We consider the family $\mathrm {MC}_d$ of monic centered polynomials of one complex variable with degree $d \geq 2$, and study the map $\widehat {\Phi }_d:\mathrm {MC}_d\to \widetilde {\Lambda }_d \subset \mathbb {C}^d / \mathfrak {S}_d$ which maps each $f \in \mathrm {MC}_d$ to its unordered collection of fixed-point multipliers. We give an explicit formula for counting the number of elements of each fiber $\widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$ for every $\bar {\unicode{x3bb} } \in \widetilde {\Lambda }_d$ except when the fiber $\widehat {\Phi }_d^{-1}(\bar {\unicode{x3bb} })$ contains polynomials having multiple fixed points. This formula is not a recursive one, and is a drastic improvement of our previous result [T. Sugiyama. The moduli space of polynomial maps and their fixed-point multipliers. Adv. Math. 322 (2017), 132–185] which gave a rather long algorithm with some induction processes.

This paper establishes the geometric rigidity of certain holomorphic correspondences in the family $(w-c)^q=z^p$ , whose post-critical set is finite in any bounded domain of $\mathbb {C}$ . In spite of being rigid on the sphere, such correspondences are J-stable by means of holomorphic motions when viewed as maps of $\mathbb {C}^2$ . The key idea is the association of a conformal iterated function system to the return branches near the critical point, giving a global description of the post-critical set and proving the hyperbolicity of these correspondences.

We investigate the location of zeros for the partition function of the anti-ferromagnetic Ising model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal. The proofs rely on describing the rational dynamical systems arising when considering ratios of partition functions on recursively defined trees.

For every $m\in \mathbb {N}$ , we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in $\mathbb {C}\setminus \{0\}$ under the $m$ th order derivatives of the iterates of a polynomials $f\in \mathbb {C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of f with pole at $\infty $ . We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field k for a sequence of effective divisors on $\mathbb {P}^1(\overline {k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of $\mathbb {C}^2$ has a given eigenvalue.

Our main result states that, under an exponential map whose Julia set is the whole complex plane, on each piecewise smooth Jordan curve there is a point whose orbit is dense. This has consequences for the boundaries of nice sets, used in induction methods to study ergodic and geometric properties of the dynamics.

The study of the dynamics of an holomorphic map near a fixed point is a central topic in complex dynamical systems. In this paper, we will consider the corresponding random setting: given a probability measure $\unicode[STIX]{x1D708}$ with compact support on the space of germs of holomorphic maps fixing the origin, we study the compositions $f_{n}\circ \cdots \circ f_{1}$, where each $f_{i}$ is chosen independently with probability $\unicode[STIX]{x1D708}$. As in the deterministic case, the stability of the family of the random iterates is mostly determined by the linear part of the germs in the support of the measure. A particularly interesting case occurs when all Lyapunov exponents vanish, in which case stability implies simultaneous linearizability of all germs in $\text{supp}(\unicode[STIX]{x1D708})$.

Understanding spatial distribution has become increasingly important in weed science. Seed dispersal, both between and within agricultural fields, is an important component of weed spatial distribution. Analysis of the effect of dispersal between fields has been relatively neglected in theoretical studies of weed population dynamics. In this paper, we present a simple landscape-level model of the influence of seed dispersal on winter wild oat population dynamics between fields. In the model, two fields are interconnected, with seeds being carried from one field to another as would occur when seeds are carried by field equipment or in irrigation water. The model is intended to characterize the effect of field-level weed management decisions on landscape-level weed population dynamics. Three scenarios were studied. The first employed no control measures in either field. The second employed annual application of herbicides in field 1 with field 2 receiving no treatment. In the third scenario, an herbicide application took place in field 1 only if the weed population in that field exceeded an action threshold. In the first scenario, the net result of the immigration and emigration processes determined the increase or decrease of the stable plant population. In the second scenario, weeds in the controlled field (field 1) were not driven to extinction as might be expected. The weed populations grew for practically all the dispersal parameter space. Each change in the parameter's values produced a new stable equilibrium. This situation might correspond to a multiplicity of stable states. The uncontrolled field (field 2) experienced an indirect control effect due to the use of control measures in field 1. In the third scenario, we observed an interesting behavior of the populations in both fields. The population in field 1 was not driven under the economic threshold, and both fields showed complex dynamics within defined combinations of migration and emigration values.

The paper presents an agent-based model to study the possible effects of different fiscal and monetary policies in the context of debt deflation. We introduce a modified Taylor rule that includes the financial position of firms as a target. Monte Carlo simulations provide a representation of the complex feedback effects generated by the interaction among the different transmission channels of monetary policy. The model also reproduces the evidence of low inflation during stock market booms and shows how it can lead to overinvestment and destabilize the system. The paper also investigates the possible reasons behind this stylized fact by testing different behavioral rules for the central bank. We find that, in a context of sticky prices and volatile expectations, endogenous credit creation can be identified as the main source of the divergent dynamics of prices in the real and financial sectors.

This paper shows that complex dynamics arises naturally in deterministic discrete choice problems. In particular, it shows that if the objective function of a maximization problem can be written as a function of a sequence of discrete variables, and if the (maximized) value function is strictly increasing in an exogenous variable, then for almost all values of the exogenous variable, any optimal path exhibits aperiodic dynamics. This result is applied to a maximization problem with indivisible durable goods, as well as to a Ramsey model with an indivisible consumption good. In each model, it is shown that optimal dynamics is almost always complex. These results are illustrated with various numerical examples.

This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by exp$({{C}_{k}}{{\left| s \right|}^{\delta }})$ in strips $|\,\text{Re}\,s\,|\,\le \,K$, where $\delta$ is the dimension of the Julia set. This leads to bounds on the number of zeros in strips (interpreted as the Pollicott–Ruelle resonances of this dynamical system). An upper bound on the number of zeros in polynomial regions $\left\{ \left| \,\operatorname{Re}s \right|\,\le \,{{\left| \,\operatorname{Im}s \right|}^{\alpha }} \right\}$ is given, followed by weaker lower bound estimates in strips $\{\text{Re}\,s\,>\,-C,\,|\,\text{IM}\,s|\,\le \,r\}$, and logarithmic neighbourhoods $\{|\text{Re}s|\le {{}_{\rho }}\log |\operatorname{Im}s|\}$. Recent numerical work of Strain-Zworski suggests the upper bounds in strips are optimal.

We investigate the dynamics of an integrated Keynesian disequilibrium model of monetary growth that allows for a variety of labor-market and employment-adjustment processes. The structure of the model is naturally nonlinear, i.e., no extrinsic nonlinear economic behavioral relationships are imposed at first. The dynamics of the model are nine-dimensional and are investigated analytically by considering appropriate subdynamics. The model generates limit cycles (via Hopf bifurcations) and more complex dynamic behavior (when a natural kink in the money-wage Phillips curve is taken into account). It exhibits hysteresis effects with respect to long-run unemployment as well as growth and implies the occurrence of steady-state depressions in particular.