Forward Orbit Research Articles

Linear and projective boundaries in HNN-extensions and distortion phenomena

Abstract Linear and projective boundaries of Cayley graphs were introduced in [Glasg. Math. J. (2014), DOI 10.1017/S0017089514000512] as quasi-isometry invariant boundaries of finitely generated groups. They consist of forward orbits g ∞ = {gi : i ∈ ℕ}, or orbits g ±∞ = {gi : i ∈ ℤ}, respectively, of non-torsion elements g of the group G, where `sufficiently close' (forward) orbits become identified, together with a metric bounded by 1. We show that for all finitely generated groups, the distance between the antipodal points g ∞ and g -∞ in the linear boundary is bounded from below by 1/21/2, and we give an example of a group which has two antipodal elements of distance at most (12/17)1/2 < 1. Our example is a derivation of the Baumslag–Gersten group. We also exhibit a group with elements g and h such that g ∞ = h ∞, but g -∞ ≠ h -∞. Furthermore, we introduce a notion of average-case-distortion—called growth—and compute explicit positive lower bounds for distances between points g ∞ and h ∞ which are limits of group elements g and h with different growth.

Schmidt games and non-dense forward orbits of certain partially hyperbolic systems

Let$f:M\rightarrow M$be a partially hyperbolic diffeomorphism with conformality on unstable manifolds. Consider a set of points with non-dense forward orbit:$E(f,y):=\{z\in M:y\notin \overline{\{f^{k}(z),k\in \mathbb{N}\}}\}$for some$y\in M$. Define$E_{x}(f,y):=E(f,y)\cap W^{u}(x)$for any$x\in M$. Following a method of Broderick, Fishman and Kleinbock [Schmidt’s game, fractals, and orbits of toral endomorphisms.Ergod. Th. & Dynam. Sys.31(2011), 1095–1107], we show that$E_{x}(f,y)$is a winning set for Schmidt games played on$W^{u}(x)$which implies that$E_{x}(f,y)$has Hausdorff dimension equal to$\dim W^{u}(x)$. Furthermore, we show that for any non-empty open set$V\subset M$,$E(f,y)\cap V$has full Hausdorff dimension equal to$\dim M$, by constructing measures supported on$E(f,y)\cap V$with lower pointwise dimension converging to$\dim M$and with conditional measures supported on$E_{x}(f,y)\cap V$. The results can be extended to the set of points with forward orbit staying away from a countable subset of$M$.

Intersections of multiplicative translates of 3-adic Cantor sets

This paper is motivated by questions concerning the discrete dynamical system on the 3 -adic integers \mathbb{Z}_{3} given by multiplication by 2 . The exceptional set \mathcal{E}(\mathbb{Z}_3) is defined to be the set of all elements of \mathbb{Z}_3 whose forward orbits under this action intersect the 3 -adic Cantor set \Sigma_{3, \bar{2}} (of 3 -adic integers whose expansions omit the digit 2 ) infinitely many times. It has been shown that this set has Hausdorff dimension at most \frac{1}{2} , and it is conjectured that it has Hausdorff dimension 0 . Upper bounds on its Hausdorff dimension can be obtained with sufficient knowledge of Hausdorff dimensions of intersections of multiplicative translates of Cantor sets by powers of 2 . This paper studies more generally the structure of finite intersections of general multiplicative translates S= \Sigma_{3, \bar{2}} \cap \frac{1}{M_1} \Sigma_{3, \bar{2}} \cap \dots \cap \frac{1}{M_n} \Sigma_{3, \bar{2}} by integers 1 < M_1 < M_2 < \dots < M_n . These sets are describable as sets of 3 -adic integers whose 3 -adic expansions have one-sided symbolic dynamics given by a finite automaton. This paper gives a method to determine the automaton for given data (M_1, \dots, M_n) and to compute the Hausdorff dimension, which is always of the form \log_{3}(\beta) where \beta is an algebraic integer. Computational examples indicate that in general the Hausdorff dimension of such sets depends in a very complicated way on the integers M_1, \dots, M_n . Exact answers are obtained for certain infinite families, which show as a corollary that a relaxed notion of generalized exceptional set has a positive Hausdorff dimension.

3x+1 inverse orbit generating functions almost always have natural boundaries

The $3x+k$ function $T_{k}(n)$ sends $n$ to $(3n+k)/2$ resp. $n/2,$ according as $n$ is odd, resp. even, where $k \equiv \pm 1~(\bmod \, 6)$. The map $T_k(\cdot)$ sends integers to integers, and for $m \ge 1$ let $n \rightarrow m$ mean that $m$ is in the forward orbit of $n$ under iteration of $T_k(\cdot).$ We consider the generating functions $f_{k,m}(z) = \sum_{n>0, n \rightarrow m} z^{n},$ which are holomorphic in the unit disk. We give sufficient conditions on $(k,m)$ for the functions $f_{k, m}(z)$ have the unit circle $\{|z|=1\}$ as a natural boundary to analytic continuation. For the $3x+1$ function these conditions hold for all $m \ge 1$ to show that $f_{1,m}(z)$ has the unit circle as a natural boundary except possibly for $m= 1, 2, 4$ and $8$. The $3x+1$ Conjecture is equivalent to the assertion that $f_{1, m}(z)$ is a rational function of $z$ for the remaining values $m=1,2, 4, 8$.

