{"title":"Iterated function systems based on the degree of nondensifiability","authors":"G. García, G. Mora","doi":"10.4171/jfg/121","DOIUrl":"https://doi.org/10.4171/jfg/121","url":null,"abstract":"","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42722763","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 consider the construction of a family ${K_N}$ of $3$-dimensional Koch-type surfaces, with a corresponding family of $3$-dimensional Koch-type ``snowflake analogues"${mathcal{C}_N}$, where $N>1$ are integers with $N notequiv 0 ,(bmod,, 3)$. We first establish that the Koch surfaces $K_N$ are $s_N$-sets with respect to the $s_N$-dimensional Hausdorff measure, for $s_N=log(N^2+2)/log(N)$ the Hausdorff dimension of each Koch-type surface $K_N$. Using self-similarity, one deduces that the same result holds for each Koch-type crystal $mathcal{C}_N$. We then develop lower and upper approximation monotonic sequences converging to the $s_N$-dimensional Hausdorff measure on each Koch-type surface $K_N$, and consequently, one obtains upper and lower bounds for the Hausdorff measure for each set $mathcal{C}_N$. As an application, we consider the realization of Robin boundary value problems over the Koch-type crystals $mathcal{C}_N$, for $N>2$.
{"title":"3D Koch-type crystals","authors":"Giovanni Ferrer, Alejandro Vélez-Santiago","doi":"10.4171/jfg/130","DOIUrl":"https://doi.org/10.4171/jfg/130","url":null,"abstract":"We consider the construction of a family ${K_N}$ of $3$-dimensional Koch-type surfaces, with a corresponding family of $3$-dimensional Koch-type ``snowflake analogues\"${mathcal{C}_N}$, where $N>1$ are integers with $N notequiv 0 ,(bmod,, 3)$. We first establish that the Koch surfaces $K_N$ are $s_N$-sets with respect to the $s_N$-dimensional Hausdorff measure, for $s_N=log(N^2+2)/log(N)$ the Hausdorff dimension of each Koch-type surface $K_N$. Using self-similarity, one deduces that the same result holds for each Koch-type crystal $mathcal{C}_N$. We then develop lower and upper approximation monotonic sequences converging to the $s_N$-dimensional Hausdorff measure on each Koch-type surface $K_N$, and consequently, one obtains upper and lower bounds for the Hausdorff measure for each set $mathcal{C}_N$. As an application, we consider the realization of Robin boundary value problems over the Koch-type crystals $mathcal{C}_N$, for $N>2$.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47129367","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}
This article concerns the dimension theory of the graphs of a family of functions which include the well-known 'popcorn function' and its pyramid-like higher-dimensional analogues. We calculate the box and Assouad dimensions of these graphs, as well as the intermediate dimensions, which are a family of dimensions interpolating between Hausdorff and box dimension. As tools in the proofs, we use the Chung$unicode{x2013}$ErdH{o}s inequality from probability theory, higher-dimensional Duffin$unicode{x2013}$Schaeffer type estimates from Diophantine approximation, and a bound for Euler's totient function. As applications we obtain bounds on the box dimension of fractional Brownian images of the graphs, and on the H"older distortion between different graphs.
{"title":"Dimensions of popcorn-like pyramid sets","authors":"Amlan Banaji, Haipeng Chen","doi":"10.4171/jfg/135","DOIUrl":"https://doi.org/10.4171/jfg/135","url":null,"abstract":"This article concerns the dimension theory of the graphs of a family of functions which include the well-known 'popcorn function' and its pyramid-like higher-dimensional analogues. We calculate the box and Assouad dimensions of these graphs, as well as the intermediate dimensions, which are a family of dimensions interpolating between Hausdorff and box dimension. As tools in the proofs, we use the Chung$unicode{x2013}$ErdH{o}s inequality from probability theory, higher-dimensional Duffin$unicode{x2013}$Schaeffer type estimates from Diophantine approximation, and a bound for Euler's totient function. As applications we obtain bounds on the box dimension of fractional Brownian images of the graphs, and on the H\"older distortion between different graphs.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47919546","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 $f$ be a generalized affine fractal interpolation function with vertical scaling function $S$. In this paper, we study $dim_B Gamma f$, the box dimension of the graph of $f$, under the assumption that $S$ is a Lipschtz function. By introducing vertical scaling matrices, we estimate the upper bound and the lower bound of oscillations of $f$. As a result, we obtain explicit formula of $dim_B Gamma f$ under certain constraint conditions.
