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.     

Stable and non-symmetric pitchfork bifurcations In Memory of Professor Shantao Liao

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 result [15] significantly. Based on our criterion,we present a class of families of non-symmetric vector fields undergoing a pitchfork bifurcation.     

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.     

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.     

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.     

Geometric Gibbs theory

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.

     

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.     

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.     

Geometric Ergodicity of Gibbs and Block Gibbs Samplers for a Hierarchical Random Effects Model

We consider fixed scan Gibbs and block Gibbs samplers for a Bayesian hierarchical random effects model with proper conjugate priors. A drift condition given in Meyn and Tweedie (1993, Chapter 15) is used to show that these Markov chains are geometrically ergodic. Showing that a Gibbs sampler is geometrically ergodic is the first step toward establishing central limit theorems, which can be used to approximate the error associated with Monte Carlo estimates of posterior quantities of interest. Thus, our results will be of practical interest to researchers using these Gibbs samplers for Bayesian data analysis.     

Geometric ergodicity of the Gibbs sampler for the Poisson change-point model

Poisson change-point models have been widely used for modelling inhomogeneous time-series of count data. There are a number of methods available for estimating the parameters in these models using iterative techniques such as MCMC. Many of these techniques share the common problem that there does not seem to be a definitive way of knowing the number of iterations required to obtain sufficient convergence. In this paper, we show that the Gibbs sampler of the Poisson change-point model is geometrically ergodic. Establishing geometric ergodicity is crucial from a practical point of view as it implies the existence of a Markov chain central limit theorem, which can be used to obtain standard error estimates. We prove that the transition kernel is a trace-class operator, which implies geometric ergodicity of the sampler. We then provide a useful application of the sampler to a model for the quarterly driver fatality counts for the state of Victoria, Australia.