Dimension profiles were introduced in [8,11] to give a formula for the box-counting and packing dimensions of the orthogonal projections of a set $R^n$ onto almost all $m$-dimensional subspaces. However, these definitions of dimension profiles are indirect and are hard to work with. Here we firstly give alternative definitions of dimension profiles in terms of capacities of $E$ with respect to certain kernels, which lead to the box-counting and packing dimensions of projections fairly easily, including estimates on the size of the exceptional sets of subspaces where the dimension of projection is smaller the typical value. Secondly, we argue that with this approach projection results for different types of dimension may be thought of in a unified way. Thirdly, we use a Fourier transform method to obtain further inequalities on the size of the exceptional subspaces.
{"title":"A capacity approach to box and packing dimensions of projections of sets and exceptional directions","authors":"K. Falconer","doi":"10.4171/jfg/96","DOIUrl":"https://doi.org/10.4171/jfg/96","url":null,"abstract":"Dimension profiles were introduced in [8,11] to give a formula for the box-counting and packing dimensions of the orthogonal projections of a set $R^n$ onto almost all $m$-dimensional subspaces. However, these definitions of dimension profiles are indirect and are hard to work with. Here we firstly give alternative definitions of dimension profiles in terms of capacities of $E$ with respect to certain kernels, which lead to the box-counting and packing dimensions of projections fairly easily, including estimates on the size of the exceptional sets of subspaces where the dimension of projection is smaller the typical value. Secondly, we argue that with this approach projection results for different types of dimension may be thought of in a unified way. Thirdly, we use a Fourier transform method to obtain further inequalities on the size of the exceptional subspaces.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46003124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Loring, W. J. Ogden, Ely Sandine, R. Strichartz
We study the analogue of polynomials (solutions to $Delta u^{n+1} =0$ for some $n$) on the Sierpinski gasket ($SG$) with respect to a family of symmetric, self-similar Laplacians constructed by Fang, King, Lee, and Strichartz, extending the work of Needleman, Strichartz, Teplyaev, and Yung on the polynomials with respect to the standard Kigami Laplacian. We define a basis for the space of polynomials, the monomials, characterized by the property that a certain "derivative" is 1 at one of the boundary points, while all other "derivatives" vanish, and we compute the values of the monomials at the boundary points of $SG$. We then present some data which suggest surprising relationships between the values of the monomials at the boundary and certain Neumann eigenvalues of the family of symmetric self-similar Laplacians. Surprisingly, the results for the general case are quite different from the results for the Kigami Laplacian.
{"title":"Polynomials on the Sierpiński gasket with respect to different Laplacians which are symmetric and self-similar","authors":"C. Loring, W. J. Ogden, Ely Sandine, R. Strichartz","doi":"10.4171/jfg/95","DOIUrl":"https://doi.org/10.4171/jfg/95","url":null,"abstract":"We study the analogue of polynomials (solutions to $Delta u^{n+1} =0$ for some $n$) on the Sierpinski gasket ($SG$) with respect to a family of symmetric, self-similar Laplacians constructed by Fang, King, Lee, and Strichartz, extending the work of Needleman, Strichartz, Teplyaev, and Yung on the polynomials with respect to the standard Kigami Laplacian. We define a basis for the space of polynomials, the monomials, characterized by the property that a certain \"derivative\" is 1 at one of the boundary points, while all other \"derivatives\" vanish, and we compute the values of the monomials at the boundary points of $SG$. We then present some data which suggest surprising relationships between the values of the monomials at the boundary and certain Neumann eigenvalues of the family of symmetric self-similar Laplacians. Surprisingly, the results for the general case are quite different from the results for the Kigami Laplacian.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44783064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $X$ be a Banach space and $f,g:Xrightarrow X$ be contractions. We investigate the set $$ C_{f,g}:={win X:m{ the attractor of IFS }F_w={f,g+w}m{ is connected}}. $$ The motivation for our research comes from papers of Mihail and Miculescu, where it was shown that $C_{f,g}$ is a countable union of compact sets, provided $f,g$ are linear bounded operators with $pa fpa,pa gpa<1$ and such that $f$ is compact. Moreover, in the case when $X$ is finitely dimensional, such sets have been intensively investigated in the last years, especially when $f$ and $g$ are affine maps. As we will be mostly interested in infinite dimensional spaces, our results can be also viewed as a next step into extending of such studies into infinite dimensional setting. In particular, unlike in the finitely dimensional case, if $X$ has infinite dimension then $C_{f,g}$ is very small set (at least nowhere dense) provided $f,g$ satisfy some natural conditions.
