Related papers: Non-statistical rational maps

Biharmonic and conformal-biharmonic maps are two fourth-order generalizations of the well-studied notion of harmonic maps in Riemannian geometry. In this article we consider maps into the Euclidean sphere and investigate a geometric…

The dominant rational maps of finite degree from a fixed variety to varieties of general type, up to birational isomorphisms, form a finite set. This has been known as the Iitaka-Severi conjecture, and is nowdays an established result, in…

Thurston maps are branched self-coverings of the sphere whose critical points have finite forward orbits. We give combinatorial and algebraic characterizations of Thurston maps that are isotopic to expanding maps as "Levy-free" maps and as…

This paper deals with both complex dynamical systems and conformal iterated function systems. We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a $d$-parameter family of…

In this paper we generalize renormalization theory for analytic critical circle maps with a cubic critical point to the case of maps with an arbitrary odd critical exponent by proving a quasiconformal rigidity statement for renormalizations…

What is the ergodic behaviour of numerically computed segments of orbits of a diffeomorphism? In this paper, we try to answer this question for a generic conservative $C^1$-diffeomorphism, and segments of orbits of Baire-generic points. The…

We provide new examples of integrable rational maps in four dimensions with two rational invariants, which have unexpected geometric properties, as for example orbits confined to non algebraic varieties, and fall outside classes studied by…

Given a multimodal interval map $f:I \to I$ and a H\"older potential $\phi:I \to \mathbb{R}$, we study the dimension spectrum for equilibrium states of $\phi$. The main tool here is inducing schemes, used to overcome the presence of…

We make the first steps towards an understanding of the ergodic properties of a rational map defined over a complete algebraically closed non-archimedean field. For such a rational map R, we construct a natural invariant probability measure…

We present a new approach to Bayesian inference that entirely avoids Markov chain simulation, by constructing a map that pushes forward the prior measure to the posterior measure. Existence and uniqueness of a suitable measure-preserving…

We demonstrate the existence of finite-system multicriticality in a qubit-boson model where biased qubits collectively coupled to a single-mode bosonic field. The interplay between biases and boson-qubit coupling produces a rich phase…

We consider a two-parameter family of maps $T_{\alpha, \beta}: [0,1] \to [0,1]$ with a neutral fixed point and a non-flat critical point. Building on a cone technique due to Baladi and Todd, we show that for a class of $L^q$ observables…

A holomorphic endomorphism of $\mathbb{CP}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study…

We study skew-product dynamics for a large class of finitely-generated semi--hyperbolic semigroups of rational maps acting on the Riemann sphere, which generalizes both the theory of iteration of a single rational map of a single complex…

We prove that the sub-Riemannian exponential map is not injective in any neighbourhood of certain critical points. Namely that it does not behave like the injective map of reals given by $f(x) = x^3$ near its critical point $x = 0$. As a…

Finite-size critical systems defined on a parallel plate geometry of finite extent along one single ($z$) direction with Dirichlet and Neumann boundary conditions at $z=0,L$ are analyzed in momentum space. We introduce a modified…

We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We provide proof that a random backward iteration algorithm to draw the pictures of the Julia sets, previously proven to work in the…

We establish an approximation of the activity current $T_c$ in the parameter space of a holomorphic family $f$ of rational functions having a marked critical point $c$ by parameters for which $c$ is periodic under $f$, i.e., is a…

The survey covers several topics related to the asymptotic structure of various combinatorial and analytic objects such as the path spaces in graded graphs (Bratteli diagrams), invariant measures with respect to countable groups, etc. The…

The behavior under iteration of the critical points of polynomial maps plays an essential role in understanding its dynamics. We study the special case where the forward orbits of the critical points are finite. Thurston's theorem tells us…