The Hausdorff dimension of some random invariant graphs

Moss, A and Walkden, C. P. (2011) The Hausdorff dimension of some random invariant graphs. [MIMS Preprint]

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Abstract

Weierstrass� example of an everywhere continuous but nowhere differentiable function is given by w(x) = P1 n=0 n cos 2bnx where 2 (0, 1), b 2, b > 1. There is a well-known and widely accepted, but as yet unproven, formula for the Hausdorff dimension of the graph of w. Hunt [H] proved that this formula holds almost surely on the addition of a random phase shift. The graphs of Weierstrass-type functions appear as repellers for a certain class of dynamical system; in this note we prove formulae analogous to those for random phase shifts of w(x) but in a dynamic context. Let T : S1 ! S1 be a uniformly expanding map of the circle. Let : S1 ! (0, 1), p : S1 ! R and define the function w(x) = P1 n=0 (x)(T(x)) · · · (Tn�1(x))p(Tn(x)). The graph of w is a repelling invariant set for the skew-product transformation T(x, y) = (T(x), (x)�1(y�p(x))) on S1�R and is continuous but typically nowhere differentiable. With the addition of a random phase shift in p, and under suitable hypotheses including a partial hyperbolicity assumption on the skew-product, we prove an almost sure formula for the Hausdorff dimension of the graph of w using a generalisation of techniques from [H] coupled with thermodynamic formalism.

Item Type:	MIMS Preprint
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theory
Depositing User:	Ms Lucy van Russelt
Date Deposited:	14 May 2011
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/1619

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