{"title":"Connectedness of attractors of a certain family of IFSs","authors":"F. Strobin, J. Swaczyna","doi":"10.4171/jfg/89","DOIUrl":"https://doi.org/10.4171/jfg/89","url":null,"abstract":"Let $X$ be a Banach space and $f,g:Xrightarrow X$ be contractions. We investigate the set $$ C_{f,g}:={win X:m{ the attractor of IFS }F_w={f,g+w}m{ is connected}}. $$ The motivation for our research comes from papers of Mihail and Miculescu, where it was shown that $C_{f,g}$ is a countable union of compact sets, provided $f,g$ are linear bounded operators with $pa fpa,pa gpa<1$ and such that $f$ is compact. Moreover, in the case when $X$ is finitely dimensional, such sets have been intensively investigated in the last years, especially when $f$ and $g$ are affine maps. As we will be mostly interested in infinite dimensional spaces, our results can be also viewed as a next step into extending of such studies into infinite dimensional setting. In particular, unlike in the finitely dimensional case, if $X$ has infinite dimension then $C_{f,g}$ is very small set (at least nowhere dense) provided $f,g$ satisfy some natural conditions.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49157888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 2016, Shmerkin and Solomyak showed that if $U subset mathbb{R}$ is an interval, and ${mu_{u}}_{u in U}$ is an analytic family of homogeneous self-similar measures on $mathbb{R}$ with similitude dimensions exceeding one, then, under a mild transversality assumption, $mu_{u} ll mathcal{L}^{1}$ for all parameters $u in U setminus E$, where $dim_{mathrm{H}} E = 0$. The purpose of this paper is to generalise the result of Shmerkin and Solomyak to non-homogeneous self-similar measures. As a corollary, we obtain new information about the absolute continuity of projections of non-homogeneous planar self-similar measures.
2016年，Shmerkin和Solomyak证明，如果$U 子集mathbb{R}$是一个区间，$ mu_{U} }_{U U U 是$mathbb{R}$上齐次自相似测度的解析族，相似维数超过1，则在温和的横向性假设下，$U set- E$中所有参数$U math_ {U} ll mathcal{L}^{1}$，其中$dim_{ mathm {H}} E = 0$。本文的目的是将Shmerkin和Solomyak的结果推广到非齐次自相似测度。作为一个推论，我们得到了关于非齐次平面自相似测度投影的绝对连续性的新信息。
{"title":"Absolute continuity in families of parametrised non-homogeneous self-similar measures","authors":"A. Käenmäki, Tuomas Orponen","doi":"10.4171/jfg/127","DOIUrl":"https://doi.org/10.4171/jfg/127","url":null,"abstract":"In 2016, Shmerkin and Solomyak showed that if $U subset mathbb{R}$ is an interval, and ${mu_{u}}_{u in U}$ is an analytic family of homogeneous self-similar measures on $mathbb{R}$ with similitude dimensions exceeding one, then, under a mild transversality assumption, $mu_{u} ll mathcal{L}^{1}$ for all parameters $u in U setminus E$, where $dim_{mathrm{H}} E = 0$. The purpose of this paper is to generalise the result of Shmerkin and Solomyak to non-homogeneous self-similar measures. As a corollary, we obtain new information about the absolute continuity of projections of non-homogeneous planar self-similar measures.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43137884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than $1$, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In particular, we prove that if $A$ has Hausdorff dimension $>1$, then the set of distances spanned by points of $A$ has Hausdorff dimension at least $40/57 > 0.7$ and there are many $yin A$ such that the pinned distance set ${ |x-y|:xin A}$ has Hausdorff dimension at least $29/42$ and lower box-counting dimension at least $40/57$. We use the approach and many results from the earlier work of Keleti and Shmerkin, but incorporate estimates from the recent work of Guth, Iosevich, Ou and Wang as additional input.
