Bidimensional linear systems

with singular dynamics

Authors:
Sylvia Novo and Rafael Obaya

Journal:
Proc. Amer. Math. Soc. **124** (1996), 3163-3172

MSC (1991):
Primary 28D05, 58F11; Secondary 34C11

DOI:
https://doi.org/10.1090/S0002-9939-96-03411-9

MathSciNet review:
1328366

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Abstract | References | Similar Articles | Additional Information

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.

**1.**A. I. Alonso and R. Obaya,*Ergodic structure of bidimensional linear systems with a linear invariant measure*, Univ. of Valladolid preprint, 1993.**2.**R. Ellis and R. Johnson,*Topological dynamics and linear differential systems*, J. Differential Equations**44**(1982), no. 1, 21--39.MR**83c:54058****3.**A. M. Fink,*Almost periodic differential equations*, Lecture Notes in Mathematics**377**, Springer-Verlag, Heidelberg, 1974. MR**57:792****4.**H. Furstenberg,*Strict ergodicity and transformations of the torus*, Amer. J. Math.**85**(1961), 573--601. MR**24:A3263****5.**S. Glasner and B. Weiss,*On the construction of minimal skew products*, Israel J. Math**34**(1979), no. 4, 321--336. MR**82f:54068****6.**R. Johnson,*On a Floquet theory for almost-periodic, two-dimensional linear systems*, J. Differential Equations**37**(1980), 184--204. MR**81j:58069****7.**------,*Almost-periodic functions with unbounded integral*, Pacific J. Math.**87**(1980), 347--362. MR**82e:42013****8.**------,*Two-dimensional, almost periodic linear systems with proximal and recurrent behavior*, Proc. Amer. Math. Soc.**82**(1981), no. 3, 417--422. MR**83c:34048****9.**------,*Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients*, J. Differential Equations**61**(1986), 54--78. MR**87e:47065****10.**M. G. Nerurkar,*Recurrent-proximal linear differential systems with almost periodic coefficients*, Proc. Amer. Math. Soc.**100**(1987), no. 4, 729--743. MR**88i:58150****11.**------,*On the construction of smooth ergodic skew-products*, Ergod. Th. Dynam. Sys.**8**(1988), 311--326. MR**89m:58123****12.**S. Novo and R. Obaya,*An ergodic classification of bidimensional linear systems*, to be published in J. Dynamics Differential Equations.**13.**R. J. Sacker and G. R. Sell,*Lifting properties in skew-product flows with applications to differential equations*, Mem. Amer. Math. Soc.**190**, Amer. Math. Soc., Providence, 1977. MR**56:6632****14.**S. Schwartzman,*Asymptotic cycles*, Ann. of Math**66**(1957), 270--284. MR**19:568**

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Additional Information

**Sylvia Novo**

Affiliation:
Departamento de Matematica Aplicada a la Ingenieria, E.T.S. de Ingenieros Industriales, Universidad de Valladolid, 47011, Valladolid, Spain

Email:
sylnov@wmatem.eis.uva.es

**Rafael Obaya**

Affiliation:
Departamento de Matematica Aplicada a la Ingenieria, E.T.S. de Ingenieros Industriales, Universidad de Valladolid, 47011, Valladolid, Spain

Email:
rafoba@wmatem.eis.uva.es

DOI:
https://doi.org/10.1090/S0002-9939-96-03411-9

Keywords:
Ergodic sheet,
recurrent-proximal system,
uniquely ergodic flow,
singular dynamics

Received by editor(s):
November 4, 1994

Received by editor(s) in revised form:
April 6, 1995

Additional Notes:
Partially supported by Junta de Castilla y León under project VA57/94

Communicated by:
Mary Rees

Article copyright:
© Copyright 1996
American Mathematical Society