Related papers: Shadowing by non uniformly hyperbolic periodic poi…

The aim of this paper is to investigate the fractional combinatorial Calabi flow for hyperbolic bordered surfaces. By Lyapunov theory, it is proved that the flow exists for all time and converges exponentially to a conformal factor that…

H-holomorphic maps are a parameter version of J-holomorphic maps into contact manifolds. They have arisen in efforts to prove the existence of higher--genus holomorphic open book decompositions and efforts to prove the existence of finite…

We show that the topological entropy is monotonic for unimodal interval maps which are obtained from the restriction of quadratic rational maps with real coefficients. This is done by ruling out the existence of certain post-critical curves…

We study expansions near the boundary of solutions to the Dirichlet problem for minimal graphs in the hyperbolic space and prove the local convergence of such expansions if the boundary is locally analytic. As a consequence, we prove a…

We introduce continuum-wise hyperbolicity, a generalization of hyperbolicity with respect to the continuum theory. We discuss similarities and differences between topological hyperbolicity and continuum-wise hyperbolicity. A shadowing lemma…

We prove a strong multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces i.e. if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then…

For non uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds…

Let $\text{G}(n)$ be equal either to $\text{PO}(n,1),\text{PU}(n,1)$ or $\text{PSp}(n,1)$ and let $\Gamma \leq \text{G}(n)$ be a uniform lattice. Denote by $\mathbb{H}^n_K$ the hyperbolic space associated to $\text{G}(n)$, where $K$ is a…

We study the dynamics of continuous maps on compact metric spaces containing a free interval (an open subset homeomorphic to the interval $(0,1)$). We provide a new proof of a result of M. Dirb\'ak, \v{L}. Snoha, V. \v{S}pitalsk\'y [Ergodic…

We show that the standard method for constructing closed hyperbolic manifolds of arbitrary dimension possessing reflective symmetries typically produces reflections whose fixed point sets are nonseparating.

A plabic graph is a planar bicolored graph embedded in a disk, which satisfies some combinatorial conditions. Postnikov's boundary measurement map takes the space of positive edge weights of a plabic graph $G$ to a positroid cell in some…

We establish new instances of the cutoff phenomenon for geodesic paths and for the Brownian motion on compact hyperbolic manifolds. We prove that for any fixed compact hyperbolic manifold, the geodesic path started on a spatially localized…

It is shown that if a non-invertible area preserving local homeomorphism on $\mathbb{T}^2$ is homotopic to a linear expanding or hyperbolic endomorphism, then it must be topologically transitive. This gives a complete characterization, in…

We show that a one-ended simply connected at infinity hyperbolic group $G$ with enough codimension-1 surface subgroups has $\partial G \cong \mathbb{S}^2$. Combined with a result of Markovic, our result gives a new characterization of…

A soft presentation of hyperbolic spaces, free of differential apparatus, is offered. Fifth Euclid's postulate in such spaces is overthrown and, among other things, it is proved that spheres (equipped with great-circle distances) and…

We prove that entropy map is upper semi-continuous for C1 nonuniformly hyperbolic systems with domination, while it is not true for C1+alpha nonuniformly hyperbolic systems in general. This goes a little against a common intuition that…

For any $\varepsilon>0$, we construct a closed hyperbolic surface of genus $g=g(\varepsilon)$ with a set of at most $\varepsilon g$ systoles that fill, meaning that each component of the complement of their union is contractible. This…

This paper is about hyperbolic properties on planar graphs. First, we study the relations among various kinds of strong isoperimetric inequalities on planar graphs and their duals. In particular, we show that a planar graph satisfies a…

We give a sufficient condition under which the fundamental group of a reglued graph of surfaces is hyperbolic. A reglued graph of surfaces is constructed by cutting a fixed graph of surfaces along the edge surfaces, then regluing by…

Let $\mathcal{C}(S_{g,p})$ denote the curve complex of the closed orientable surface of genus $g$ with $p$ punctures. Masur-Minksy and subsequently Bowditch showed that $\mathcal{C}(S_{g,p})$ is $\delta$-hyperbolic for some…