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A Positive Operator-Valued Measure for an Iterated Function System

Abstract

Given an iterated function system (IFS) on a complete and separable metric space \(Y\), there exists a unique compact subset \(X \subseteq Y\) satisfying a fixed point relation with respect to the IFS. This subset is called the attractor set, or fractal set, associated to the IFS. The attractor set supports a specific Borel probability measure, called the Hutchinson measure, which itself satisfies a fixed point relation. P. Jorgensen generalized the Hutchinson measure to a projection-valued measure, under the assumption that the IFS does not have essential overlap (Jorgensen in Adv. Appl. Math. 34(3):561–590, 2005; Operator Theory, Operator Algebras, and Applications, pp. 13–26, 2006). In previous work, we developed an alternative approach to proving the existence of this projection-valued measure (Davison in Acta Appl. Math. 140(1):11–22, 2015; Acta Appl. Math. 140(1):23–25, 2015; Generalizing the Kantorovich metric to projection-valued measures: with an application to iterated function systems, 2015). The situation when the IFS exhibits essential overlap has been studied by Jorgensen and colleagues in Jorgenson et al. (J. Math. Phys. 48(8):083511, 35, 2007). We build off their work to generalize the Hutchinson measure to a positive-operator valued measure for an arbitrary IFS, that may exhibit essential overlap. This work hinges on using a generalized Kantorovich metric to define a distance between positive operator-valued measures. It is noteworthy to mention that this generalized metric, which we use in our previous work as well, was also introduced by R.F. Werner to study the position and momentum observables, which are central objects of study in the area of quantum theory (Werner in J. Quantum Inf. Comput. 4(6):546–562, 2004). We conclude with a discussion of Naimark’s dilation theorem with respect to this positive operator-valued measure, and at the beginning of the paper, we prove a metric space completion result regarding the classical Kantorovich metric.

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References

  1. 1.

    Akerlund-Bistrom, C.: A generalization of Hutchinson distance and applications. Random Comput. Dyn. 5(2–3), 159–176 (1997)

  2. 2.

    Barnsley, M.: Fractals Everywhere, 2nd edn. Academic Press, San Diego (1993)

  3. 3.

    Billingsley, P.P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)

  4. 4.

    Bogachev, V.: Measure Theory: Volume II. Springer, New York (2000)

  5. 5.

    Conway, J.: A Course in Functional Analysis, 2nd edn. Springer, New York (2000)

  6. 6.

    Davison, T.: Generalizing the Kantorovich metric to projection-valued measures. Acta Appl. Math. 140(1), 11–22 (2015). https://doi.org/10.1007/s10440-014-9976-y

  7. 7.

    Davison, T.: Erratum to: Generalizing the Kantorovich metric to projection-valued measures. Acta Appl. Math. 140(1), 23–25 (2015). https://doi.org/10.1007/s10440-015-0018-1

  8. 8.

    Davison, T.: Generalizing the Kantorovich metric to projection-valued measures: with an application to iterated function systems. University of Colorado at Boulder, ProQuest Dissertations Publishing (2015)

  9. 9.

    Hutchinson, J.: Fractals and self similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)

  10. 10.

    Jorgensen, P.: Iterated function systems, representations, and Hilbert space. Int. J. Math. 15, 813 (2004)

  11. 11.

    Jorgensen, P.: Measures in wavelet decompositions. Adv. Appl. Math. 34(3), 561–590 (2005)

  12. 12.

    Jorgensen, P.: Use of operator algebras in the analysis of measures from wavelets and iterated function system. In: Operator Theory, Operator Algebras, and Applications. Contemp. Math., vol. 414, pp. 13–26. Am. Math. Soc., Providence (2006)

  13. 13.

    Jorgensen, P.: The measure of a measurement. J. Math. Phys. 48(10), 103506, 15 (2007)

  14. 14.

    Jorgenson, P., Kornelson, K., Shuman, K.: Harmonic analysis of iterated function systems with overlap. J. Math. Phys. 48(8), 083511, 35 (2007)

  15. 15.

    Kravchenko, A.S.: Completeness of the space of separable measures in the Kantorovich-Rubinshtein metric. Sib. Math. J. 47(1), 68–76 (2006)

  16. 16.

    Kribs, D.W.: A quantum computing primer for operator theorists. Linear Algebra Appl. 400, 147–167 (2005)

  17. 17.

    Kribs, D.W., Laflamme, R., Poulin, D., Lesosky, M.: Operator quantum error correction. Quantum Inf. Comput. 6(4–5), 382–398 (2006)

  18. 18.

    Werner, R.F.: The uncertainty relation for joint measurement of position and momentum. Quantum Inf. Comput. 4(6), 546–562 (2004)

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Acknowledgements

The author would like to thank Judith Packer (University of Colorado) for her careful review of this material, and her guidance on this research.

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Correspondence to Trubee Davison.

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Davison, T. A Positive Operator-Valued Measure for an Iterated Function System. Acta Appl Math 157, 1–24 (2018). https://doi.org/10.1007/s10440-018-0161-6

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Keywords

  • Kantorovich metric
  • Operator-valued measure
  • Cuntz algebra
  • Fixed point

Mathematics Subject Classification

  • 46C99
  • 46L05