Related papers: SRB measures for some diffeomorphisms with dominat…
We provide two robust examples of globally partially hyperbolic systems with a multi one-dimensional center splitting, for which all Gibbs u-states are hyperbolic and the number of physical measures is fixed. In the second example, the…
We obtain measure rigidity results for stationary measures of random walks generated by diffeomorphisms, and for actions of $\operatorname{SL}(2,\mathbb{R})$ on smooth manifolds. Our main technical result, from which the rest of the…
In this paper, we prove that for every $C^1$ vector field preserving an ergodic hyperbolic invariant measure which is not supported on singularities, if the Oseledec splitting of the ergodic hyperbolic invariant measure is a dominated…
We show that a stably ergodic diffeomorphism can be $C^1$ approximated by a diffeomorphism having stably non-zero Lyapunov exponents.
We consider partially hyperbolic diffeomorphisms $f$ with a one-dimensional central direction such that the unstable entropy exceeds the stable entropy. Our main result proves that such maps have a finite number of ergodic measures of…
We consider random perturbations of discrete-time dynamical systems. We give sufficient conditions for the stochastic stability of certain classes of maps, in a strong sense. This improves the main result in J. F. Alves, V. Araujo, Random…
Let $M$ be a compact manifold and $f:\,M\to M$ be a $C^1$ diffeomorphism on $M$. If $\mu$ is an $f$-invariant probability measure which is absolutely continuous relative to Lebesgue measure and for $\mu$ $a.\,\,e.\,\,x\in M,$ there is a…
In this paper, we mainly study the long-time dynamical behaviors of 2D nonlocal stochastic Swift-Hohenberg equations with multiplicative noise from two perspectives. Firstly, by adopting the analytic semigroup theory, we prove the upper…
We consider the set of points with high pointwise emergence for $C^{1+\alpha}$ diffeomorphisms preserving a hyperbolic measure. We find a lower bound on the Hausdorff dimension of this set in terms of unstable Hausdorff dimension of the…
Let $f$ be a $C^1$-diffeomorphism and $\mu$ be a hyperbolic ergodic $f$-invariant Borel probability measure with positive measure-theoretic entropy. Assume that the Oseledec splitting $$T_xM=E_1(x) \oplus\cdots\oplus E_s(x) \oplus…
We consider the "thermodynamic limit"of a d-dimensional lattice of hyperbolic dynamical systems on the 2-torus, interacting via weak and nearest neighbor coupling. We prove that the SRB measure is analytic in the strength of the coupling.…
We establish existence of an ergodic invariant measure on $H^1(D,\mathbb{R}^3)\cap L^2(D,\mathbb{S}^2)$ for the stochastic Landau-Lifschitz-Gilbert equation on a bounded one dimensional interval $D$. The conclusion is achieved by employing…
In this paper, we study two properties of the Lyapunov exponents under small perturbations: one is when we can remove zero Lyapunov exponents and the other is when we can distinguish all the Lyapunov exponents. The first result shows that…
Given any compact connected manifold $M$, we describe $C^2$-open sets of iterated functions systems (IFS's) admitting fully-supported ergodic measures whose Lyapunov exponents along $M$ are all zero. Moreover, these measures are…
For any continuous map f on a compact manifold M, we define the SRB-like (or observable) probabilities as a generalization of Sinai-Ruelle-Bowen (i.e. physical) measures. We prove that f has observable measures, even if SRB measures do not…
We study $C^1$-robustly transitive and nonhyperbolic diffeomorphisms having a partially hyperbolic splitting with one-dimensional central bundle whose strong un-/stable foliations are both minimal. {In dimension $3$, an important class of…
We prove robustness and uniqueness of equilibrium states for a class of partially hyperbolic diffeomorphisms with dominated splittings and H\"older continuous potentials with not very large oscillation.
Using scanning tunneling spectroscopy (STS) at 5 K in B-fields up to 7 T, we investigate the local density of states of a two-dimensional electron system (2DES) created by Cs adsorption on p-type InSb(110). The 2DES, which in contrast to…
We obtain new entropy rigidity results for $u$-Gibbs measures by showing that whenever a $u$-Gibbs measure of a partially hyperbolic diffeomorphism admits an unstable Margulis family, the unstable Jacobian data of the system must to be…
In the case of smooth non-invertible maps which are hyperbolic on folded basic sets $\Lambda$, we give approximations for the Gibbs states (equilibrium measures) of arbitrary H\"{o}lder potentials, with the help of weighted sums of atomic…