Shaobo Gan

Peking University | PKU · School of Mathematical Sciences

Let f be a non-invertible irreducible Anosov map on d-torus. We show that if the stable bundle of f is one-dimensional, then f has the integrable unstable bundle, if and only if, every periodic point of f admits the same Lyapunov exponent on the stable bundle as its linearization. For higher-dimensional stable bundle case, we get the same result on...

Let $f$ be a non-invertible irreducible Anosov map on $d$-torus. We show that if the stable bundle of $f$ is one-dimensional, then $f$ has the integrable unstable bundle, if and only if, every periodic point of $f$ admits the same Lyapunov exponent on the stable bundle with its linearization. For higher-dimensional stable bundle case, we get the sa...

We show that for a $C^1$ generic vector field $X$ away from homoclinic tangencies, a nontrivial Lyapunov stable chain recurrence class is a homoclinic class. The proof uses an argument with $C^2$ vector fields approaching $X$ in $C^1$ topology, with their Gibbs $F$-states converging to a Gibbs $F$-state of $X$.

In this paper we consider the semi-continuity of the physical-like measures for diffeomorphisms with dominated splittings. We prove that any weak-* limit of physical-like measures along a sequence of C ¹ diffeomorphisms {f n } must be a Gibbs F-state for the limiting map f. As a consequence, we establish the statistical stability for the C ¹ pertur...

We call a partially hyperbolic diffeomorphism partially volume expanding if the Jacobian restricted to any hyperplane that contains the unstable bundle \(E^u\) is larger than 1. This is a \(C^1\) open property. We show that any \(C^{1+}\) partially volume expanding diffeomorphisms admits finitely many physical measures, and the union of their basin...

In this paper we consider the semi-continuity of the physical-like measures for diffeomorphisms with dominated splittings. We prove that any weak-* limit of physical-like measures along a sequence of $C^1$ diffeomorphisms $\{f_n\}$ must be a Gibbs $F$-state for the limiting map $f$. As a consequence, we establish the statistical stability for the $...

We call a partially hyperbolic diffeomorphism \emph{partially volume expanding} if the Jacobian restricted to any hyperplane that contains the unstable bundle $E^u$ is larger than $1$. This is a $C^1$ open property. We show that any $C^{1+}$ partially volume expanding diffeomorphisms admits finitely many physical measures, the union of whose basins...

Let f be an Anosov diffeomorphism on a nilmanifold. We consider Birkhoff sums for a Holder continuous observation along periodic orbits. We show that if there are two Birkhoff sums distributed at both sides of zero, then the set of Birkhoff sums of all periodic points is dense in the whole set of real numbers.

Let $f$ be a conservative partially hyperbolic diffeomorphism, which is homotopic to an Anosov automorphism $A$ on $\mathbb{T}^3$. We show that the stable and unstable bundles of $f$ are jointly integrable if and only if every periodic point of $f$ admits the same center Lyapunov exponent with $A$. In particular, $f$ is Anosov. Thus every conservat...

In 1994, I. Kan constructed a smooth map on the annulus admitting two physical measures, whose basins are intermingled. In this paper, we prove that Kan's map is C² robustly topologically mixing.

We call that a flow has the orbital shadowing property if for any ε>0 there is d>0 such that, for any d-pseudo orbit g(t) there exists an orbit Orb(x) satisfying distH(g(t),Orb(x))<ε. In this paper, we show that the C1-interior of the set of 3-dimensional flows having the orbital shadowing property is contained in the set of Ω-stable 3-flows.

We construct a family of partially hyperbolic skew-product diffeomorphisms on $\mathbb{T}^3$ that are robustly transitive and admitting two physical measures with intermingled basins. In particularly, all these diffeomorphisms are not topologically mixing. Moreover, for every such example, it exhibits a dichotomy under perturbation: every perturbat...

In this paper, we construct a partially hyperbolic skew-product diffeomorphism $f$ on $\mathbb{T}^3$, such that $f$ is accessible and chain transitive, but not transitive.

We give three equivalent conditions for non-accessibility of an Anosov diffeomorphism on the 3-torus with a partially hyperbolic splitting. Since accessibility is an open property, this gives a negative answer to Hammerlindl’s question about homology boundedness of strong unstable foliation.

