Quasi-shadowing Property And Entropy For Partially Hyperbolic Systems

In the study of differentiable dynamical system, partially hyperbolic system is cur-rently one of the most active branches. For partially hyperbolic systems, a center direction is allowed in addition to the hyperbolic directions. Partially hyperbolic system has more important theoretical significance and more extensive application value. The presence of center direction permits a more complex and rich type of structure in these systems, also makes the research more difficult and challenging.In this paper we study the quasi-shadowing property and its application in the study of entropy for partially hyperbolic systems. The main content includes the following four parts:In the first part, we prove that a diffeomorphism on a compact Riemannian manifold has the quasi-shadowing property in a neighborhood of a partially hyperbolic set in the following sense:Let f be a partially hyperbolic diffeomorphism on a compact Riemannian manifold M, and ∧ (?) M be a partially hyperbolic set of f. There exists the neighborhood O(∧) of ∧ such that for any ε there exists δ> 0 such that for any δ-pseudo orbit {xk}k∈z (?) O(∧) of f, there exist a sequence of points{yk}k∈z, and a sequence of vectors {uk ∈ Exkc}k∈Z such that d{xk,yk)< ε, where yk= expxk(expxk-1 (f(yk-1))+uk).In the second part, the robustness of the orbit structure is investigated for a par-tially hyperbolic endomorphism f on a compact manifold M. It is first proved that, the dynamical structure of its orbit space (inverse limit space) Mf of f is topologically quasi-stable under C0-small perturbations in the following sense:for any covering endo-morphism g C0-close to f, there is a continuous map φ from Mg to Π-∞∞ M such that for any{yi}i∈z ∈φ(Mg), yi+1 and f (yi) differ only by a motion along the center direction. It is then proved that f has quasi-shadowing property in the following sense:for any pseudo orbit{xi}i∈z, there is a sequence of points{yi}i∈z tracing it in which yi+1 is obtained from f(yi) by a motion along the center direction.In the third part, we prove that a C1+T diffeomorphism with integrable center fo-liation has the quasi-shadowing property in the following sense:Let f be a C1+T dif-feomorphism, then there is a non-empty regular points set ∧. For any α> 0, there exists a sequence (δk)k=1+∞ such that for any (δk)k=1+∞ -pesudo-orbit{xn}-∞+∞ ∈∧ there exists at least one α-quasi-shadowing sequence (yn)-∞+∞ satisfying:d(xn,yn)≤α/lsn and yn∈Wα/lsnc(f(yn-1)) (?)n ∈ Z, where{lk}k=1+∞ is a real number sequence. As an application, we show that if f is center integrable and plaque expansive with respect to the center foliation, then for all k ≥ 1,0< α< 1, there exists β=β(k,α)> 0 such that:if x, fp(x) ∈ ∧k and p ∈ N with d(x, fpx)< β, there exists a periodic center leaf Wc(z) of period p satisfying d((z),x)< α.In the fourth part, we investigate the relationship among topological entropy h(f), entropy of the restriction of f on the center foliation h(f, Wc) and the growth rate of periodic center leaves pc(f) for a partially hyperbolic diffeomorphism with a uniformly compact center foliation Wc. It is first shown that, if a compact locally maximal invariant center set A is center topologically mixing then f|∧ has the center specification property, i.e., any specification with a large spacing can be center shadowed by a periodic center leaf with a fine precision. Applying the center spectral decomposition and the center specification property, we show that h(f) ≤ h(f,Wc)+pc(f). Moreover, if the center foliation Wc is of dimension one, we obtain an equality h(f)= pc(f).In the end, the continuity of topological entropy is investigated for a class of partially hyperbolic diffeomorphisms on a compact Riemannian manifold. Let f :M â†' M be a partially hyperbolic diffeomorphism with uniformly compact center foliation. If the center foliation Wc is of dimension one, then there exists a C1 neighborhood u of f, in the space of C1 diffeomorphisms of M, such that the topological entropy is locally constant in U.