Narrow Results

Let M be a positive integer and let f be a holomorphic mapping from a ball Δⁿ = {x ∈ ℂⁿ;|x| < δ} into ℂⁿ such that the origin 0 is an isolated fixed point of both f and the M-th iteration f^M of f. Then one can define the number , which can be interpreted to be the number of periodic orbits of f with period M hidden at the fixed point 0.

For a 3 × 3 matrix A, of which the eigenvalues are all distinct primitive M-th roots of unity, we will give a sufficient and necessary condition for A such that for any holomorphic mapping f: Δ ³ → ℂ³ with f(0) = 0 and Df(0) = A, if 0 is an isolated fixed point of the M-th iteration f^M, then .

We use generating function techniques developed by Givental, Théret and ourselves to deduce a proof in $ℂ{\text{P}}^d$ of the homological generalization of Franks theorem due to Shelukhin. This result proves in particular the Hofer–Zehnder conjecture in the nondegenerate case: every Hamiltonian diffeomorphism of $ℂ{\text{P}}^d$ that has at least $d+2$ nondegenerate periodic points has infinitely many periodic points. Our proof does not appeal to Floer homology or the theory of $J$-holomorphic curves. An appendix written by Shelukhin contains a new proof of the Smith-type inequality for barcodes of Hamiltonian diffeomorphisms that arise from Floer theory, which lends itself to adaptation to the setting of generating functions.

Let $X$ be a klt projective variety with numerically trivial canonical divisor. A surjective endomorphism $f:X\to X$ is amplified (respectively, quasi-amplified) if $f^{\ast }D-D$ is ample (respectively, big) for some Cartier divisor $D$. We show that after iteration and equivariant birational contractions, a quasi-amplified endomorphism will descend to an amplified endomorphism. As an application, when $X$ is Hyperkähler, $f$ is quasi-amplified if and only if it is of positive entropy. In both cases, $f$ has Zariski dense periodic points. When $X$ is an abelian variety, we give and compare several cohomological and geometric criteria of amplified endomorphisms and endomorphisms with countable and Zariski dense periodic points (after an uncountable field extension).

Interval maps reveal precious information about the chaotic behavior of general nonlinear systems. If an interval map f:I→I is chaotic, then its iterates fⁿ will display heightened oscillatory behavior or profiles as n→∞. This manifestation is quite intuitive and is, here in this paper, studied analytically in terms of the total variations of fⁿ on subintervals. There are four distinctive cases of the growth of total variations of fⁿ as n→∞:

(i) the total variations of fⁿ on I remain bounded;

(ii) they grow unbounded, but not exponentially with respect to n;

(iii) they grow with an exponential rate with respect to n;

(iv) they grow unbounded on every subinterval of I.

We study in detail these four cases in relations to the well-known notions such as sensitive dependence on initial data, topological entropy, homoclinic orbits, nonwandering sets, etc. This paper is divided into three parts. There are eight main theorems, which show that when the oscillatory profiles of the graphs of fⁿ are more extreme, the more complex is the behavior of the system.

We consider the behavior of piecewise isometries in Euclidean spaces. We show that if n is odd and the system contains no orientation reversing isometries then recurrent orbits with rational coding are not expected. More precisely, a prevalent set of piecewise isometries do not have recurrent points having rational coding. This implies that when all atoms are convex no periodic points exist for almost every piecewise isometry.

By contrast, if n≥2 is even then periodic points are stable for almost every piecewise isometry whose set of defining isometries are not orientation reversing. If, in addition, the defining isometries satisfy an incommensurability condition then all unbounded orbits must be irrationally coded.

As is well known, a continuous function f : ℝ → ℝ that has a cycle of period 3 will have cycles of any given period n. Let c_n be the minimum of the numbers of distinct cycles of period n, over all possible f with a 3-cycle. We show how c_n can be calculated from a simple recursive formula, through elementary and geometrical considerations.

In this paper certain identities are generated for minimal polynomials Θ_n(z) of numbers 2cos(2π/n) and for Chebyshev polynomials of the first kind. Against the background of these identities, sets of cyclic points of some modified versions of these polynomials are derived. The discussion leads to the analysis of a certain new counting sequence of numbers connected with Carmichael function.

It is well-known that binary-valued cellular automata, which are defined by simple local rules, have the amazing feature of generating very complex patterns and having complicated dynamical behaviors. In this paper, we present a new type of cellular automaton based on real-valued states which produce an even greater amount of interesting structures such as fractal, chaotic and hypercyclic. We also give proofs to real-valued cellular systems which have fixed points and periodic solutions.

