Positive operators and Hausdorff dimension of invariant sets

Nussbaum, Roger; Priyadarshi, Amit; Verduyn Lunel, Sjoerd

doi:10.1090/S0002-9947-2011-05484-X

Positive operators and Hausdorff dimension of invariant sets
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by Roger D. Nussbaum, Amit Priyadarshi and Sjoerd Verduyn Lunel

Trans. Amer. Math. Soc. 364 (2012), 1029-1066

DOI: https://doi.org/10.1090/S0002-9947-2011-05484-X

Published electronically: October 5, 2011

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Abstract:

In this paper we obtain theorems which give the Hausdorff dimension of the invariant set for a finite family of contraction mappings which are “infinitesimal similitudes” on a complete, perfect metric space. Our work generalizes the graph-directed construction of Mauldin and Williams (1988) and is related in its general setting to results of Schief (1996), but differs crucially in that the mappings need not be similitudes. We use the theory of positive linear operators and generalizations of the Krein-Rutman theorem to characterize the Hausdorff dimension as the unique value of $\sigma > 0$ for which $r(L_\sigma )=1$, where $L_\sigma$, $\sigma >0$, is a naturally associated family of positive linear operators and $r(L_\sigma )$ denotes the spectral radius of $L_\sigma$. We also indicate how these results can be generalized to countable families of infinitesimal similitudes. The intent here is foundational: to derive a basic formula in its proper generality and to emphasize the utility of the theory of positive linear operators in this setting. Later work will explore the usefulness of the basic theorem and its functional analytic setting in studying questions about Hausdorff dimension.

References

F. F. Bonsall, Linear operators in complete positive cones, Proc. London Math. Soc. (3) 8 (1958), 53–75. MR 92938, DOI 10.1112/plms/s3-8.1.53
Richard T. Bumby, Hausdorff dimensions of Cantor sets, J. Reine Angew. Math. 331 (1982), 192–206. MR 647383, DOI 10.1515/crll.1982.331.192
Richard T. Bumby, Hausdorff dimension of sets arising in number theory, Number theory (New York, 1983–84) Lecture Notes in Math., vol. 1135, Springer, Berlin, 1985, pp. 1–8. MR 803348, DOI 10.1007/BFb0074599
Gerald Edgar, Measure, topology, and fractal geometry, 2nd ed., Undergraduate Texts in Mathematics, Springer, New York, 2008. MR 2356043, DOI 10.1007/978-0-387-74749-1
K. J. Falconer, The Geometry of Fractal Sets, Cambridge Univ. Press, Cambridge, 1985.
Kazimierz Goebel and Simeon Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, Inc., New York, 1984. MR 744194
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
M. G. Kreĭn and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspehi Matem. Nauk (N.S.) 3 (1948), no. 1(23), 3–95 (Russian). MR 0027128
John Mallet-Paret and Roger D. Nussbaum, Eigenvalues for a class of homogeneous cone maps arising from max-plus operators, Discrete Contin. Dyn. Syst. 8 (2002), no. 3, 519–562. MR 1897866, DOI 10.3934/dcds.2002.8.519
—, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theory and Applications 7 (2010), 103–143.
Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
R. Daniel Mauldin and Mariusz Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3) 73 (1996), no. 1, 105–154. MR 1387085, DOI 10.1112/plms/s3-73.1.105
R. Daniel Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988), no. 2, 811–829. MR 961615, DOI 10.1090/S0002-9947-1988-0961615-4
S. Mazur and S. Ulam, Sur les transformations isométriques d’espaces vectoriels normés, C. R. Acad. Sci. Paris 194 (1932), 946–948.
P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc. 42 (1946), 15–23. MR 14397, DOI 10.1017/s0305004100022684
J. R. Munkres, Topology, second ed., Prentice Hall, 2000.
Roger D. Nussbaum, Periodic solutions of some nonlinear integral equations, Dynamical systems (Proc. Internat. Sympos., Univ. Florida, Gainesville, Fla., 1976) Academic Press, New York, 1977, pp. 221–249. MR 0463844
Roger D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Kreĭn-Rutman theorem, Fixed point theory (Sherbrooke, Que., 1980) Lecture Notes in Math., vol. 886, Springer, Berlin-New York, 1981, pp. 309–330. MR 643014
Roger Nussbaum, Periodic points of positive linear operators and Perron-Frobenius operators, Integral Equations Operator Theory 39 (2001), no. 1, 41–97. MR 1806843, DOI 10.1007/BF01192149
C. E. Rickart, General Theory of Banach Algebras, Robert E. Krieger Publishing Co., New York, 1974.
Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039
Andreas Schief, Self-similar sets in complete metric spaces, Proc. Amer. Math. Soc. 124 (1996), no. 2, 481–490. MR 1301047, DOI 10.1090/S0002-9939-96-03158-9
Jussi Väisälä, A proof of the Mazur-Ulam theorem, Amer. Math. Monthly 110 (2003), no. 7, 633–635. MR 2001155, DOI 10.2307/3647749