scholarly journals On the Brownian separable permuton

2019 ◽  
Vol 29 (2) ◽  
pp. 241-266
Author(s):  
Mickaël Maazoun
Keyword(s):  
Hausdorff Dimension ◽  
Probability Measure ◽  
Scaling Limit ◽  
Random Measure ◽  
Local Minima ◽  
Self Similarity ◽  
Brownian Excursion ◽  
Random Probability ◽  
Dimension One ◽  
Totally Disconnected

AbstractThe Brownian separable permuton is a random probability measure on the unit square, which was introduced by Bassino, Bouvel, Féray, Gerin and Pierrot (2016) as the scaling limit of the diagram of the uniform separable permutation as size grows to infinity. We show that, almost surely, the permuton is the pushforward of the Lebesgue measure on the graph of a random measure-preserving function associated to a Brownian excursion whose strict local minima are decorated with independent and identically distributed signs. As a consequence, its support is almost surely totally disconnected, has Hausdorff dimension one, and enjoys self-similarity properties inherited from those of the Brownian excursion. The density function of the averaged permuton is computed and a connection with the shuffling of the Brownian continuum random tree is explored.

scholarly journals NEI Modelling of the ISM - Turbulent Dissipation and Hausdorff Dimension

2009 ◽  
Vol 5 (H15) ◽  
pp. 468-469 ◽  
Author(s):  
Miguel A. de Avillez ◽  
Dieter Breitschwerdt
Keyword(s):  
High Resolution ◽  
Hausdorff Dimension ◽  
Dissipative Structures ◽  
Self Similarity ◽  
Extended Self ◽  
Turbulent Dissipation ◽  
Independent Information ◽  
Ionization Structure ◽  
Wide Range ◽  
Self Gravity

AbstractHigh-resolution non-ideal magnetohydrodynamical simulations of the turbulent magnetized ISM, powered by supernovae types Ia and II at Galactic rate, including self-gravity and non-equilibriuim ionization (NEI), taking into account the time evolution of the ionization structure of H, He, C, N, O, Ne, Mg, Si, S and Fe, were carried out. These runs cover a wide range (from kpc to sub-parsec) of scales, providing resolution independent information on the injection scale, extended self-similarity and the fractal dmension of the most dissipative structures.

scholarly journals The Infinite limit of random permutations avoiding patterns of length three

2019 ◽  
Vol 29 (1) ◽  
pp. 137-152 ◽  
Author(s):  
Ross G. Pinsky
Keyword(s):  
Probability Measure ◽  
Metric Space ◽  
The Other ◽  
Random Permutations ◽  
Compact Metric Space ◽  
Sigma 1 ◽  
Limiting Behaviour ◽  
Random Probability

AbstractFor $$\tau \in {S_3}$$, let $$\mu _n^\tau $$ denote the uniformly random probability measure on the set of $$\tau $$-avoiding permutations in $${S_n}$$. Let $${\mathbb {N}^*} = {\mathbb {N}} \cup \{ \infty \} $$ with an appropriate metric and denote by $$S({\mathbb{N}},{\mathbb{N}^*})$$ the compact metric space consisting of functions $$\sigma {\rm{ = }}\{ {\sigma _i}\} _{i = 1}^\infty {\rm{ }}$$ from $$\mathbb {N}$$ to $${\mathbb {N}^ * }$$ which are injections when restricted to $${\sigma ^{ - 1}}(\mathbb {N})$$; that is, if $${\sigma _i}{\rm{ = }}{\sigma _j}$$, $$i \ne j$$, then $${\sigma _i} = \infty $$. Extending permutations $$\sigma \in {S_n}$$ by defining $${\sigma _j} = j$$, for $$j \gt n$$, we have $${S_n} \subset S({\mathbb{N}},{{\mathbb{N}}^*})$$. For each $$\tau \in {S_3}$$, we study the limiting behaviour of the measures $$\{ \mu _n^\tau \} _{n = 1}^\infty $$ on $$S({\mathbb{N}},{\mathbb{N}^*})$$. We obtain partial results for the permutation $$\tau = 321$$ and complete results for the other five permutations $$\tau \in {S_3}$$.

scholarly journals Dimension, entropy and Lyapunov exponents

1982 ◽  
Vol 2 (1) ◽  
pp. 109-124 ◽  
Author(s):  
Lai-Sang Young
Keyword(s):  
Hausdorff Dimension ◽  
Probability Measure ◽  
Lyapunov Exponents ◽  
Borel Probability Measure ◽  
Fractional Dimension ◽  
Present Context ◽  
Rényi Dimension ◽  
Borel Probability

AbstractWe consider diffeomorphisms of surfaces leaving invariant an ergodic Borel probability measure μ. Define HD (μ) to be the infimum of Hausdorff dimension of sets having full μ-measure. We prove a formula relating HD (μ) to the entropy and Lyapunov exponents of the map. Other classical notions of fractional dimension such as capacity and Rényi dimension are discussed. They are shown to be equal to Hausdorff dimension in the present context.

Local self-similarity and the Hausdorff dimension

2003 ◽  
Vol 336 (3) ◽  
pp. 267-272 ◽  
Author(s):  
Albert Benassi ◽  
Serge Cohen ◽  
Jacques Istas
Keyword(s):  
Hausdorff Dimension ◽  
Self Similarity

scholarly journals Poisson random balls: self-similarity and X-ray images

2006 ◽  
Vol 38 (4) ◽  
pp. 853-872 ◽  
Author(s):  
Hermine Biermé ◽  
Anne Estrade
Keyword(s):  
Porous Media ◽  
Random Field ◽  
Random Measure ◽  
Second Order ◽  
Poisson Random Measure ◽  
Media Images ◽  
Self Similarity ◽  
X Ray ◽  
Ray Transform

We study a random field obtained by counting the number of balls containing a given point when overlapping balls are thrown at random according to a Poisson random measure. We describe a microscopic process which exhibits multifractional behavior. We are particularly interested in the local asymptotic self-similarity (LASS) properties of the field, as well as in its X-ray transform. We obtain two different LASS properties when considering the asymptotics either in law or in the sense of second-order moments, and prove a relationship between the LASS behavior of the field and the LASS behavior of its X-ray transform. These results can be used to model and analyze porous media, images, or connection networks.

scholarly journals Relative bifurcation sets and the local dimension of univoque bases

2020 ◽  
pp. 1-33
Author(s):  
PIETER ALLAART ◽  
DERONG KONG
Keyword(s):  
Hausdorff Dimension ◽  
Explicit Expression ◽  
Lebesgue Measure ◽  
Pairwise Disjoint ◽  
Measure Zero ◽  
Lebesgue Measure Zero ◽  
Local Dimension ◽  
Dimension Zero ◽  
Image Position ◽  
Dimension One

Fix an alphabet $A=\{0,1,\ldots ,M\}$ with $M\in \mathbb{N}$ . The univoque set $\mathscr{U}$ of bases $q\in (1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper describes how the points in the set $\mathscr{U}$ are distributed over the interval $(1,M+1)$ by determining the limit $$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$ for all $q\in (1,M+1)$ . We show in particular that $f(q)>0$ if and only if $q\in \overline{\mathscr{U}}\backslash \mathscr{C}$ , where $\mathscr{C}$ is an uncountable set of Hausdorff dimension zero, and $f$ is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of $\mathscr{U}$ called relative bifurcation sets, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval, answering a question of Kalle et al [On the bifurcation set of unique expansions. Acta Arith. 188 (2019), 367–399]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [On univoque and strongly univoque sets. Adv. Math.308 (2017), 575–598] on strongly univoque sets.

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