Integral orbits over function fields

Over the function field of a smooth projective curve over an algebraically closed field, we investigate the set of S-integral elements in a forward orbit under a rational function by establishing some analogues of the classical Siegel theorem.

Attracting cycles in p-adic dynamics and height bounds for postcritically finite maps

A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a non-archimedean version of Fatou's classical result that every attracting cycle of a rational function over the complex numbers attracts a critical point.

Law of iterated logarithm and invariance principle for one-parameter families of interval maps

We show that for almost every map in a transversal one-parameter family of piecewise expanding unimodal maps the Birkhoff sum of suitable observables along the forward orbit of the turning point satisfies the law of iterated logarithm. This result will follow from an almost sure invariance principle for the Birkhoff sum, as a function on the parameter space. Furthermore, we obtain a similar result for general one-parameter families of piecewise expanding maps on the interval.

Some family of center manifolds of a fixed indeterminate point

In this article, we study the local dynamical structure of a rational mapping F of P2 at a fixed indeterminate point p. In the previous paper, using a sequence of points which is defined by blow-ups, we have constructed an invariant family of holomorphic curves at p. In this paper, using the same sequence of points, we approximate a set of points whose forward orbits stay in a neighborhood of p. Moreover, for a specific rational mapping we construct a family {Wj}j$\in$ {1,2}N of center manifolds of p. The main result of this paper is to give the asymptotic expansion of the defining function of Wj.

Determination of all rational preperiodic points for morphisms of PN

For a morphism f : P N → P N f:\mathbb {P}^N \to \mathbb {P}^N , the points whose forward orbit by f f is finite are called preperiodic points for f f . This article presents an algorithm to effectively determine all the rational preperiodic points for f f defined over a given number field K K . This algorithm is implemented in the open-source software Sage for Q \mathbb {Q} . Additionally, the notion of a dynatomic zero-cycle is generalized to preperiodic points. Along with examining their basic properties, these generalized dynatomic cycles are shown to be effective.

The structure of limit sets for $\mathbb{Z}^d$ actions

Central to the study of $\mathbb{Z}$ actions on compact metric spaces is the $\omega$-limit set, the set of all limit points of a forward orbit. A closed set $K$ is <em>internally chain transitive</em> provided for every $x,y\in K$ there is an $\epsilon$-pseudo-orbit of points from $K$ that starts with $x$ and ends with $y$. It is known in several settings that the property of internal chain transitivity characterizes $\omega$-limit sets. In this paper, we consider actions of $\mathbb{Z}^d$ on compact metric spaces. We give a general definition for shadowing and limit sets in this setting. We characterize limit sets in terms of a more general internal property which we call <em>internal mesh transitivity</em>.

On Schmidt's game and the set of points with non-dense orbits under a class of expanding maps

Given an expanding interval map, we consider sets of points whose forward orbits are bounded away from a given point. It is shown that such sets are 1/2-winning in the sense of Schmidt's game for a class of expanding interval maps including the Gauss transformation and all beta transformations. This improves upon several results dating back to a work of W. Schmidt.

Galois theory of quadratic rational functions

For a number field K with absolute Galois group G_K , we consider the action of G_K on the infinite tree of preimages of \alpha \in K under a degree-two rational function \phi \in K(x) , with particular attention to the case when \phi commutes with a non-trivial Möbius transformation. In a sense this is a dynamical systems analogue to the \ell -adic Galois representation attached to an elliptic curve, with particular attention to the CM case. Using a result about the discriminants of numerators of iterates of \phi , we give a criterion for the image of the action to be as large as possible. This criterion is in terms of the arithmetic of the forward orbits of the two critical points of \phi . In the case where \phi commutes with a non-trivial Möbius transformation, there is in effect only one critical orbit, and we give a modified version of our maximality criterion. We prove a Serre-type finite-index result in many cases of this latter setting.

Examples of dynamical degree equals arithmetic degree

Let f : X --> X be a dominant rational map of a projective variety defined over a number field. An important geometric-dynamical invariant of f is its (first) dynamical degree d_f= lim SpecRadius((f^n)^*)^{1/n}. For algebraic points P of X whose forward orbits are well-defined, there is an analogous (upper) arithmetic degree a_f(P) = limsup h_X(f^n(P))^{1/n}, where h_X is an ample Weil height on X. In an earlier paper, we proved the fundamental inequality a_f(P) \le d_f and conjectured that a_f(P) = d_f whenever the orbit of P is Zariski dense. In this paper we show that the conjecture is true for several types of maps. In other cases, we provide support for the conjecture by proving that there is a Zariski dense set of points with disjoint orbits and satisfying a_f(P) = d_f.