{"title":"Improved bounds for the dimensions of planar distance sets","authors":"Pablo Shmerkin","doi":"10.4171/jfg/97","DOIUrl":"https://doi.org/10.4171/jfg/97","url":null,"abstract":"We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than $1$, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In particular, we prove that if $A$ has Hausdorff dimension $>1$, then the set of distances spanned by points of $A$ has Hausdorff dimension at least $40/57 > 0.7$ and there are many $yin A$ such that the pinned distance set ${ |x-y|:xin A}$ has Hausdorff dimension at least $29/42$ and lower box-counting dimension at least $40/57$. We use the approach and many results from the earlier work of Keleti and Shmerkin, but incorporate estimates from the recent work of Guth, Iosevich, Ou and Wang as additional input.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41649382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The attractor is a central object of an iterated function system (IFS), and fractal transformations are the natural maps from the attractor of one IFS to the attractor of another. This paper presents a global point of view, showing how to extend the domain of a fractal transformation from an attractor with non-empty interior to the ambient space. Intimitely related is the extension of addressing from such an attractor to the set of points of the ambient space. Properties of such global fractal transformations are obtained, and tilings are constructed based on global addresses.
{"title":"Global fractal transformations and global addressing","authors":"A. Vince","doi":"10.4171/JFG/65","DOIUrl":"https://doi.org/10.4171/JFG/65","url":null,"abstract":"The attractor is a central object of an iterated function system (IFS), and fractal transformations are the natural maps from the attractor of one IFS to the attractor of another. This paper presents a global point of view, showing how to extend the domain of a fractal transformation from an attractor with non-empty interior to the ambient space. Intimitely related is the extension of addressing from such an attractor to the set of points of the ambient space. Properties of such global fractal transformations are obtained, and tilings are constructed based on global addresses.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/65","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48460902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove estimates on the Hölder exponent of the density of states measure for discrete Schrödinger operators with potential of the form V (n)= λ (b(n+1)βc−bnβc), with λ large enough, and conclude that for almost all values of β , the density of states measure is not Hölder continuous.
{"title":"Frequency dependence of Hölder continuity for quasiperiodic Schrödinger operators","authors":"P. Munger","doi":"10.4171/JFG/68","DOIUrl":"https://doi.org/10.4171/JFG/68","url":null,"abstract":"We prove estimates on the Hölder exponent of the density of states measure for discrete Schrödinger operators with potential of the form V (n)= λ (b(n+1)βc−bnβc), with λ large enough, and conclude that for almost all values of β , the density of states measure is not Hölder continuous.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/68","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47076340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Assouad and quasi-Assouad dimensions of a metric space provide information about the extreme local geometric nature of the set. The Assouad dimension of a set has a measure theoretic analogue, which is also known as the upper regularity dimension. One reason for the interest in this notion is that a measure has finite Assouad dimension if and only if it is doubling. Motivated by recent progress on both the Assouad dimension of measures that satisfy a strong separation condition and the quasi-Assouad dimension of metric spaces, we introduce the notion of the quasi-Assouad dimension of a measure. As with sets, the quasi-Assouad dimension of a measure is dominated by its Assouad dimension. It dominates both the quasi-Assouad dimension of its support and the supremal local dimension of the measure, with strict inequalities possible in all cases. Our main focus is on self-similar measures in $mathbb{R}$ whose support is an interval and which may have `overlaps'. For measures that satisfy a weaker condition than the weak separation condition we prove that finite quasi-Assouad dimension is equivalent to quasi-doubling of the measure, a strictly less restrictive property than doubling. Further, we exhibit a large class of such measures for which the quasi-Assouad dimension coincides with the maximum of the local dimension at the endpoints of the support. This class includes all regular, equicontractive self-similar measures satisfying the weak separation condition, such as convolutions of uniform Cantor measures with integer ratio of dissection. Other properties of this dimension are also established and many examples are given.