Let X be a C 1 vector field on a compact boundaryless Riemannian manifold M (dim M ≥ 2), and Λ a compact invariant set of X. Suppose that Λ has a hyperbolic splitting, i.e., T ΛM = E s ⊕〈X〉⊕E u with E s uniformly contracting and E u uniformly expanding. We prove that if, in addition, Λ is chain transitive, then the hyperbolic splitting is continuou...

We prove that, for $C^1$-generic diffeomorphisms, if a homoclinic class is not hyperbolic, then there is a non-hyperbolic ergodic measure supported on it. This proves a conjecture by D\'iaz and Gorodetski [28]. We also discuss the conjectured existence of periodic points with different stable dimension in the class.

Let f be a homeomorphism on 2-torus, homotopic to the identity and preserving two transverse foliations. If both foliations are minimal, or one is minimal and f has the skew-product form, then f is conjugate to a translation. In particular, the rotation set of f consists of a single point.

We prove transitivity for volume preserving $C^{1+}$ diffeomorphisms on $\mathbb{T}^3$ which are isotopic to a linear Anosov automorphism along a path of weakly partially hyperbolic diffeomorphisms.

We construct a diffeomorphism f on a 2-torus with dominated splitting E ⊕ F such that there exists an open neighbourhood satisfying that for any , neither Eg nor Fg is integrable.

We call that a vector field has the oriented shadowing property if for any $\varepsilon>0$ there is $d>0$ such that each $d$-pseudo orbit is $\varepsilon$-oriented shadowed by some real orbit. In this paper, we show that the $C^1$-interior of the set of vector fields with the oriented shadowing property is contained in the set of vector fields with...

Let $f: \mathbb{T}^3\to\mathbb{T}^3$ be a partially hyperbolic diffeomorphism on the 3-torus $\mathbb{T}^3$. In his thesis, Hammerlindl proved that for lifted center foliation $\mathcal{F}^c_f$, there exists $R>0$, such that for any $x\in \mathbb{R}^3$, ${\cal F}^c_f(x)\subset B_R (x+E^c)$, where $\mathbb{R}^3=E^s\oplus E^c\oplus E^u$ is the partia...

We construct a diffeomorphism $f$ on 2-torus with a dominated splitting $E \oplus F$ such that there exists an open neighborhood $\mathcal{U} \ni f$ satisfying that for any $g \in \mathcal{U}$, neither $E_g$ nor $F_g$ is integrable.

We prove for a generic star vector field $X$ that, if for every chain recurrent class $C$ of $X$ all singularities in $C$ have the same index, then the chain recurrent set of $X$ is singular hyperbolic. We also prove that every Lyapunov stable chain recurrent class of $X$ is singular hyperbolic. As a corollary, we prove that the chain recurrent set...

We prove that for any regular endomorphism f on a 2-torus T2 which is not one to one, there is a regular map g homotopic to f such that g is C1 robustly non-hyperbolic transitive. We also introduce interesting blender phenomena (a fat horseshoe) of 2-dimensional endomorphisms, which play an important role in our construction of some examples.

We prove that for every three-dimensional vector field, either it can be accumulated by Morse-Smale ones, or it can be accumulated by ones with a transverse homoclinic intersection of some hyperbolic periodic orbit in the $C^1$ topology.

We prove that for generic three-dimensional vector fields, domination implies singular hyperbolicity.

Let f be a diffeomorphism on a closed manifold, and p be a hyperbolic periodic point of f. Denote Cf(p) the chain component of f that contains p. We say Cf(p) is C1-stably shadowable if there is a C1-neighborhood U of f such that for every g∈U, Cg(pg) has the shadowing property, where pg is the continuation of p. We prove in this paper that if Cf(p...

We prove that for $C^1$ generic diffeomorphisms, every expansive homoclinic class is hyperbolic. Comment: 7 pages

We give a sufficient criterion for the hyperbolicity of a homoclinic class. More precisely, if the homoclinic class H(p) admits a partially hyperbolic splitting TH(p)M = Es ⊕lt; F, where Es is uniformly contracting and dimEs = ind(p), and all periodic points homoclinically related with p are uniformly E u-expanding at the period, then H(p) is hyper...

It was proved recently in [M. Li, S. Gan and L. Wen, ibid. 13, No. 2, 239–269 (2005; Zbl 1115.37022)] that any robustly transitive singular set that is strongly homogeneous must be partially hyperbolic, as long as the indices of singularities and periodic orbits satisfy certain condition. We prove that this index-condition is automatically satisfie...

