Stable and non-symmetric pitchfork bifurcations

Science China Mathematics - In this paper, we present a criterion for pitchfork bifurcations of smooth vector fields based on a topological argument. Our result expands Rajapakse and Smale℉s...     

Geometric Gibbs theory In Memory of Professor Shantao Liao

We extend the classical Gibbs theory for smooth potentials to the geometric Gibbs theory for certain continuous potentials. We study the existence and uniqueness and the compatibility of geometric Gibbs measures associated with these continuous potentials. We introduce a complex Banach manifold structure on the space of these continuous potentials as well as on the space of all geometric Gibbs measures. We prove that with this complex Banach manifold structure, the space is complete and, moreover, is the completion of the space of all smooth potentials as well as the space of all classical Gibbs measures. There is a maximum metric on the space,which is incomplete. We prove that the topology induced by the newly introduced complex Banach manifold structure and the topology induced by the maximal metric are the same. We prove that a geometric Gibbs measure is an equilibrium state, and the infimum of the metric entropy function on the space is zero.     

Inverse shadowing and related measures In Memory of Professor Shantao Liao

We study various weaker forms of the inverse shadowing property for discrete dynamical systems on a smooth compact manifold. First, we introduce the so-called ergodic inverse shadowing property(Birkhoff averages of continuous functions along an exact trajectory and the approximating one are close). We demonstrate that this property implies the continuity of the set of invariant measures in the Hausdorff metric. We show that the class of systems with ergodic inverse shadowing is quite broad; it includes all diffeomorphisms with hyperbolic nonwandering sets. Second, we study the so-called individual inverse shadowing(any exact trajectory can be traced by approximate ones, but this shadowing is not uniform with respect to the initial point of the trajectory).We demonstrate that this property is closely related to structural stability and ?-stability of diffeomorphisms.     

Large coupling asymptotics for the entropy of quasi-periodic operators In Memory of Professor Shantao Liao

In this paper, we give an asymptotic estimate for the entropy, i.e., the sum of all positive Lyapunov exponents, of the quasi-periodic finite-range operator with a large trigonometric polynomial potential and Diophantine frequency.     

Anosov and expanding attractors Dedicated to the Memory of Professor Shantao Liao

An earlier conjecture is settled with an immersion of a 2-dimensional branched manifold. Possible obstructions in linear algebra and tiling theory are studied first.     

Towards mesoscopic ergodic theory Dedicated with Admiration to the Memory of Professor Shantao Liao

The present paper is devoted to a preliminary study towards the establishment of an ergodic theory for stochastic differential equations(SDEs) with less regular coefficients and degenerate noises. These equations are often derived as mesoscopic limits of complex or huge microscopic systems. By studying the associated Fokker-Planck equation(FPE), we prove the convergence of the time average of globally defined weak solutions of such an SDE to the set of stationary measures of the FPE under Lyapunov conditions. In the case where the set of stationary measures consists of a single element, the unique stationary measure is shown to be physical.Similar convergence results for the solutions of the FPE are established as well. Some of our convergence results, while being special cases of those contained in Ji et al.(2019) for SDEs with periodic coefficients, have weaken the required Lyapunov conditions and are of much simplified proofs. Applications to stochastic damping Hamiltonian systems and stochastic slow-fast systems are given.     

Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center To the Memory of Professor Shantao Liao

In this paper, we study transitive partially hyperbolic diffeomorphisms with one-dimensional topologically neutral center, meaning that the length of the iterate of small center segments remains small. Such systems are dynamically coherent. We show that there exists a continuous metric along the center foliation which is invariant under the dynamics. As an application, we classify the transitive partially hyperbolic diffeomorphisms on 3-manifolds with topologically neutral center.     

Lyapunov exponent,Liao perturbation and persistence Dedicated to the Memory of Professor Shantao Liao at the Centenary of His Birth

Consider a C~1 vector field together with an ergodic invariant probability that has ? nonzero Lyapunov exponents. Using orthonormal moving frames along a generic orbit we construct a linear system of ?differential equations which is a linearized Liao standard system. We show that Lyapunov exponents of this linear system coincide with all the nonzero exponents of the given vector field with respect to the given ergodic probability. Moreover, we prove that these Lyapunov exponents have a persistence property meaning that a small perturbation to the linear system(Liao perturbation) preserves both the sign and the value of the nonzero Lyapunov exponents.     

