Diffeomorphisms approximated by Anosov on the 2-torus and their SBR measures

Sumi, Naoya

doi:10.1090/S0002-9947-99-02426-5

Diffeomorphisms approximated by Anosov on the 2-torus and their SBR measures
HTML articles powered by AMS MathViewer

by Naoya Sumi PDF

Trans. Amer. Math. Soc. 351 (1999), 3373-3385 Request permission

Abstract:

We consider the $C^{2}$ set of $C^{2}$ diffeomorphisms of the 2-torus $\mathbb {T}^{2}$, provided the conditions that the tangent bundle splits into the directed sum $T\mathbb {T}^{2}=E^{s}\oplus E^{u}$ of $Df$-invariant subbundles $E^{s}$, $E^{u}$ and there is $0<\lambda <1$ such that $\Vert Df|_{E^{s}}\Vert <\lambda$ and $\Vert Df|_{E^{u}}\Vert \ge 1$. Then we prove that the set is the union of Anosov diffeomorphisms and diffeomorphisms approximated by Anosov, and moreover every diffeomorphism approximated by Anosov in the $C^{2}$ set has no SBR measures. This is related to a result of Hu-Young.

References

D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature. , Proceedings of the Steklov Institute of Mathematics, No. 90 (1967), American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by S. Feder. MR 0242194
N. Aoki and K. Hiraide, Topological theory of dynamical systems, North-Holland Mathematical Library, vol. 52, North-Holland Publishing Co., Amsterdam, 1994. Recent advances. MR 1289410, DOI 10.1016/S0924-6509(08)70166-1
Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989, DOI 10.1007/BFb0081279
Koichi Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math. 27 (1990), no. 1, 117–162. MR 1049828
Hu Yi Hu and Lai-Sang Young, Nonexistence of SBR measures for some diffeomorphisms that are “almost Anosov”, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 67–76. MR 1314969, DOI 10.1017/S0143385700008245
Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 133–163. MR 0271991
F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 163–188 (French). MR 743818, DOI 10.1007/BF02698772
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, DOI 10.2307/1971328
Ricardo Mañé, Contributions to the stability conjecture, Topology 17 (1978), no. 4, 383–396. MR 516217, DOI 10.1016/0040-9383(78)90005-8
Ja. B. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 6, 1332–1379, 1440 (Russian). MR 0458490
J. W. Robbin, A structural stability theorem, Ann. of Math. (2) 94 (1971), 447–493. MR 287580, DOI 10.2307/1970766
Clark Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mountain J. Math. 7 (1977), no. 3, 425–437. MR 494300, DOI 10.1216/RMJ-1977-7-3-425
V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 3–56 (Russian). MR 0217258
Ja. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no. 4(166), 21–64 (Russian). MR 0399421
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1