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A description of harmonic functions via properties of the group representation of the Cayley tree. (English) Zbl 1187.43005
Math. Notes 79, No. 3, 399-407 (2006); translation from Mat. Zametki 79, No. 3, 434-443 (2006).
Summary: We introduce a natural generalization of the notion of harmonic functions on a Cayley tree and use some properties of the group representation of the Cayley tree to describe the set of harmonic functions periodic with respect to normal subgroups of finite index.

43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
20E08 Groups acting on trees
05C05 Trees
Full Text: DOI
[1] N. N. Ganikhodzhaev, ”The group representation and automorphisms of the Cayley tree,” Dokl. Akad. Nauk Resp. Uzbekistan (1994), no. 4, 3–5. · Zbl 0838.93062
[2] U. A. Rozikov, ”Representability and some applications of trees,” Mat. Zametki [Math. Notes], 72 (2002), no. 4, 516–527. · Zbl 1028.82008
[3] U. A. Rozikov, ”Structure of partitions of the Cayley tree and their applications to the description of periodic Gibbs distributions,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 112 (1997), no. 1, 170–175. · Zbl 0978.82506
[4] U. A. Rozikov, ”Construction of uncountably many Gibbs measures in the inhomogeneous Ising model,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 118 (1999), no. 1, 95–104. · Zbl 0957.82007
[5] N. N. Ganikhodzhaev and U. A. Rozikov, ”Description of nonperiodic extreme Gibbs measures of some models on a Cayley tree,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 111 (1997), no. 1, 109–117. · Zbl 0964.82504
[6] N. N. Ganikhodzhaev and U. A. Rozikov, ”Group representations of the Cayley forest and some of its applications,” Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 67 (2003), no. 1, 17–27. · Zbl 1065.82005
[7] U. A. Rozikov, ”Random walks in random media on metric groups,” Mat. Zametki [Math. Notes], 67 (2000), no. 1, 129–135. · Zbl 0969.60011
[8] B. Bollobás, Modern Graph Theory, Graduate Texts in Math., vol. 184, Springer-Verlag, New York, 1998.
[9] N. N. Ganikhodjaev and U. A. Rozikov, ”On disordered phase in the ferromagnetic Potts model on the Bethe lattice,” Osaka J. Math., 37 (2000), no. 2, 373–383. · Zbl 0980.82006
[10] U. A. Rozikov, ”Random walks in random media on a Cayley tree,” Ukrain. Mat. Zh. [Ukrainian Math. J.], 53 (2001), no. 10, 1391–1401. · Zbl 1007.34033 · doi:10.1023/A:1014350221388
[11] J. Aczel, Lectures on Functional Equations and Their Applications, Academic Press, New York–London, 1966.
[12] M. Kuczma, ”Functional Equations in a Single Variable,” in: Monografie Matematyczne, vol. 46, Państwowe Wydawnictwo Naukowe, Warsaw, 1968. · Zbl 0196.16403
[13] M. Kuczma, B. Choczewski, and R. Ger, Iterative Functional Equations, Cambridge Univ. Press, Cambridge, 1990. · Zbl 0703.39005
[14] Sh. A. Shoyusupov, On the Index of Normal Subgroups in the Group Representation of the Cayley Tree [in Russian], Magister Dissertation, Uzbekistan National University, Tashkent, 2003.
[15] Ya. G. Sinai, Topics in Ergodic Theory, Princeton Univ. Press, Princeton, 1994. · Zbl 0805.58005
[16] A. N. Sharkovskii, ”Coexistence of cycles of a continuous mapping of the line into itself,” Ukrain. Mat. Zh. [Ukrainian Math. J.], 16 (1964), no. 1, 61–71.
[17] A. N. Sharkovskii, ”A classification of fixed points,” Ukrain. Mat. Zh. [Ukrainian Math. J.], 17 (1965), no. 5, 80–95. · Zbl 0174.54303 · doi:10.1007/BF02527088
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