arXiv subjects

Amir Algom

Publications and source records attributed to Amir Algom.

26 records · Page 2Linked to original sources

arXiv2021

Pointwise normality and Fourier decay for self-conformal measures

Let $Φ$ be a $C^{1+γ}$ smooth IFS on $\mathbb{R}$, where $γ>0$. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points $x$ that are absolutely normal. That is, for integer $p\geq 2$ the sequence $\lbrace p^k x \rbrace_{k\in \mathbb{N}}$ equidistributes modulo $1$. We thus extend several state of the art results of Hochman and Shmerkin about the prevalence of normal numbers in fractals. When $Φ$ is self-similar we show that the set of absolutely normal numbers has full Hausdorff dimension in its attractor, unless $Φ$ has an explicit structure that is associated with some integer $n\geq 2$. These conditions on the derivative cocycle are also shown to imply that every self conformal measure is a Rajchman measure, that is, its Fourier transform decays to $0$ at infinity. When $Φ$ is self similar and satisfies a certain Diophantine condition, we establish a logarithmic rate of decay.

math.DS↗

arXiv2021

Improved versions of some Furstenberg type slicing Theorems for self-affine carpets

Let $F$ be a Bedford-McMullen carpet defined by independent integer exponents. We prove that for every line $\ell \subseteq \mathbb{R}^2$ not parallel to the major axes, $$ \dim_H (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_H F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace$$ and $$ \dim_P (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_P F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace$$ where $\dim^*$ is Furstenberg's star dimension (maximal dimension of microsets). This improves the state of art results on Furstenberg type slicing Theorems for affine invariant carpets.

math.DS↗

arXiv2020

Actions of diagonal endomorphisms on conformally invariant measures on the 2-torus

Let $ν$ be a probability measure that is ergodic under the endomorphism $(\times p, \times p)$ of the torus $\mathbb{T}^2$, such that $\dim πμ< \dim μ$ for some non-principal projection $π$. We show that, if both $m\neq n$ are independent of $p$, the $(\times m, \times n)$ orbits of $ν$ typical points will equidistribute towards the Lebesgue measure. If $m>p$ then typically the $(\times m, \times p)$ orbits will equidistribute towards the product of the Lebesgue measure with the marginal of $μ$ on the $y$-axis. We also prove results in the same spirit for certain self similar measures $ν$. These are higher dimensional analogues of results due (among others) to Host, Lindenstrauss, and Hochman-Shmerkin.

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arXiv2019

A simultaneous version of Host's equidistribution Theorem

Let $μ$ be a probability measure on $\mathbb{R}/\mathbb{Z}$ that is ergodic under the $\times p$ map, with positive entropy. In 1995, Host showed that if $\gcd(m,p)=1$ then $μ$ almost every point is normal in base $m$. In 2001, Lindenstrauss showed that the conclusion holds under the weaker assumption that $p$ does not divide any power of $m$. In 2015, Hochman and Shmerkin showed that this holds in the "correct" generality, i.e. if $m$ and $p$ are independent. We prove a simultaneous version of this result: for $μ$ typical $x$, if $m>p$ are independent, we show that the orbit of $(x,x)$ under $(\times m, \times p)$ equidistributes for the product of the Lebesgue measure with $μ$. We also show that if $m>n>1$ and $n$ is independent of $p$ as well, then the orbit of $(x,x)$ under $(\times m, \times n)$ equidistributes for the Lebesgue measure.

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arXiv2018

Slicing Theorems and rigidity phenomena for self affine carpets

Let $F$ be a Bedford-McMullen carpet defined by independent exponents. We prove that $\overline{\dim}_B (\ell \cap F) \leq \max \lbrace \dim^* F -1,0 \rbrace$ for all lines $\ell$ not parallel to the principal axes, where $\dim^*$ is Furstenberg's star dimension (maximal dimension of a microset). We also prove several rigidity results for incommensurable Bedford-McMullen carpets, that is, carpets $F$ and $E$ such that all defining exponents are independent: Assuming various conditions, we find bounds on the dimension of the intersection of such carpets, show that self affine measures on them are mutually singular, and prove that they do not embed affinely into each other. We obtain these results as an application of a slicing Theorem for products of certain Cantor sets. This Theorem is a generlization of the results of Shmerkin and Wu, that proved Furstenberg's slicing Conjecture.

math.DS↗

arXiv2018

Affine embeddings of Cantor sets in the plane

Let $F,E\subseteq \mathbb{R}^2$ be two self similar sets. First, assuming $F$ is generated by an IFS $Φ$ with strong separation, we characterize the affine maps $g:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $g(F)\subseteq F$. Our analysis depends on the cardinality of the group $G_Φ$ generated by the orthogonal parts of the similarities in $Φ$. When $|G_Φ|=\infty$ we show that any such self embedding must be a similarity, and so (by the results of Elekes, Keleti and Máthé) some power of its orthogonal part lies in $G_Φ$. When $|G_Φ| < \infty$ and $Φ$ has a uniform contraction $λ$, we show that the linear part of any such embedding is diagonalizable, and the norm of each of its eigenvalues is a rational power of $λ$. We also study the existence and properties of affine maps $g$ such that $g(F)\subseteq E$, where $E$ is generated by an IFS $Ψ$. In this direction, we provide more evidence for a Conjecture of Feng, Huang and Rao, that such an embedding exists only if the contraction ratios of the maps in $Φ$ are algebraically dependent on the contraction ratios of the maps in $Ψ$. Furthermore, we show that, under some conditions, if $|G_Φ|=\infty$ then $|G_Ψ|=\infty$ and if $|G_Φ|<\infty$ then $|G_Ψ|<\infty$.

math.DS↗

arXiv2016

Affine embeddings of Cantor sets on the line

Let $s\in (0,1)$, and let $F\subset \mathbb{R}$ be a self similar set such that $0 < \dim_H F \leq s$ . We prove that there exists $δ= δ(s) >0$ such that if $F$ admits an affine embedding into a homogeneous self similar set $E$ and $0 \leq \dim_H E - \dim_H F < δ$ then (under some mild conditions on $E$ and $F$) the contraction ratios of $E$ and $F$ are logarithmically commensurable. This provides more evidence for a Conjecture of Feng, Huang, and Rao, that states that these contraction ratios are logarithmically commensurable whenever $F$ admits an affine embedding into $E$ (under some mild conditions). Our method is a combination of an argument based on the approach of Feng, Huang, and Rao, with a new result by Hochman, which is related to the increase of entropy of measures under convolutions.

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arXiv2016

Self embeddings of Bedford-McMullen carpets

Let $F \subseteq \mathbb{R}^2$ be a Bedford-McMullen carpet defined by multiplicatively independent exponents, and suppose that either $F$ is not a product set, or it is a product set with marginals of dimension strictly between $0$ and $1$. We prove that any similarity $g$ such that $g(F) \subseteq F$ is an isometry composed of reflections about lines parallel to the axes. Our approach utilizes the structure of tangent sets of $F$, obtained by "zooming in" on points of $F$, projection theorems for products of self-similar sets, and logarithmic commensurability type results for self similar sets in the line.

math.DS↗