Foundations for an iteration theory of entire quasiregular maps

The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire functions. Here the Julia set is defined as the set of all points such that complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.

Invariant varieties for polynomial dynamical systems

We study algebraic dynamical systems (and, more generally, s -varieties) F:A n C ?A n C given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials. More precisely, we find a nearly canonical way to write a polynomial as a composition of �clusters� from which one may easily read off possible compositional identities. Our main result is an explicit description of the (weakly) skew-invariant varieties, that is, for a fixed field automorphism s:C?C those algebraic varieties X?A n C for which F(X)?X s . As a special case, we show that if f(x)?C[x] is a polynomial of degree at least two that is not conjugate to a monomial, Chebyshev polynomial or a negative Chebyshev polynomial, and X?A 2 C is an irreducible curve that is invariant under the action of (x,y)?(f(x),f(y)) and projects dominantly in both directions, then X must be the graph of a polynomial that commutes with f under composition. As consequences, we deduce a variant of a conjecture of Zhang on the existence of rational points with Zariski dense forward orbits and a strong form of the dynamical Manin-Mumford conjecture for liftings of the Frobenius. We also show that in models of ACFA 0 , a disintegrated set defined by s(x)=f(x) for a polynomial f has Morley rank one and is usually strongly minimal, that model theoretic algebraic closure is a locally finite closure operator on the nonalgebraic points of this set unless the skew-conjugacy class of f is defined over a fixed field of a power of s , and that nonorthogonality between two such sets is definable in families if the skew-conjugacy class of f is defined over a fixed field of a power of s

On the Hyperbolicity of Lorenz Renormalization

We consider infinitely renormalizable Lorenz maps with real critical exponent $\alpha>1$ and combinatorial type which is monotone and satisfies a long return condition. For these combinatorial types we prove the existence of periodic points of the renormalization operator, and that each map in the limit set of renormalization has an associated unstable manifold. An unstable manifold defines a family of Lorenz maps and we prove that each infinitely renormalizable combinatorial type (satisfying the above conditions) has a unique representative within such a family. We also prove that each infinitely renormalizable map has no wandering intervals and that the closure of the forward orbits of its critical values is a Cantor attractor of measure zero.

Chaotic dynamics of a quasiregular sine mapping

This article studies the iterative behaviour of a quasiregular mapping that is an analogue of a sine function. We prove that the periodic points of S form a dense subset of . We also show that the Julia set of this map is in the sense that the forward orbit under S of any non-empty open set is the whole space . The map S was constructed by Bergweiler and Eremenko (Ann. Acad. Sci. Fenn. Math. 36 (2011), pp. 165–175) who proved that the escaping set is also dense in .

STABILITY OF ORBITS VIA LYAPUNOV EXPONENTS IN AUTONOMOUS AND NONAUTONOMOUS SYSTEMS

We construct examples proving that there exist C1-dynamical systems in [0, 1) having forward orbits unstable in the sense of Lyapunov with negative Lyapunov exponent and forward orbits stable in the same sense, but having positive Lyapunov exponents. Such examples complete some results from Demir and Koçak on one-dimensional systems and prove the appearance of Perron effects in such systems. Additionally we consider similar problems in two-dimensional dynamical systems and also in the setting of nonautonomous systems.

Galois representations from pre-image trees: an arboreal survey

— Given a global fieldK and a rational function φ ∈ K(x), one may take pre-images of 0 under successive iterates of φ, and thus obtain an infinite rooted tree T∞ by assigning edges according to the action of φ. The absolute Galois group of K acts on T∞ by tree automorphisms, giving a subgroup G∞(φ) of the group Aut(T∞) of all tree automorphisms. Beginning in the 1980s with work of Odoni, and developing especially over the past decade, a significant body of work has emerged on the size and structure of this Galois representation. These inquiries arose in part because knowledge of G∞(φ) allows one to prove density results on the set of primes of K that divide at least one element of a given orbit of φ. Following an overview of the history of the subject and two of its fundamental questions, we survey in Section 2 cases where G∞(φ) is known to have finite index in Aut(T∞). While it is tempting to conjecture that such behavior should hold in general, we exhibit in Section 3 four classes of rational functions where it does not, illustrating the difficulties in formulating the proper conjecture. Fortunately, one can achieve the aforementioned density results with comparatively little information about G∞(φ), thanks in part to a surprising application of probability theory, as we discuss in Section 4. Underlying all of this analysis are results on the factorization into irreducibles of the numerators of iterates of φ, which we survey briefly in Section 5. We find that for each of these matters, the arithmetic of the forward orbits of the critical points of φ proves decisive, just as the topology of these orbits is decisive in complex dynamics. 2010 Mathematics Subject Classification. — 11R32, 37P15, 11F80.

Non-Typical Points for β-Shifts

We study sets of non-typical points under the map fβ↦βx mod 1 for non-integer β and extend our results from [Fund. Math. 209 (2010)] in several directions. In particular, we prove that sets of points whose forward orbit avoid certain Cantor sets, and the set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition β>1.541 found in the above paper can be removed.