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Assouad Type Dimensions and Homogeneity of Fractals Assouad type dimensions and homogeneity of fractals Jonathan M. Fraser The University of St Andrews Jonathan M. Fraser Assouad type dimensions In particular, it usually studies how much space the set takes up on small scales. Common examples of `dimensions' are the Hausdorff, packing and box dimensions. Fractals are sets with a complex structure on small scales and thus they may have fractional dimension! People working in dimension theory and fractal geometry are often concerned with the rigorous computation of the dimensions of abstract classes of fractal sets. Dimension theory A `dimension' is a function that assigns a (usually positive, finite real) number to a metric space which attempts to quantify how `large' the set is. Jonathan M. Fraser Assouad type dimensions Common examples of `dimensions' are the Hausdorff, packing and box dimensions. Fractals are sets with a complex structure on small scales and thus they may have fractional dimension! People working in dimension theory and fractal geometry are often concerned with the rigorous computation of the dimensions of abstract classes of fractal sets. Dimension theory A `dimension' is a function that assigns a (usually positive, finite real) number to a metric space which attempts to quantify how `large' the set is. In particular, it usually studies how much space the set takes up on small scales. Jonathan M. Fraser Assouad type dimensions Fractals are sets with a complex structure on small scales and thus they may have fractional dimension! People working in dimension theory and fractal geometry are often concerned with the rigorous computation of the dimensions of abstract classes of fractal sets. Dimension theory A `dimension' is a function that assigns a (usually positive, finite real) number to a metric space which attempts to quantify how `large' the set is. In particular, it usually studies how much space the set takes up on small scales. Common examples of `dimensions' are the Hausdorff, packing and box dimensions. Jonathan M. Fraser Assouad type dimensions People working in dimension theory and fractal geometry are often concerned with the rigorous computation of the dimensions of abstract classes of fractal sets. Dimension theory A `dimension' is a function that assigns a (usually positive, finite real) number to a metric space which attempts to quantify how `large' the set is. In particular, it usually studies how much space the set takes up on small scales. Common examples of `dimensions' are the Hausdorff, packing and box dimensions. Fractals are sets with a complex structure on small scales and thus they may have fractional dimension! Jonathan M. Fraser Assouad type dimensions Dimension theory A `dimension' is a function that assigns a (usually positive, finite real) number to a metric space which attempts to quantify how `large' the set is. In particular, it usually studies how much space the set takes up on small scales. Common examples of `dimensions' are the Hausdorff, packing and box dimensions. Fractals are sets with a complex structure on small scales and thus they may have fractional dimension! People working in dimension theory and fractal geometry are often concerned with the rigorous computation of the dimensions of abstract classes of fractal sets. Jonathan M. Fraser Assouad type dimensions Dimension theory However, fractals and dimensions often crop up in a wide variety of contexts, with links and applications being found in diverse areas of mathematics, for example, geometric measure theory, dynamical systems, probability theory and differential equations. Jonathan M. Fraser Assouad type dimensions Geometric measure theory Figure: A Kakeya needle set. Jonathan M. Fraser Assouad type dimensions Dynamical systems Figure: Chaotic solution to the Lorenz system. Jonathan M. Fraser Assouad type dimensions Probability theory Figure: Fractal percolation. Jonathan M. Fraser Assouad type dimensions Differential equations: fluid dynamics Figure: Kelvin-Helmholtz instability Jonathan M. Fraser Assouad type dimensions How many balls of diameter 1/8 do we need to cover it? In general we will need roughly r −1 balls of diameter r to cover the line segment ::: ::: and the `dimension' of the line segment is 1. Dimension Consider a unit line segment. Jonathan M. Fraser Assouad type dimensions Dimension Consider a unit line segment. How many balls of diameter 1/8 do we need to cover it? In general we will need roughly r −1 balls of diameter r to cover the line segment ::: ::: and the `dimension' of the line segment is 1. Jonathan M. Fraser Assouad type dimensions In general we will need roughly r −1 balls of diameter r to cover the line segment ::: ::: and the `dimension' of the line segment is 1. Dimension Consider a unit line segment. How many balls of diameter 1/8 do we need to cover it? ::: 8 = (1=8)−1 Jonathan M. Fraser Assouad type dimensions ::: and the `dimension' of the line segment is 1. Dimension Consider a unit line segment. How many balls of diameter 1/8 do we need to cover it? ::: 8 = (1=8)−1 In general we will need roughly r −1 