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1、E39?E3m?aa?Vol.39, No. 3 1996d5?ACTA MATHEMATICA SINICAMay, 1996 ?HGIBOTJ?EM?ANY ?v? (?a?b?bB?p?510631) ?f?p?x (?s?b?b?bB?b230026) VQ:?wo?e?b?bg?k?e?o?b ?l?j?bSarkovskiiez?drb?Z?qnd rb?Z?c?drb?Z?b?i?a?4,?s?b? ?u?b?db?h?2bm? ?o?g?Sarkovskiiyt?i 1SP ?t(Warsaw circle) W?i?R2?mC?g?z? W0= (x,y) R2| x = 0
2、, 1 y 1, W1= (x,y) R2| x = 0, 3 y 0?fn(z) L1.y?z,f(z) 2(f) P(f),?dj? p?z,fn(z) W0xz,fn(z) L.r?V? z = 1(z) ? L?X? ? fn(? z) ? L.? ? y (?f).?Yy (?f). F?O? x = 1(x).? ? K? x?hg?L?K = (?K) M.?x?hg?L ?K1 M?K1W0= K.y?x 4(f),m?K1?u 3(f)x?m 0? fm(u) L1.y?u,fm(u) 3(f)P(f),?dj?p?u,fm(u) W0xu,fm(u) K. r?V? u =
3、 1(u) ? K?X? ? fm(? u) ? K.j?pS?S? u (?f).?X?J ? x 2(?f).?Yx (2(?f).yYy?2(?f) = P(?f),m?x (P(?f),?X?D? KUX? r?X?J4(f) P(f). ?u?vnAP(f) 4(f)?E?K? ? ?3 ?f : W W?tW?hgE?u?f?lF?qF ?K?P(f) = R(f). WCj?K?2?X? P(f) R(f) 4(f) = P(f). ?YP(f) = R(f). R?3?f : X X?hgE?u?n?X?hg?RI? h(f) =sup xR(f) h(f |(x,f). WC
4、?hg?E?S 15. R?4?f : I I?hgE?u?n?I?IR?hgE?tf?l F?lP?2?Z?V?Whgx I, (x,f)? WC?x P(f).?x P(f)?Ey(x,f)?hg?z?mC?x P(f)P(f). j?K?C,?f?MFe I? (f2n(x) x)(x e) 0x(f2n+1(x) e)(x e) 0?r?X? f(x,f) y I | y e) = (x,f) y I | y e. ?QE?u?f2,j?K?B, f2?lF?lgP?2?Z?m?dj?K?C? ?e1,e2 I?e1 0. ?oc? z R(?f)m? ? f |(? z,?f)= ?
5、 f |(? z,?f). ?mh(?f |(? z,?f) 0. Oz = (? z).?X?z R(f),m?( |(? z,?f) ) (?f |(? z,?f) = (f |(z,f) ( |(?z,?f). y?(? z,?f)?m |(? z,?f): (? z, ? f) (z,f)? hg?h?mh(f |(z,f) = h(?f |(? z,?f) 0. ?j?s?3, h(f) 0. ?J?K?cV?c?X?D?V?yy R(f),?y1 (R(?f)? ?(y,f) = (y1,f).?r?V?y R(f)?X?y,f(y), W0,?r? |0,1/8: 0,1/8 W
6、0?hg?h?my (R(?f);?n 0?fn(y) / W0,y? |(1/8,1): (1/8,1) W W0?hg?hmy1= fn(y) (R(?f),?q(y,f) = (y1,f). F?f?lF?lP?2?Z?o? ? f?lF?lgP?2?Z? ?p?sy R(f),?X?h(f |(y,f).j?pS?X?Ky (R(?f),?q O? y = 1(y) R(?f).j?s?4, (? y,?f)?hg?z?J = a,b 0,1)?(? y,?f)? ?0,1)?v(? y,?f)?O?E?Kqu? : 0,1) J?K?eV? c?oc?P( ? f) P(?f)x(?
7、 y,?f) = (? y, ? f),?mu? ? f : J J?l F?lP?2?Z?Yj?K?Dxs?3,?X?h(?f |(? y,?f) = 0. ?v?K? eV?c?S?h(f |(y,f) = 0.?j?s?3,?h(f) = 0. ?KL 3mX?B?u?F?v?RNBW?299 1 Nadler S B Jr. Continuum Theory. New York: Marcel Dekker, Inc, 1992. 2 Block L, Coppel W A. Dynamics in one dimension.Lecture Notes in Mathematics
8、No.1513, Berlin: Springer, 1992. 3Y?JTG?w?a?b?1984, 29: 518520. 4 Minc P, Transue W R R. Sarkovskii theorem for hereditarily decomposable chainable continua. Trans Amer Math Soc, 1989, 315: 173188. 5Y?JTw?OC?B?a?k?n?R?b?1988, 17: 110. 6?Y?w?J?i?OCB?Q?1992. 7 Adler R L, Koheim A G, McAndrew M H. Topo
9、logical entropy. Trans Amer Math Soc, 1965, 4: 309319. 8 Sarkovskii A N. Non-wandering Points and the Centre of a Continuous Mapping of the Line into Itself. Dopovidi Akad Nauk Ukrain RSR, 1964. 865868. 9Y?W?G?w?f, (f | (f) = P(f).?b?1982, 27: 513514. 10 Sarkovskii A N. Coexistence of cycles of a co
10、ntinuous mapping of the line into itself. Ukrain Mat Z, 1964, 16: 6171. 11 Xiong J. Sets of recurrent points of continuous maps of the interval.Proc Amer Math Soc, 1985, 95: 491494. 12 Misiurewicz M. Horseshoes for mappings of the interval.Bull Acad Polon Sci Ser Sci Math, 1979, 27: 167169. 13?N?P?i
11、h?s?b(A?), 1985, 10: 883889. 14Y?MisuirewiczihL?f?s?b?bb?1989, 19: 2124. 15 Walters P. An Introduction to Ergodic Theory. New York: Springer-Verlag, 1982. Some Dynamical Properties of Continuous Maps on the Warsaw Circle Xiong Jincheng (Department of Mathematics, South China Normal University, Guang
12、zhou 510631, China) Ye XiangdongZhang ZhiqiangHuang Jun (Department of Mathematics, University of Science and Technology of China, Hefei 230026, China) Abstract: In the present paper we study the dynamical properties of continuous maps on the Warsaw circle and show that for such a map a theorem simi
13、lar to the Sarkovskii Theorem for interval maps holds, the closure of the set of periodic points coincides with the closure of the set of recurrent points, the depth of the center is not greater than 4, and the topological entropy is zero if and only if the periods of periodic points are the powers of 2. Keywords: Warsaw circle, dynamical property, Sarkovskii space, center, depth of the center
