Critical points of almost periodic functions

• Autorzy: Janeczko S. , Zając M.
• Języki publikacji: EN

Abstrakty

EN

The spaces of almost periodic and quasi-periodic functions are considered. We show that the local critical point properties can continue along the domain with the relative density property, and vanishing of the appropriate derivatives is only a "virtual" property. The stable virtual critical points of quasi-periodic functions are classified and the density result for the virtual critical points of type A(k) is proved. The special construction of almost periodic functions with prescribed properties is provided and applied to construct an almost periodic function which nas no density property of the nondegenerate critical points, althougth all its derivatives are almost periodic functions.

Słowa kluczowe

EN

almost periodic functions; quasi-periodic functions; critical points; stable critical points

PL

funkcja prawie okresowa; funkcja quasi-okresowa; punkt krytyczny; stabilny punkt krytyczny

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2003
• Tom: Vol. 51, no 1
• Strony: 107--120
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Janeczko S.

• Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warszawa, Poland

autor

Zając M.

• Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warszawa, Poland
• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland

Bibliografia

• [1] V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko, Singularities of differentiable maps, 1, Birkhäuser, Boston 1985.
• [2] V. I. Arnold, Remarks on quasicrystallic symmetries, Phys. D, 33 (1988) 21-25.
• [3] S. Bochner, Abstrakte fastperiodische Funktionen, Acta Math., 61 (1933) 149-184.
• [4] H. Bohr, Zur Theorie der fastperiodischen Funktionen, Acta Math., 45 (1925) 29-127.
• [5] N. G. de Bruijn, Algebraic theory of Penrose non-periodic tilings, Nederl. Akad. Wetensch. Proc., A84 (1981) 39-66.
• [6] C. Corduneanu, Almost periodic functions, Chelsea Publ. Co., New York 1989.
• [7] M. Golubitsky, V. Guillemin, Stable mappings and their singularities, GTM 14, Springer-Verlag, New York 1973.
• [8] S. M. Gusein-Zade, The number of critical points of a quasi-periodic potential, Funct. Anal. Appl., 23 (2) (1989) 129-130.
• [9] J. Martinet, Singularities of smooth functions and maps, Cambridge Univ. Press, Cambridge 1982.
• [10] R. V. Moody, J. Patera, Dynamical generation of quasicrystals, Lett. Math. Phys., 36 (1996) 291-300.
• [11] R. Penrose, Pentaplexity, Eureka, 39 (1978) 16-22.
• [12] G. Wassermann, Stability of unfoldings, Lecture Notes in Math., 393, Springer-Verlag, Berlin 1974.
• [13] S. Zaidman, Almost periodic functions in abstract spaces, Pitman Publ. Ltd., London 1985.
• [14] M. Zając, Quasicrystals and almost periodic functions, An. Polon. Math., LXXII (3) (1999) 251-259.