By Bloch S.J., et al. (eds.)
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Extra info for Applications of algebraic K-theory to algebraic geometry and number theory, Part 1
That cp; carries infinitesimal circles to infinitesimal circles. This is important because Bowen's method works essentially by using the dynamics of the maps to take efficient coverings to efficient coverings by smaller sets. H the maps cp; are affine then they take circles to ellipses: coverings with ellipses are usually poor which is why Bowen's method does not work in a non-conformal situation. We'll say more about this problem soon. (e) Recurrent curves The fractal recurrent curves of Dekking [Del], [De2] (see his lectures) can be put into the framework of example (b).
M. Dekking (1982), Recurrent sets: a fractal formalism, Delft University of Technology Report of the Department of Mathematics 82-32. J. Falconer (1985), The Geometry of Fractal Sets, Cambridge University Press,Cambridge. J. Falconer (1988), A subadditive thermodynamic formalism for mixing repellers, J. Phys. 21A, 737-742. E. Hutchinson (1981), Fractals and self-similarity, Indiana Univ. Math. J. 30, 713747. [Ke] M. Math. 16, 309-324. [Ma) B. H. Freeman [Me) C. McMullen (1984), The Hausdorff dimension of general Sierpnski carpets, Nagoya Math J.
The following Lemma is easy to check. Xn-11 < a -1 () 2 llxo .. xn-11 > 0, where I~2. xn_1 is a Cantor interval of C;(i = 1, 2). :, n > 0 and i = 1,2. 2 is very surprising because it breaks two very general rules of dynamical systems theory: (1) Differentiable conjugacy classes are usually hard to characterize. (2) Many results are proven first for "linear" systems and then with various degrees of difficulty are extended to "non-linear" systems. ··:• ' ' I ' ' I o I ,.. ··:·' : .. · .... · . ·'.
Applications of algebraic K-theory to algebraic geometry and number theory, Part 1 by Bloch S.J., et al. (eds.)