We study the existence of harmonic maps and Dirac-harmonic from degenerating surfaces to non-positive curved manifold via scheme Sacks Uhlenbeck. By choosing a suitable sequence $\alpha$-(Dirac-)harmonic closed hyperbolic surface, we get convergence cleaner energy identity under uniformly bounded assumption. In this identity, there is no loss near punctures. As an application, obtain result abo...

Let ρ be a Sinai-Ruelle-Bowen (SRB or 'physical') measure for the discrete time evolution given by a map f, and let ρ(A) denote the expectation value of a smooth function A. If f depends on a parameter, the derivative δρ(A) of ρ(A) with respect to the parameter is formally given by the value of the so-called susceptibility function Ψ(z) at z=1. When f is a uniformly hyperbolic diffeomorphism, i...

We start by reviewing recent probabilistic results on ergodic sums in a large class of (non-uniformly) hyperbolic dynamical systems. Namely, we describe the central limit theorem, the almost-sure convergence to the gaussian and other stable laws, and large deviations. Next, we describe a new branch in the study of probabilistic properties of dynamical systems, namely concentration inequalities....

In this paper we study a special family of Lorentz gas with infinite horizon. The periodic scatterers have C3 smooth boundary with positive curvature except on finitely many flat points. In addition there exists a trajectory with infinite free path and tangentially touching the scatterers only at some flat points. The singularity set of the system is analyzed in detail. And we prove that the fr...

We study an Ishikawa type algorithm for two multi-valued quasinonexpansive maps on a special class of nonlinear spaces namely hyperbolic metric spaces; in particular, strong and 4−convergence theorems for the proposed algorithms are established in a uniformly convex hyperbolic space which improve and extend the corresponding known results in uniformly convex Banach spaces. Our new results are a...

The paper is a non-technical survey and is aimed to illustrate Dolgopyat’s profound contributions to smooth ergodic theory. I will discuss some of Dolgopyat’s work on partial hyperbolicity and nonuniform hyperbolicity with emphasis on interaction between the two – the class of dynamical systems with mixed hyperbolicity. On the one hand, this includes uniformly partially hyperbolic diffeomorphis...

We consider partially hyperbolic diieomorphisms preserving a splitting of the tangent bundle into a strong-unstable subbundle E uu (uniformly expanding) and a subbundle E c , dominated by E uu. We prove that if the central direction E c is mostly contracting for the diieomorphism (negative Lyapunov exponents), then the ergodic Gibbs u-states are the Sinai-Ruelle-Bowen measures, there are nitely...

which gives an isometric embedding of the hyperbolic space H into R. Hano and Nomizu [11] were probably the first to observe the non-uniqueness of isometric embeddings of H in R by constructing other (geometrically distinct) entire solutions of (1.1)–(1.2) for n 1⁄4 2 (and c1 1) using methods of ordinary di¤erential equations. Using the theory of Monge-Ampère equations, A.-M. Li [12] studied en...

We consider an autonomous system of partial differential equations for a onedimensional distributed medium with periodic boundary conditions. Dynamics in time consists of alternating birth and death of patterns with spatial phases transformed from one stage of activity to another by the doubly expanding circle map. So, the attractor in the Poincaré section is uniformly hyperbolic, a kind of Sma...

We extend some of the theory of multifractal analysis for conformal expanding systems to two new cases: The non-uniformly hyperbolic example of the Manneville– Pomeau equation and the continued fraction transformation. A common point in the analysis is the use of thermodynamic formalism for transformations with infinitely many branches. We effect a complete multifractal analysis of the Lyapunov...