Related papers: Random complex dynamics and semigroups of holomorp…
Random matrix theory (RMT) universality is the defining property of quantum mechanical chaotic systems, and can be probed by observables like the spectral form factor (SFF). In this paper, we describe systematic deviations from RMT…
There are many classical results, related to the Denjoy--Wolff Theorem, concerning the relationship between orbits of interior points and orbits of boundary points under iterates of holomorphic self-maps of the unit disc. Here, for the…
We study bifurcation behavior in periodic perturbations of two-dimensional symmetric systems exhibiting codimension-two bifurcations with a double eigenvalue when the frequencies of the perturbation terms are small. We transform the…
For polynomials and rational maps of fixed degree over a finite field, we bound both the average number of connected components of their functional graphs as well as the average number of periodic points of their associated dynamical…
We evaluate the variance of coefficients of the characteristic polynomial for binary quantum graphs using a dynamical approach. This is the first example where a spectral statistic can be evaluated in terms of periodic orbits for a system…
We investigate the nonlinear holomorphic supersymmetry for quantum-mechanical systems on Riemann surfaces subjected to an external magnetic field. The realization is shown to be possible only for Riemann surfaces with constant curvature…
Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in…
We study time-reversal symmetry in dynamical systems with finite phase space, with applications to birational maps reduced over finite fields. For a polynomial automorphism with a single family of reversing symmetries, a universal (i.e.,…
The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such…
The dynamics of one dimensional iterative maps in the regime of fully developed chaos is studied in detail. Motivated by the observation of dynamical structures around the unstable fixed point we introduce the geometrical concept of a…
Given a random sequence of holomorphic maps $f_1,f_2,f_3,...$ of the unit disk $\Delta$ to a subdomain $X$, we consider the compositions $$F_n=f_1 \circ f_{2} \circ ... f_{n-1} \circ f_n.$$ The sequence $\{F_n\}$ is called the {\em iterated…
Dynamical systems at the edge of chaos, which have been considered as models of self-organization phenomena, are marked by their ability to perform nontrivial computations. To distinguish them from systems with limited computing power, we…
Quantum chaotic systems exhibit certain universal statistical properties that closely resemble predictions from random matrix theory (RMT). With respect to observables, it has recently been conjectured that, when truncated to a sufficiently…
We show that the classical kernel and domain functions associated to an n-connected domain in the plane are all given by rational combinations of three or fewer holomorphic functions of one complex variable. We characterize those domains…
We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both…
We investigate a randomly evolving process of subgraphs in an underlying host graph using the spectral theory of semigroups related to the Tsetlin library and hyperplane arrangements. Starting with some initial subgraph, at each iteration,…
We show that the dynamical degree of an (i.i.d) random sequence of dominant, rational self-maps on projective space is almost surely constant. We then apply this result to height growth and height counting problems in random orbits.
We analyze the infinitesimal effect of holomorphic perturbations of the dynamics of a structurally stable rational map on a neighborhood of its Julia set. This implies some restrictions on the behavior of critical points.
We define polygonal dynamics as a family of dynamical systems acting on points in projective spaces. The most famous example is the pentagram map. Similar collapsing phenomena seem to occur in most of these systems. We prove it in some…
The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a…