{"title":"Box dimension of generalized affine fractal interpolation functions","authors":"Lai Jiang, H. Ruan","doi":"10.4171/jfg/136","DOIUrl":"https://doi.org/10.4171/jfg/136","url":null,"abstract":"Let $f$ be a generalized affine fractal interpolation function with vertical scaling function $S$. In this paper, we study $dim_B Gamma f$, the box dimension of the graph of $f$, under the assumption that $S$ is a Lipschtz function. By introducing vertical scaling matrices, we estimate the upper bound and the lower bound of oscillations of $f$. As a result, we obtain explicit formula of $dim_B Gamma f$ under certain constraint conditions.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46034326","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}
{"title":"On the Hausdorff dimension of the recurrent sets induced from endomorphisms of free groups","authors":"Y. Ishii, Tatsuya Oka","doi":"10.4171/jfg/120","DOIUrl":"https://doi.org/10.4171/jfg/120","url":null,"abstract":"","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41425863","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 this paper, we determine the almost sure values of the $Phi$-dimensions of random measures $mu$ supported on random Moran sets in $R^d$ that satisfy a uniform separation condition. This paper generalizes earlier work done on random measures on homogeneous Moran sets cite{HM} to the case of unequal scaling factors. The $Phi$-dimensions are intermediate Assouad-like dimensions with the (quasi-)Assouad dimensions and the $theta$-Assouad spectrum being special cases. The almost sure value of $dim_Phi mu$ exhibits a threshold phenomena, with one value for ``large'' $Phi$ (with the quasi-Assouad dimension as an example of a ``large'' dimension) and another for ``small'' $Phi$ (with the Assouad dimension as an example of a ``small'' dimension). We give many applications, including where the scaling factors are fixed and the probabilities are uniformly distributed. The almost sure $Phi$ dimension of the underlying random set is also a consequence of our results.
{"title":"Assouad-like dimensions of a class of random Moran measures. II. Non-homogeneous Moran sets","authors":"K. Hare, F. Mendivil","doi":"10.4171/jfg/133","DOIUrl":"https://doi.org/10.4171/jfg/133","url":null,"abstract":"In this paper, we determine the almost sure values of the $Phi$-dimensions of random measures $mu$ supported on random Moran sets in $R^d$ that satisfy a uniform separation condition. This paper generalizes earlier work done on random measures on homogeneous Moran sets cite{HM} to the case of unequal scaling factors. The $Phi$-dimensions are intermediate Assouad-like dimensions with the (quasi-)Assouad dimensions and the $theta$-Assouad spectrum being special cases. The almost sure value of $dim_Phi mu$ exhibits a threshold phenomena, with one value for ``large'' $Phi$ (with the quasi-Assouad dimension as an example of a ``large'' dimension) and another for ``small'' $Phi$ (with the Assouad dimension as an example of a ``small'' dimension). We give many applications, including where the scaling factors are fixed and the probabilities are uniformly distributed. The almost sure $Phi$ dimension of the underlying random set is also a consequence of our results.","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47989374","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}
{"title":"Baker domains and non-convergent deformations","authors":"Rodrigo Robles, G. Sienra","doi":"10.4171/jfg/115","DOIUrl":"https://doi.org/10.4171/jfg/115","url":null,"abstract":"","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46631718","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}
{"title":"Invariant measures for iterated function systems with inverses","authors":"Yuki Takahashi","doi":"10.4171/jfg/114","DOIUrl":"https://doi.org/10.4171/jfg/114","url":null,"abstract":"","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44600610","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}
{"title":"A fractal interpolation scheme for a possible sizeable set of data","authors":"Radu Miculescu, Alexandru Mihail, C. Păcurar","doi":"10.4171/jfg/117","DOIUrl":"https://doi.org/10.4171/jfg/117","url":null,"abstract":"","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48394186","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}
{"title":"An upper bound for the intermediate dimensions of Bedford–McMullen carpets","authors":"I. Kolossváry","doi":"10.4171/jfg/118","DOIUrl":"https://doi.org/10.4171/jfg/118","url":null,"abstract":"","PeriodicalId":48484,"journal":{"name":"Journal of Fractal Geometry","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47421101","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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