Related papers: Equilibrium measure for one-dimensional Lorenz-lik…

For a closed subset $K$ of a compact metric space $A$ possessing an $\alpha$-regular measure $\mu$ with $\mu(K)>0$, we prove that whenever $s>\alpha$, any sequence of weighted minimal Riesz $s$-energy configurations…

In the context of smooth interval maps, we study an inducing scheme approach to prove existence and uniqueness of equilibrium states for potentials $\phi$ with he `bounded range' condition $\sup \phi - \inf \phi < \htop$, first used by…

We prove strong statistical stability of a large class of one-dimensional maps which may have an arbitrary finite number of discontinuities and of non-degenerate critical points and/or singular points with infinite derivative, and satisfy…

We examine uniqueness of equilibrium states for the natural extension of a topologically exact, non-uniformly expanding, local homeomorphism with a H\"older continuous potential function. We do this by applying general techniques developed…

We consider families of multimodal interval maps with polynomial growth of the derivative along the critical orbits. For these maps Bruin and Todd have shown the existence and uniqueness of equilibrium states for the potential…

We prove that, for every $\alpha > -1$, the pull-back measure $\phi ({\cal A}_\alpha)$ of the measure $d{\cal A}_\alpha (z) = (\alpha + 1) (1 - |z|^2)^\alpha \, d{\cal A} (z)$, where ${\cal A}$ is the normalized area measure on the unit…

Let $(\mathcal{X}, \rho, \mu)$ be a metric measure space of homogeneous type which supports a certain Poincar\'e inequality. Denote by the symbol $\mathcal{C}_{\mathrm{c}}^\ast(\mathcal{X})$ the space of all continuous functions $f$ with…

Length-constrained expander decompositions are a new graph decomposition that has led to several recent breakthroughs in fast graph algorithms. Roughly, an $(h, s)$-length $\phi$-expander decomposition is a small collection of length…

We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight H\"older-behavior for this operation under minimal assumptions. Our…

We develop a thermodynamic formalism for a strongly dissipative H\'enon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. For any $t\in\mathbb R$…

We study the equilibrium measure for a logarithmic potential in the presence of an external field V*(x) + tp(x), where t is a parameter, V*(x) is a smooth function and p(x) a monic polynomial. When p(x) is of an odd degree, the equilibrium…

We prove a couple of results on local continuous extension of proper holomorphic maps $F:D \rightarrow \Omega$, $D, \Omega \varsubsetneq \mathbb{C}^n$, making local assumptions on $\partial{D}$ and $\partial{\Omega}$. The first result…

We investigate the behaviour of solutions $\phi = \phi^{(p)}$ to the one-dimensional nonlinear wave equation $-\phi_{tt} + \phi_{xx} = -|\phi|^{p-1} \phi$ with initial data $\phi(0,x) = \phi_0(x)$, $\phi_t(0,x) = \phi_1(x)$, in the high…

Let $I_1=[a_0,a_1),\ldots,I_{k}= [a_{k-1},a_k)$ be a partition of the interval $I=[0,1)$ into $k$ subintervals. Let $f:I\to I$ be a map such that each restriction $f|_{I_i}$ is an increasing Lipschitz contraction. We prove that any $f$…

Let $E \subset \mathbb R^d$, $d \ge 2$, be compact, and let $\phi(x,y)$ be a smooth function satisfying the Phong--Stein rotational curvature condition on $\{\phi(x,y)=1\}$. We prove that if $\dim_{\mathcal H}(E)>1$, then $$…

Let $\mathcal{A}$ and $\mathcal{B}$ be unital finite-dimensional complex algebras, each equipped with the unique Hausdorff vector topology. Denote by $\mathrm{Max}(\mathcal{A})=\{\mathcal{M}_1, \ldots, \mathcal{M}_p\}$ and…

Let $\phi(x) = |1 - \frac 1x|$ for all $x > 0$. Then we extend $\phi(x)$ in the usual way to become a continuous map from the compact topological (but not metric) space $[0, \infty]$ onto itself which also maps the set of irrational points…

Suppose that $p \in (1,\infty]$, $\nu \in [1/2,\infty)$, $\mathcal{S}_\nu = \left\{ (x_1,x_2) \in \mathbb{R}^2 \setminus \{(0, 0)\}: |\phi| < \frac{\pi}{2\nu}\right\}$, where $\phi$ is the polar angle of $(x_1,x_2)$. Let $R>0$ and…

We show that, for a robust ($C^2$-open) class of random non-uniformly expanding maps, there exists equilibrium states for a large class of potentials.In particular, these sytems have measures of maximal entropy. These results also give a…

We consider the regularity of measurable solutions $\chi$ to the cohomological equation \[ \phi = \chi \circ T -\chi, \] where $(T,X,\mu)$ is a dynamical system and $\phi \colon X\rightarrow \R$ is a $C^k$ valued cocycle in the setting in…