This paper is concerned with the study of the iteration of the quadratic coquaternionic map $f_{\mathrm{\text{c}}}(\mathrm{\text{q}})={\mathrm{\text{q}}}^2+\mathrm{\text{c}}$, where c is a fixed coquaternionic parameter. The fixed points and periodic points of period two are determined, revealing the existence of a type of sets of these points which do not occur in the classical complex case: sets of nonisolated points. This brings the need to consider a different concept of stability. The analysis of the stability, in this new sense, of the sets of fixed points and periodic points is performed and a discussion of certain type of bifurcations which occur, in the case of a real parameter c, is also presented.

It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free language.

We introduce the notion of exotic periodic points of a meromorphic self-map. We then establish the expected asymptotic for the number of isolated or exotic periodic points for holomorphic self-maps with a simple action on the cohomology groups on a compact Kähler manifold.

Let ${𝒜}$ be a $G{L}_d(ℝ)$-valued cocycle over a subshift of finite type. Under a certain twisting assumption, we prove that ${𝒜}$ has a uniform Lyapunov exponent if and only if the largest Lyapunov exponent of ${𝒜}$ at all periodic points equals. Under the typicality assumption, we give two checkable criteria for deciding whether ${𝒜}$ has uniform singular value exponents.

In this paper, we study orbits and fixed points of polynomials in a general skew polynomial ring $D[x,\sigma ,\delta ]$. We extend results of the first author and Vishkautsan on polynomial dynamics in $D[x]$. In particular, we show that if $a\in D$ and $f\in D[x,\sigma ,\delta ]$ satisfy $f(a)=a$, then $f^{\circ n}(a)=a$ for every formal power of $f$. More generally, we give a sufficient condition for a point $a$ to be $r$-periodic with respect to a polynomial $f$. Our proofs build upon foundational results on skew polynomial rings due to Lam and Leroy.

Let F be a global field, let φ ∈ F(x) be a rational map of degree at least 2, and let α ∈ F. We say that α is periodic if φⁿ(α) = α for some n ≥ 1. A Hasse principle is the idea, or hope, that a phenomenon which happens everywhere locally should happen globally as well. The principle is well known to be true in some situations and false in others. We show that a Hasse principle holds for periodic points, and further show that it is sufficient to know that α is periodic on residue fields for every prime in a set of natural density 1 to know that α is periodic in F.

Explicit solutions of the cubic Fermat equation are constructed in ring class fields ${\mathrm{\Omega }}_f$, with conductor $f$ prime to $3$, of any imaginary quadratic field $K$ whose discriminant satisfies $d_K\equiv 1$ (mod $3$), in terms of the Dedekind $\eta$-function. As $K$ and $f$ vary, the set of coordinates of all solutions is shown to be the exact set of periodic points of a single algebraic function and its inverse defined on natural subsets of the maximal unramified, algebraic extension $K_3$ of the $3$-adic field ${\mathbb{Q}}_3$. This is used to give a dynamical proof of a class number relation of Deuring. These solutions are then used to give an unconditional proof of part of Aigner’s conjecture: the cubic Fermat equation has a nontrivial solution in $K=\mathbb{Q}(\sqrt{-d})$ if $d_K\equiv 1$ (mod $3$) and the class number $h(K)$ is not divisible by $3$. If $3|h(K)$, congruence conditions for the trace of specific elements of ${\mathrm{\Omega }}_f$ are exhibited which imply the existence of a point of infinite order in ${\mathrm{\text{Fer}}}_3(K)$.

For every nonconstant rational function $\varphi \in \mathbb{Q}(x)$, the Galois groups of the dynatomic polynomials of $\varphi$ encode various properties of $\varphi$ are of interest in the subject of arithmetic dynamics. We study here the structure of these Galois groups as $\varphi$ varies in a particular one-parameter family of maps, namely, the quadratic rational maps having a critical point of period 2. In particular, we provide explicit descriptions of the third and fourth dynatomic Galois groups for maps in this family.

For a prime p, positive integers $r,n$, and a polynomial f with coefficients in ${𝔽}_{p^r}$, let $W_{p,r,n}(f)=f^n\left({𝔽}_{p^r}\right)\backslash f^{n+1}\left({𝔽}_{p^r}\right)$. As n varies, the $W_{p,r,n}(f)$ partition the set of strictly preperiodic points of the dynamical system induced by the action of f on ${𝔽}_{p^r}$. In this paper, we compute statistics of strictly preperiodic points of dynamical systems induced by unicritical polynomials over finite fields by obtaining effective upper bounds for the proportion of ${𝔽}_{p^r}$ lying in a given $W_{p,r,n}(f)$. Moreover, when we generalize our definition of $W_{p,r,n}(f)$, we obtain both upper and lower bounds for the resulting averages.

We investigate the dynamics of the Nash better response map for a family of games with two players and two strategies. This family contains the games of Coordination, Stag Hunt and Chicken. Each map is a piecewise rational map of the unit square to itself. We describe completely the dynamics for all maps from the family. All trajectories converge to fixed points or period 2 orbits. We create tools that should be applicable to other systems with similar behavior.