Bidimensional linear systems with singular dynamics

Novo, Sylvia; Obaya, Rafael

doi:10.1090/S0002-9939-96-03411-9

Bidimensional linear systems with singular dynamics
HTML articles powered by AMS MathViewer

by Sylvia Novo and Rafael Obaya PDF

Proc. Amer. Math. Soc. 124 (1996), 3163-3172 Request permission

Abstract:

We analyze a class of bidimensional linear systems for which the following characteristics are generic: the system is recurrent and there exists a unique ergodic measure which is concentrated in one ergodic sheet. The trajectories exhibit an oscillatory behaviour from one to the other side of the ergodic sheet which assures the proximal character of the flow.

References

A. I. Alonso and R. Obaya, Ergodic structure of bidimensional linear systems with a linear invariant measure, Univ. of Valladolid preprint, 1993.
Robert Ellis and Russell A. Johnson, Topological dynamics and linear differential systems, J. Differential Equations 44 (1982), no. 1, 21–39. MR 651685, DOI 10.1016/0022-0396(82)90023-7
A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR 0460799, DOI 10.1007/BFb0070324
H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math. 83 (1961), 573–601. MR 133429, DOI 10.2307/2372899
S. Glasner and B. Weiss, On the construction of minimal skew products, Israel J. Math. 34 (1979), no. 4, 321–336 (1980). MR 570889, DOI 10.1007/BF02760611
Russell A. Johnson, On a Floquet theory for almost-periodic, two-dimensional linear systems, J. Differential Equations 37 (1980), no. 2, 184–205. MR 587221, DOI 10.1016/0022-0396(80)90094-7
Russell A. Johnson, Almost-periodic functions with unbounded integral, Pacific J. Math. 87 (1980), no. 2, 347–362. MR 592741, DOI 10.2140/pjm.1980.87.347
Russell A. Johnson, Two-dimensional, almost periodic linear systems with proximal and recurrent behavior, Proc. Amer. Math. Soc. 82 (1981), no. 3, 417–422. MR 612732, DOI 10.1090/S0002-9939-1981-0612732-9
Russell A. Johnson, Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Differential Equations 61 (1986), no. 1, 54–78. MR 818861, DOI 10.1016/0022-0396(86)90125-7
Mahesh G. Nerurkar, Recurrent-proximal linear differential systems with almost periodic coefficients, Proc. Amer. Math. Soc. 100 (1987), no. 4, 739–743. MR 894447, DOI 10.1090/S0002-9939-1987-0894447-0
Mahesh G. Nerurkar, On the construction of smooth ergodic skew-products, Ergodic Theory Dynam. Systems 8 (1988), no. 2, 311–326. MR 951272, DOI 10.1017/S0143385700004454
S. Novo and R. Obaya, An ergodic classification of bidimensional linear systems, to be published in J. Dynamics Differential Equations.
Robert J. Sacker and George R. Sell, Lifting properties in skew-product flows with applications to differential equations, Mem. Amer. Math. Soc. 11 (1977), no. 190, iv+67. MR 448325, DOI 10.1090/memo/0190
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335