We give an affirmative answer to a problem of Liao and Mañé which asks whether, for a nonsingular flow to loose the Ω-stability, it must go through a critical-element-bifurcation. More precisely, a vector field S on a compact boundaryless manifold is called a star system if S has a C1 neighborhood U\mathcal{U} in the set of C1 vector fields such t...

The authors give a quick account for the theory of quasi-hyperbolicity and linear transversality, due independently to Mañé, Sac ker-Sell, and Selgrade. In the last section, the authors explain the correspondence between this theory and the obstruction sets theory of Liao. In fact, this account serves also as an illustration of Liao's theory.

We show that, for C 1 -generic diffeomorphisms, every chain recurrent class C that has a partially hyperbolic splitting E s ⊕E c ⊕E u with dim E c = 1 is either an isolated hyperbolic periodic orbit, or is accumulated by non-trivial homoclinic classes. We also proves that, for C 1 -generic diffeomorphisms, any chain recurrent class that has a domin...

Morales, Pacifico and Pujals proved recently that every robustly transitive singular set for a 3-dimensional flow must be partially hyperbolic. In this paper we generalize the result to higher dimensions. By definition, an isolated invariant set Λ of a C1 vector field S is called robustly transitive if there exist an isolating neighborhood U of Λ i...

It seems that in Mañé’s proof of the C 1Ω-stability conjecture containing in the famous paper which published in I. H. E. S. (1988), there exists a deficiency in the main lemma which says that for f∈ \({\fancyscript F}\) 1(M) there exists a dominated splitting \( TM_{{\left| {\overline{P} _{i} {\left( f \right)}} \right.}} = \widetilde{E}^{s}_{i} \...

We prove some C 1 generic results about orbit-connecting, in particular about heteroclinic cycles and homoclinic closures. As a consequence we obtain a three-ways C 1 density theorem: Diffeomorphisms with either infinitely many weakly transitive components or a heterodimensional cycle are C 1 dense in the complement of the C 1 closure of Axiom A an...

In this paper, we prove a generalized shadowing lemma. Let f ∈ Diff(M). Assume that ∧ is a closed invariant set of f and there is a continuous invariant splitting T∧M = E ⊕ F on ∧. For any λ ∈ (0,1) there exist L > 0, d0 > 0 such that for any d ∈ (0, d0] and any λ-quasi-hyperbolic d-pseudoorbit {xi,ni}i=-∞∞, there exists a point x which Ld-shadows...

Let M be a two-dimensional closed Riemannian manifold and denote by Diff1(M) the set of C1 diffeomorphisms on M. Then, the C1 interior of {f in Diff1(M):h(f) = 0} is equal to the C1 interior of the closure of the Morse-Smale systems and equal to the C1 interior of the set of diffeomorphisms having no horseshoe.

In this paper, we give a complete description for the lexicographic world £= {(x,y) ∈ ∑ × ∑ : ∑xy ≠ θ} = {(x,y) : y ≥ φ(x)}, where ∑ = {0,1}N, ∑ab = {x ∈ ∑ : a ≤ σi(x) ≤ b, for all i ≥ 0}, φ : ∑ → ∑ is defined by φ(a) = inf {b : ∑ab ≠ θ} and the order ≤ is the lexicographic order on ∑. The main result is that b = φ(a) for some a = 0x if and only if...

Based on the characterization of periodic eigenvalues using rotation numbers, we analyse the second and the third periodic eigenvalues of one-dimensional Schrödinger operators with certain step potentials. This gives counter-examples to the Alikakos– Fusco conjecture on the second periodic eigenvalues. Using this simple model, we can also construct...

In this paper we give a new proof for the C1Ω-stability conjecture for flows. Our approach is considerably different from that of Hayashi or Wen, and a good deal of Liao's obstruction sets theory is used.

In this paper, we use the rotation number approach to study in detail the charac-teristic values of Hill's equations with two-step periodic potentials. As a result, the global structure of resonance pockets is described completely. The results in this paper show that resonance pockets behave in a sensible and fairly rich way even in this simplest c...

TheC 1 structural stability conjecture for flows byC 1 connecting lemma and obstruction sets is proved.

A C1 difieomorphismon a compact boundaryless manifold is said to exhibit an i-eigenvalue gap if for every periodic point x of `, the modulus of i-th eigenvalue of Dn(x) is strictly less than the modulus of (i + 1)-th eigenvalue of Dn(x), where n is the period of x. We prove that ` has a dominated splitting of index i over the set of preperiodic poi...