{"title":"Quasi-doubling of self-similar measures with overlaps","authors":"K. Hare, K. Hare, Sascha Troscheit","doi":"10.4171/jfg/91","DOIUrl":"https://doi.org/10.4171/jfg/91","url":null,"abstract":"The Assouad and quasi-Assouad dimensions of a metric space provide information about the extreme local geometric nature of the set. The Assouad dimension of a set has a measure theoretic analogue, which is also known as the upper regularity dimension. One reason for the interest in this notion is that a measure has finite Assouad dimension if and only if it is doubling. \u0000Motivated by recent progress on both the Assouad dimension of measures that satisfy a strong separation condition and the quasi-Assouad dimension of metric spaces, we introduce the notion of the quasi-Assouad dimension of a measure. As with sets, the quasi-Assouad dimension of a measure is dominated by its Assouad dimension. It dominates both the quasi-Assouad dimension of its support and the supremal local dimension of the measure, with strict inequalities possible in all cases. \u0000Our main focus is on self-similar measures in $mathbb{R}$ whose support is an interval and which may have `overlaps'. For measures that satisfy a weaker condition than the weak separation condition we prove that finite quasi-Assouad dimension is equivalent to quasi-doubling of the measure, a strictly less restrictive property than doubling. Further, we exhibit a large class of such measures for which the quasi-Assouad dimension coincides with the maximum of the local dimension at the endpoints of the support. This class includes all regular, equicontractive self-similar measures satisfying the weak separation condition, such as convolutions of uniform Cantor measures with integer ratio of dissection. Other properties of this dimension are also established and many examples are given.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44969257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By a metric fractal we understand a compact metric space $K$ endowed with a finite family $mathcal F$ of contracting self-maps of $K$ such that $K=bigcup_{finmathcal F}f(K)$. If $K$ is a subset of a metric space $X$ and each $finmathcal F$ extends to a contracting self-map of $X$, then we say that $(K,mathcal F)$ is a fractal in $X$. We prove that each metric fractal $(K,mathcal F)$ is $bullet$ isometrically equivalent to a fractal in the Banach spaces $C[0,1]$ and $ell_infty$; $bullet$ bi-Lipschitz equivalent to a fractal in the Banach space $c_0$; $bullet$ isometrically equivalent to a fractal in the Hilbert space $ell_2$ if $K$ is an ultrametric space. We prove that for a metric fractal $(K,mathcal F)$ with the doubling property there exists $kinmathbb N$ such that the metric fractal $(K,mathcal F^{circ k})$ endowed with the fractal structure $mathcal F^{circ k}={f_1circdotscirc f_k:f_1,dots,f_kinmathcal F}$ is equi-H"older equivalent to a fractal in a Euclidean space $mathbb R^d$. This result is used to prove our main result saying that each finite-dimensional compact metrizable space $K$ containing an open uncountable zero-dimensional space $Z$ is homeomorphic to a fractal in a Euclidean space $mathbb R^d$. For $Z$, being a copy of the Cantor set, this embedding result was proved by Duvall and Husch in 1992.