SRB measures for pointwise hyperbolic systems on open regions Dedicated to the Memory of Professor Shantao Liao

A diffeomorphism f:M→M is pointwise partially hyperbolic on an open invariant subset N?M if there is an invariant decomposition TNM=E~u⊕E~c⊕E~ssuch that Dxf is strictly expanding on ■ and contracting on ■ at each x∈N.We show that under certain conditions f has unstable and stable manifolds,and admits a finite or an infinite u-Gibbs measureμ.If f is pointwise hyperbolic on N,thenμis a SinaiRuelle-Bowen (SRB) measure or an infinite SRB measure.As applications,we show that some almost Anosov diffeomorphisms and gentle perturbations of Katok’s map have the properties.     

On the notions of singular domination and(multi-)singular hyperbolicity To the Memory of Professor Shantao Liao

The properties of uniform hyperbolicity and dominated splitting have been introduced to study the stability of the dynamics of diffeomorphisms. One meets difficulties when trying to extend these definitions to vector fields and Shantao Liao has shown that it is more relevant to consider the linear Poincaré flow rather than the tangent flow in order to study the properties of the derivative. In this paper, we define the notion of singular domination, an analog of the dominated splitting for the linear Poincaré flow which is robust under perturbations. Based on this, we give a new definition of multi-singular hyperbolicity which is equivalent to the one recently introduced by Bonatti and da Luz(2017). The novelty of our definition is that it does not involve the blow-up of the singular set and the rescaling cocycle of the linear flows.     

Imperfect transcritical and pitchfork bifurcations

Imperfect bifurcation phenomena are formulated in framework of analytical bifurcation theory on Banach spaces. In particular the perturbations of transcritical and pitchfork bifurcations at a simple eigenvalue are examined, and two-parameter unfoldings of singularities are rigorously established. Applications include semilinear elliptic equations, imperfect Euler buckling beam problem and perturbed diffusive logistic equation.     

Regionally proximal relation of order d along arithmetic progressions and nilsystems Dedicated to Professor Shantao Liao

The regionally proximal relation of order d along arithmetic progressions,namely AP~([d])for d∈N,is introduced and investigated.It turns out that if (X,T) is a topological dynamical system with AP~([d])=?,then each ergodic measure of (X,T) is isomorphic to a d-step pro-nilsystem,and thus (X,T) has zero entropy.Moreover,it is shown that if (X,T) is a strictly ergodic distal system with the property that the maximal topological and measurable d-step pro-nilsystems are isomorphic,then AP~([d])=RP~([d])for each d∈N.It follows that for a minimal∞-pro-nilsystem,AP~([d])=RP~([d])for each d∈N.An example which is a strictly ergodic distal system with discrete spectrum whose maximal equicontinuous factor is not isomorphic to the Kronecker factor is constructed.     

Smoothness of invariant manifolds and foliations for infinite dimensional random dynamical systems Dedicated to Professor Shantao Liao

In this paper, we investigate the smoothness of invariant manifolds and foliations for random dynamical systems with nonuniform pseudo-hyperbolicity in Hilbert spaces. We discuss on the effect of temperedness and the spectral gaps in the nonuniform pseudo-hyperbolicity so as to prove the existence of invariant manifolds and invariant foliations, which preserve the C~(N,τ(ω))Holder smoothness of the random system in the space variable and the measurability of the random system in the sample point. Moreover, we also prove that the stable foliation is C~(N-1,τ(ω))in the base point.     

Pathwise description of dynamic pitchfork bifurcations with additive noise

 The slow drift (with speed ɛ) of a parameter through a pitchfork bifurcation point, known as the dynamic pitchfork bifurcation, is characterized by a significant delay of the transition from the unstable to the stable state. We describe the effect of an additive noise, of intensity σ, by giving precise estimates on the behaviour of the individual paths. We show that until time after the bifurcation, the paths are concentrated in a region of size around the bifurcating equilibrium. With high probability, they leave a neighbourhood of this equilibrium during a time interval , after which they are likely to stay close to the corresponding deterministic solution. We derive exponentially small upper bounds for the probability of the sets of exceptional paths, with explicit values for the exponents. Received: 7 August 2000 / Revised version: 19 April 2001 / Published online: 20 December 2001     

回忆Kai-Lai Chung

Kai Lai Chung （1917-2009）, a Chinese-born mathematician, a world renowned expert in Probability Theory. He was born in Shanghai, and graduated from South-western Associated University, 1940 in Kunming, and got Ph.D. from Princeton University （USA） in 1947. He had worked many years as a full professor at Math. Dept. of Stanford University （USA） since 1960’s, and became a professor emeritus after his retirement. He made various unforgettable important contributions to Markoff chains as well as to the theory of stochastic processes. He had been the adviser of our JMRE （Journal of Mathematical Research and Exposition） for 29 years, and as a matter of fact, the name of our journal was devised and suggested by him in 1981. Certainly, the passing-away of Chung is really a great loss to the community of Chinese mathematicians and also to the world of probabilists.