between these representations by adding an extra dimension and describing the system as a diagonally affine IFS on $\mathbb R^2$. For the associated two dimensional level sets of digit frequencies we give the Birkhoff spectrum and an expression for the Hausdorff dimension. To obtain these results we first prove a more general result on the Hausdorff dimension of level sets for Birkhoff averages of continuous potentials for a certain family of diagonally affine IFS's. We also study the Hausdorff dimension of digit frequency sets along fibres.Comment: 21 pages, 1 figur

Matching for random systems with an application to minimal weight expansions

We extend the notion of matching for one-dimensional dynamical systems to random matching for random dynamical systems on an interval. We prove that for a large family of piecewise affine random systems of the interval the property of random matching implies that any invariant density is piecewise constant. We further introduce a one-parameter family of random dynamical systems that produce signed binary expansions of numbers in the interval [-1,1]. This family has random matching for Lebesgue almost every parameter. We use this to prove that the frequency of the digit 0 in the associated signed binary expansions never exceeds 1/2

The random continued fraction transformation

We introduce a random dynamical system related to continued fraction expansions. It uses random combination of the Gauss map and the R\'enyi (or backwards) continued fraction map. We explore the continued fraction expansions that this system produces as well as the dynamical properties of the system.PostprintPeer reviewe