Conditions for the Existence of SBR Measures for ``Almost Anosov'' Diffeomorphisms
Author:
Huyi Hu
Journal:
Trans. Amer. Math. Soc. 352 (2000), 23312367
MSC (1991):
Primary 58F11, 58F15; Secondary 28D05
Published electronically:
December 10, 1999
MathSciNet review:
1661238
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A diffeomorphism of a compact manifold is called ``almost Anosov'' if it is uniformly hyperbolic away from a finite set of points. We show that under some nondegeneracy condition, every almost Anosov diffeomorphism admits an invariant measure that has absolutely continuous conditional measures on unstable manifolds. The measure is either finite or infinite, and is called SBR measure or infinite SBR measure respectively. Therefore, tends to either an SBR measure or for almost every with respect to Lebesgue measure. ( is the Dirac measure at .) For each case, we give sufficient conditions by using coefficients of the third order terms in the Taylor expansion of at .
 [B]
Rufus
Bowen, Equilibrium states and the ergodic theory of Anosov
diffeomorphisms, Lecture Notes in Mathematics, Vol. 470,
SpringerVerlag, Berlin, 1975. MR 0442989
(56 #1364)
 [BY]
Michael
Benedicks and LaiSang
Young, SinaĭBowenRuelle measures for certain Hénon
maps, Invent. Math. 112 (1993), no. 3,
541–576. MR 1218323
(94e:58074), http://dx.doi.org/10.1007/BF01232446
 [C]
Maria
Carvalho, SinaĭRuelleBowen measures for
𝑁dimensional [𝑁 dimensions] derived from Anosov
diffeomorphisms, Ergodic Theory Dynam. Systems 13
(1993), no. 1, 21–44. MR 1213077
(94h:58102), http://dx.doi.org/10.1017/S0143385700007185
 [HY]
Hu
Yi Hu and LaiSang
Young, Nonexistence of SBR measures for some diffeomorphisms that
are “almost Anosov”, Ergodic Theory Dynam. Systems
15 (1995), no. 1, 67–76. MR 1314969
(95j:58096), http://dx.doi.org/10.1017/S0143385700008245
 [L]
F.
Ledrappier, Propriétés ergodiques des mesures de
Sinaï, Inst. Hautes Études Sci. Publ. Math.
59 (1984), 163–188 (French). MR 743818
(86f:58092)
 [LS]
François
Ledrappier and JeanMarie
Strelcyn, A proof of the estimation from below in Pesin’s
entropy formula, Ergodic Theory Dynam. Systems 2
(1982), no. 2, 203–219 (1983). MR 693976
(85f:58070), http://dx.doi.org/10.1017/S0143385700001528
 [LY]
F.
Ledrappier and L.S.
Young, The metric entropy of diffeomorphisms. I. Characterization
of measures satisfying Pesin’s entropy formula, Ann. of Math.
(2) 122 (1985), no. 3, 509–539. MR 819556
(87i:58101a), http://dx.doi.org/10.2307/1971328
 [O]
V. I. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197221.
 [P1]
Ya. B. Pesin, Families of invariant manifolds corresponding to nonzero characteristics exponents, Math. USSRIzv. 10 (1978), 12611305
 [P2]
Ya.
B. Pesin, Dynamical systems with generalized hyperbolic attractors:
hyperbolic, ergodic and topological properties, Ergodic Theory Dynam.
Systems 12 (1992), no. 1, 123–151. MR 1162404
(93b:58095), http://dx.doi.org/10.1017/S0143385700006635
 [R]
V. A. Rohlin, Lectures on the theory of entropy of transformations with invariant measures, Russian Math. Surveys 22 (1967), 154.
 [S]
Ja.
G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat.
Nauk 27 (1972), no. 4(166), 21–64 (Russian). MR 0399421
(53 #3265)
 [B]
 R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math. 470 Springer, New York, 1975. MR 56:1364
 [BY]
 M. Benedicks and L.S. Young, SinaiBowenRuelle measure for certain Hénon maps, Invent. Math. 112 (1993), 541576. MR 94e:58074
 [C]
 M. Carvalho, SinaiRuelleBowen measures for dimensional derived from Anosov diffeomorphisms, Ergodic Theory Dynamical Systems 13 (1993), 2144. MR 94h:58102
 [HY]
 H. Huyi and L.S. Young, Nonexistence of SBR measures for some diffeomorphisms that are ``almost Anosov'', Ergodic Theory Dynamical Systems 15 (1995), 6776. MR 95j:58096
 [L]
 F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Inst. Hautes. Études Sci. Publ. Math. 59 (1984), 163188. MR 86f:58092
 [LS]
 F. Ledrappier and J.M. Strelcyn, A proof of the estimation from below in Pesin's entropy formula, Ergodic Theory Dynamical Systems 2 (1982), 203219. MR 85f:58070
 [LY]
 F. Ledrappier and L.S. Young, The metric entropy of diffeomorphisms. I, Ann. of Math. 122 (1985), 509574. MR 87i:58101a
 [O]
 V. I. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (1968), 197221.
 [P1]
 Ya. B. Pesin, Families of invariant manifolds corresponding to nonzero characteristics exponents, Math. USSRIzv. 10 (1978), 12611305
 [P2]
 , Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties, Ergodic Theory Dynamical Systems 12 (1992), 123151. MR 93b:58095
 [R]
 V. A. Rohlin, Lectures on the theory of entropy of transformations with invariant measures, Russian Math. Surveys 22 (1967), 154.
 [S]
 Ya. G. Sinai, Gibbs measure in ergodic theory, Uspehi Mat. Nauk 27 (1972), 2164. MR 53:3265
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (1991):
58F11,
58F15,
28D05
Retrieve articles in all journals
with MSC (1991):
58F11,
58F15,
28D05
Additional Information
Huyi Hu
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742
Address at time of publication:
Department of Mathematics, Penn State University, University Park, Pennsylvania 16801
Email:
hu@math.psu.edu
DOI:
http://dx.doi.org/10.1090/S0002994799024770
PII:
S 00029947(99)024770
Keywords:
Almost Anosov diffeomorphism,
SBR measure,
infinite SBR measure,
local H\"{o}lder condition
Received by editor(s):
November 23, 1997
Published electronically:
December 10, 1999
Additional Notes:
The author of this work was supported by NSF under grants DMS8802593 and DMS9116391, and by DOE (Office of Scientific Computing).
Article copyright:
© Copyright 2000 American Mathematical Society