{"title":"Embedding fractals in Banach, Hilbert or Euclidean spaces","authors":"T. Banakh, M. Nowak, F. Strobin","doi":"10.4171/JFG/94","DOIUrl":"https://doi.org/10.4171/JFG/94","url":null,"abstract":"By a metric fractal we understand a compact metric space $K$ endowed with a finite family $mathcal F$ of contracting self-maps of $K$ such that $K=bigcup_{finmathcal F}f(K)$. If $K$ is a subset of a metric space $X$ and each $finmathcal F$ extends to a contracting self-map of $X$, then we say that $(K,mathcal F)$ is a fractal in $X$. We prove that each metric fractal $(K,mathcal F)$ is \u0000$bullet$ isometrically equivalent to a fractal in the Banach spaces $C[0,1]$ and $ell_infty$; \u0000$bullet$ bi-Lipschitz equivalent to a fractal in the Banach space $c_0$; \u0000$bullet$ isometrically equivalent to a fractal in the Hilbert space $ell_2$ if $K$ is an ultrametric space. \u0000We prove that for a metric fractal $(K,mathcal F)$ with the doubling property there exists $kinmathbb N$ such that the metric fractal $(K,mathcal F^{circ k})$ endowed with the fractal structure $mathcal F^{circ k}={f_1circdotscirc f_k:f_1,dots,f_kinmathcal F}$ is equi-H\"older equivalent to a fractal in a Euclidean space $mathbb R^d$. This result is used to prove our main result saying that each finite-dimensional compact metrizable space $K$ containing an open uncountable zero-dimensional space $Z$ is homeomorphic to a fractal in a Euclidean space $mathbb R^d$. For $Z$, being a copy of the Cantor set, this embedding result was proved by Duvall and Husch in 1992.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43637647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
During the late spring, most of the Arctic Ocean is covered by sea ice with a layer of snow on top. As the snow and sea ice begin to melt, water collects on the surface to form melt ponds. As melting progresses, sparse, disconnected ponds coalesce to form complex, self-similar structures which are connected over large length scales. The boundaries of the ponds undergo a transition in fractal dimension from 1 to about 2 around a critical length scale of 100 square meters, as found previously from area–perimeter data. Melt pond geometry depends strongly on sea ice and snow topography. Here we construct a rather simple model of melt pond boundaries as the intersection of a horizontal plane, representing the water level, with a random surface representing the topography. We show that an autoregressive class of anisotropic random Fourier surfaces provides topographies that yield the observed fractal dimension transition, with the ponds evolving and growing as the plane rises. The results are compared with a partial differential equation model of melt pond evolution that includes much of the physics of the system. Properties of the shift in fractal dimension, such as its amplitude, phase and rate, are shown to depend on the surface anisotropy and autocorrelation length scales in the models. Melting-driven differences between the two models are highlighted. Mathematics Subject Classification (2010). 51, 35, 42, 86.
{"title":"Modeling the fractal geometry of Arctic melt ponds using the level sets of random surfaces","authors":"B. Bowen, C. Strong, K. Golden","doi":"10.4171/JFG/58","DOIUrl":"https://doi.org/10.4171/JFG/58","url":null,"abstract":"During the late spring, most of the Arctic Ocean is covered by sea ice with a layer of snow on top. As the snow and sea ice begin to melt, water collects on the surface to form melt ponds. As melting progresses, sparse, disconnected ponds coalesce to form complex, self-similar structures which are connected over large length scales. The boundaries of the ponds undergo a transition in fractal dimension from 1 to about 2 around a critical length scale of 100 square meters, as found previously from area–perimeter data. Melt pond geometry depends strongly on sea ice and snow topography. Here we construct a rather simple model of melt pond boundaries as the intersection of a horizontal plane, representing the water level, with a random surface representing the topography. We show that an autoregressive class of anisotropic random Fourier surfaces provides topographies that yield the observed fractal dimension transition, with the ponds evolving and growing as the plane rises. The results are compared with a partial differential equation model of melt pond evolution that includes much of the physics of the system. Properties of the shift in fractal dimension, such as its amplitude, phase and rate, are shown to depend on the surface anisotropy and autocorrelation length scales in the models. Melting-driven differences between the two models are highlighted. Mathematics Subject Classification (2010). 51, 35, 42, 86.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/JFG/58","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45646360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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