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Register a new userHidden Positivity and a New Approach to Numerical Computation of Hausdorff Dimension: Higher Order Methods
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Richard Falk
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In [14], the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend this approach to incorporate high order approximation methods. We again rely on the fact that we can associate to the IFS a parametrized family of positive, linear, Perron-Frobenius operators $L_s$, an idea known in varying degrees of generality for many years. Although $L_s$ is not compact in the setting we consider, it possesses a strictly positive $C^m$ eigenfunction $v_s$ with eigenvalue $R(L_s)$ for arbitrary $m$ and all other points $z$ in the spectrum of $L_s$ satisfy $|z| le b$ for some constant $b < R(L_s)$. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value $s=s_*$ for which $R(L_s) =1$. This eigenvalue problem is then approximated by a collocation method at the extended Chebyshev points of each subinterval using continuous piecewise polynomials of arbitrary degree $r$. Using an extension of the Perron theory of positive matrices to matrices that map a cone $K$ to its interior and explicit a priori bounds on the derivatives of the strictly positive eigenfunction $v_s$, we give rigorous upper and lower bounds for the Hausdorff dimension $s_*$, and these bounds converge rapidly to $s_*$ as the mesh size decreases and/or the polynomial degree increases.
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We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case, our methods require only C^3 regularity of the maps in the IFS. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of positive, linear, Perron-Frobenius operators L_s. The operators L_s can typically be studied in many different Banach spaces. Here, unlike most of the literature, we study L_s in a Banach space of real-valued, C^k functions, k >= 2; and we note that L_s is not compact, but has a strictly positive eigenfunction v_s with positive eigenvalue lambda_s equal to the spectral radius of L_s. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value s=s_* for which lambda_s =1. This eigenvalue problem is then approximated by a collocation method using continuous piecewise linear functions (in one dimension) or bilinear functions (in two dimensions). Using the theory of positive linear operators and explicit a priori bounds on the derivatives of the strictly positive eigenfunction v_s, we give rigorous upper and lower bounds for the Hausdorff dimension s_*, and these bounds converge to s_* as the mesh size approaches zero.
We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case that we consider here, our methods require only $C^3$ regularity of the maps in the IFS. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of positive, linear, Perron-Frobenius operators $L_s$. The operators $L_s$ can typically be studied in many different Banach spaces. Here, unlike most of the literature, we study $L_s$ in a Banach space of real-valued, $C^k$ functions, $k ge 2$. We note that $L_s$ is not compact, but has essential spectral radius $rho_s$ strictly less than the spectral radius $lambda_s$ and possesses a strictly positive $C^k$ eigenfunction $v_s$ with eigenvalue $lambda_s$. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value $s=s_*$ for which $lambda_s =1$. This eigenvalue problem is then approximated by a collocation method using continuous piecewise linear functions. Using the theory of positive linear operators and explicit a priori bounds on the derivatives of the strictly positive eigenfunction $v_s$, we give rigorous upper and lower bounds for the Hausdorff dimension $s_*$, and these bounds converge to $s_*$ as the mesh size approaches zero.
In a previous paper, dealing with Applications in $mathbb{R}^1$, the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS and studied some applications in one dimension. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of positive, linear, Perron-Frobenius operators $L_s$. In our context, $L_s$ is studied in a space of $C^m$ functions and is not compact. Nevertheless, it is has a strictly positive $C^m$ eigenfunction $v_s$ with positive eigenvalue $lambda_s$ equal to the spectral radius of $L_s$. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value $s=s_*$ for which $lambda_s =1$. To compute the Hausdorff dimension of an IFS associated to complex continued fractions, (which may arise from an infinite iterated function system), we again approximate the eigenvalue problem by a collocation method, but now using continuous piecewise bilinear functions. Using the theory of positive linear operators and explicit a priori bounds on the partial derivatives of the strictly positive eigenfunction $v_s$, we are able to give rigorous upper and lower bounds for the Hausdorff dimension $s_*$, and these bounds converge to $s_*$ as the mesh size approaches zero. We also demonstrate by numerical computations that improved estimates can be obtained by the use of higher order piecewise tensor product polynomial approximations, although the present theory does not guarantee that these are strict upper and lower bounds. An important feature of our approach is that it also applies to the much more general problem of computing approximations to the spectral radius of positive transfer operators, which arise in many other applications.
In this paper we prove the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent $mu$ $in$ (1/2, 1) is 2(1 -- $mu$) when $mu$ $ge$ $sqrt$ 2/2, whereas for $mu$ textless{} $sqrt$ 2/2 it is greater than 2(1 -- $mu$) and at most (3 -- 2$mu$)(1 -- $mu$)/(1 + $mu$ + $mu$ 2). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) when $mu$ tends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. We also prove a lower bound on the packing dimension that is strictly greater than the Hausdorff dimension for $mu$ $ge$ 0.565. .. .
We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension. Let a be any real number greater than or equal to 2 and let b be any non-negative real less than or equal to 2/a. We show that there is a Cantor-like set with Hausdorff dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.
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