• Corpus ID: 222125110

Self-similar measures associated to a homogeneous system of three maps

  title={Self-similar measures associated to a homogeneous system of three maps},
  author={Ariel Rapaport and P'eter P'al Varj'u},
  journal={arXiv: Dynamical Systems},
We study the dimension of self-similar measures associated to a homogeneous iterated function system of three contracting similarities on $\bf R$ and other more general IFS's. We extend some of the theory recently developed for Bernoulli convolutions to this setting. In the setting of three maps a new phenomenon occurs, which has been highlighted by recent examples of Baker, and Barany, Kaenmaki. To overcome the difficulties stemming form these, we develop novel techniques, including an… 

Estimates on the dimension of self‐similar measures with overlaps

In this paper, we provide an algorithm to estimate from below the dimension of self‐similar measures with overlaps. As an application, we show that for any β∈(1,2)$ \beta \in (1,2)$ , the dimension

Self-similar sets and measures on the line

We discuss the problem of determining the dimension of self-similar sets and measures on R. We focus on the developments of the last four years. At the end of the paper, we survey recent results

Pointwise normality and Fourier decay for self-conformal measures



Absolute Continuity of Bernoulli Convolutions, A Simple Proof

A bstract . The distribution νλ of the random series ∑ ±λn has been studied by many authors since the two seminal papers by Erdős in 1939 and 1940. Works of Alexander and Yorke, Przytycki and

Entropy of Bernoulli convolutions and uniform exponential growth for linear groups

The exponential growth rate of non-polynomially growing subgroups of GLrf is conjectured to admit a uniform lower bound. This is known for non-amenable subgroups, while for amenable subgroups it is

Iterated function systems with super-exponentially close cylinders II

  • S. Baker
  • Mathematics
    Proceedings of the American Mathematical Society
  • 2019

On self-similar sets with overlaps and inverse theorems for entropy in $\mathbb{R}^d$

We study self-similar sets and measures on $\mathbb{R}^{d}$. Assuming that the defining iterated function system $\Phi$ does not preserve a proper affine subspace, we show that one of the following

Dimension of invariant measures for affine iterated function systems

Let $\{S_i\}_{i\in \Lambda}$ be a finite contracting affine iterated function system (IFS) on ${\Bbb R}^d$. Let $(\Sigma,\sigma)$ denote the two-sided full shift over the alphabet $\Lambda$, and

Dimension theory of iterated function systems

Let {Si}  i = 1𝓁 be an iterated function system (IFS) on ℝd with attractor K. Let (Σ, σ) denote the one‐sided full shift over the alphabet {1, …, 𝓁}. We define the projection entropy function hπ on

Additive functions of intervals and Hausdorff measure

  • P. Moran
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1946
Consider bounded sets of points in a Euclidean space Rq of q dimensions. Let h(t) be a continuous increasing function, positive for t>0, and such that h(0) = 0. Then the Hausdroff measure h–mE of a

Convolutions of cantor measures without resonance

Denote by µa the distribution of the random sum $$(1 - a)\sum\nolimits_{j = 0}^\infty {{w_j}{a^j}} $$, where P(ωj = 0) = P(ωj = 1) = 1/2 and all the choices are independent. For 0 < a < 1/2, the

On the dimension of Bernoulli convolutions for all transcendental parameters

The Bernoulli convolution $\nu_\lambda$ with parameter $\lambda\in(0,1)$ is the probability measure supported on $\mathbf{R}$ that is the law of the random variable $\sum\pm\lambda^n$, where the