balls of diameter r to cover the line segment ::: Jonathan M. Fraser Assouad type dimensions Dimension Consider a unit line segment. How many balls of diameter 1/8 do we need to cover it? ::: 8 = (1=8)−1 In general we will need roughly r −1 balls of diameter r to cover the line segment ::: ::: and the `dimension' of the line segment is 1. Jonathan M. Fraser Assouad type dimensions Heuristically, − dim F Nr (F ) ≈ r : In fact, solving for dim F formally yields the upper and lower box dimensions. Dimension Let (X ; d) be a compact metric space. For any non-empty subset F ⊆ X and r > 0, let Nr (F ) be the smallest number of open sets with diameter less than or equal to r required to cover F . Jonathan M. Fraser Assouad type dimensions In fact, solving for dim F formally yields the upper and lower box dimensions. Dimension Let (X ; d) be a compact metric space. For any non-empty subset F ⊆ X and r > 0, let Nr (F ) be the smallest number of open sets with diameter less than or equal to r required to cover F . Heuristically, − dim F Nr (F ) ≈ r : Jonathan M. Fraser Assouad type dimensions Dimension Let (X ; d) be a compact metric space. For any non-empty subset F ⊆ X and r > 0, let Nr (F ) be the smallest number of open sets with diameter less than or equal to r required to cover F . Heuristically, − dim F Nr (F ) ≈ r : In fact, solving for dim F formally yields the upper and lower box dimensions. Jonathan M. Fraser Assouad type dimensions I Important tool in the study of quasi-conformal mappings, embeddability problems and PDEs I In fact the initial motivation was to prove the following theorem: a metric space can be quasisymmetrically embedded into some Euclidean space if and only if it has finite Assouad dimension. Robinson: Dimensions, Embeddings, and Attractors Heinonen: Lectures on Analysis on Metric Spaces. The Assouad dimension I The Assouad dimension was introduced by Assouad in the 1970s Jonathan M. Fraser Assouad type dimensions I In fact the initial motivation was to prove the following theorem: a metric space can be quasisymmetrically embedded into some Euclidean space if and only if it has finite Assouad dimension. Robinson: Dimensions, Embeddings, and Attractors Heinonen: Lectures on Analysis on Metric Spaces. The Assouad dimension I The Assouad dimension was introduced by Assouad in the 1970s I Important tool in the study of quasi-conformal mappings, embeddability problems and PDEs Jonathan M. Fraser Assouad type dimensions Robinson: Dimensions, Embeddings, and Attractors Heinonen: Lectures on Analysis on Metric Spaces. The Assouad dimension I The Assouad dimension was introduced by Assouad in the 1970s I Important tool in the study of quasi-conformal mappings, embeddability problems and PDEs I In fact the initial motivation was to prove the following theorem: a metric space can be quasisymmetrically embedded into some Euclidean space if and only if it has finite Assouad dimension. Jonathan M. Fraser Assouad type dimensions The Assouad dimension I The Assouad dimension was introduced by Assouad in the 1970s I Important tool in the study of quasi-conformal mappings, embeddability problems and PDEs I In fact the initial motivation was to prove the following theorem: a metric space can be quasisymmetrically embedded into some Euclidean space if and only if it has finite Assouad dimension. Robinson: Dimensions, Embeddings, and Attractors Heinonen: Lectures on Analysis on Metric Spaces. Jonathan M. Fraser Assouad type dimensions until recently... 2011 - Mackay: Assouad dimension of Lalley-Gatzouras carpets 2011 - Olsen: Assouad dimension of graph-directed Moran fractals 2012 - K¨aenm¨aki-Lehrb¨ack-Vuorinen: Relationships with Whitney covers and tubular neighbourhoods 2013 - F.: Assouad dimension of Bara´nskicarpets, quasi-self-similar sets and self-similar sets with overlaps I The Assouad dimension gives `coarse but local' information about a set, unlike the Hausdorff dimension which gives ‘fine but global' information. The Assouad dimension I Minimal attention in the literature on fractals, Jonathan M. Fraser Assouad type dimensions 2011 - Mackay: Assouad dimension of Lalley-Gatzouras carpets 2011 - Olsen: Assouad dimension of graph-directed Moran fractals 2012 - K¨aenm¨aki-Lehrb¨ack-Vuorinen: Relationships with Whitney covers and tubular neighbourhoods 2013 - F.: Assouad dimension of Bara´nskicarpets, quasi-self-similar sets and self-similar sets with overlaps I The Assouad dimension gives `coarse but local' information about a set, unlike the Hausdorff dimension which gives ‘fine but global' information. The Assouad dimension I Minimal attention in the literature on fractals, until recently... Jonathan M. Fraser Assouad type dimensions 2011 - Olsen: Assouad dimension of graph-directed Moran fractals 2012 - K¨aenm¨aki-Lehrb¨ack-Vuorinen: Relationships with Whitney covers and tubular neighbourhoods 2013 - F.: Assouad dimension of Bara´nskicarpets, quasi-self-similar sets and self-similar sets with overlaps I The Assouad dimension gives `coarse but local' information about a set, unlike the Hausdorff dimension which gives ‘fine but global' information.
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