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I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close together (super-exponentially in their lengths). Applications include resolution of some conjectures of Furstenberg on the dimension of sumsets and, together with work of Shmerkin, progress on the absolute continuity of Bernoulli convolutions. The main new ingredient is a statement in additive combinatorics concerning the structure of measures whose entropy does not grow very much under convolution. If time permits I will discuss the analogous results in higher dimensions.
[-] I will discuss recent progress on understanding the dimension of self-similar sets and measures. The main conjecture in this field is that the only way that the dimension of such a fractal can be "non-full" is if the semigroup of contractions which define it is not free. The result I will discuss is that "non-full" dimension implies "almost non-freeness", in the sense that there are distinct words in the semigroup which are extremely close ...
[+] 28A80 ; 37A10 ; 03D99 ; 54H20
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y
Given $x\in(0, 1]$, let ${\mathcal U}(x)$ be the set of bases $\beta\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^{\infty}{{d_i}/{\beta^i}}$. In 2014, Lü, Tan and Wu proved that ${\mathcal U}(x)$ is a Lebesgue null set of full Hausdorff dimension. In this talk, we will show that the algebraic sum ${\mathcal U}(x)+\lambda {\mathcal U}(x)$, and the product ${\mathcal U}(x)\cdot {\mathcal U}(x)^{\lambda}$ contain an interval for all $x\in (0, 1]$ and $\lambda\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters associated with the family of symmetric binary expansions studied recently by the first speaker and C. Kalle.
This is joint work with V. Komornik, D. Kong and W. Li.
[-] Given $x\in(0, 1]$, let ${\mathcal U}(x)$ be the set of bases $\beta\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^{\infty}{{d_i}/{\beta^i}}$. In 2014, Lü, Tan and Wu proved that ${\mathcal U}(x)$ is a Lebesgue null set of full Hausdorff dimension. In this talk, we will show that the algebraic sum ${\mathcal U}(x)+\lambda {\mathcal U}(x)$, and the product ${\mathcal U}(x)\cdot {\mathcal ...
[+] 28A80 ; 11A63 ; 37B10
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y
In the way of Arnoux-Ito, we give a general geometric criterion for a subshift to be measurably conjugated to a domain exchange and to a translation on a torus. For a subshift coming from an unit Pisot irreducible substitution, we will see that it becomes a simple topological criterion. More precisely, we define a topology on $\mathbb{Z}^d$ for which the subshift has pure discrete spectrum if and only if there exists a domain of the domain exchange on the discrete line that has non-empty interior. We will see how we can compute exactly such interior using regular languages. This gives a way to decide the Pisot conjecture for any example of unit Pisot irreducible substitution.
Joint work with Shigeki Akiyama.
[-] In the way of Arnoux-Ito, we give a general geometric criterion for a subshift to be measurably conjugated to a domain exchange and to a translation on a torus. For a subshift coming from an unit Pisot irreducible substitution, we will see that it becomes a simple topological criterion. More precisely, we define a topology on $\mathbb{Z}^d$ for which the subshift has pure discrete spectrum if and only if there exists a domain of the domain ...
[+] 37B10 ; 28A80 ; 11A63 ; 68R15
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In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter.
[-] In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for ...
[+] 28A80 ; 37C45
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y
In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter.
[-] In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for ...
[+] 28A80 ; 37C45
Déposez votre fichier ici pour le déplacer vers cet enregistrement.
y
In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter.
[-] In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for ...
[+] 28A80 ; 37C45
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y
In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter.
[-] In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for ...
[+] 28A80 ; 37C45
Déposez votre fichier ici pour le déplacer vers cet enregistrement.
y
In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter.
[-] In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for ...
[+] 28A80 ; 37C45
English
