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Nevanlinna functions are meromorphic functions with a finite number of asymptotic values and no critical values. In [KK2] it was proved that if the orbits of all the asymptotic values accumulate on a compact set on which the function acts as a repeller, then the function acts ergodically on its Julia set. In [CJK4] we proved the action of the function on its Julia set is still ergodic if some, but not all of the asymptotic values land on infinity, and the remaining ones land on a compact repeller. In this paper, we complete the characterization of ergodicity for Nevanlinna functions by proving that if all the asymptotic values land on infinity, then the Julia set is the whole sphere and the action of the map there is non-ergodic., Comment: 18 pages, 4 figures

We study Nevanlinna functions f that are transcendental meromorphic functions having N asymptotic values and no critical values. In [KK] it was proved that if the orbits of all the asymptotic values have accumulation sets that are compact and on which f is a repeller, then f acts ergodically on its Julia set. In this paper, we prove that if some, but not all of the asymptotic values have this property, while the others are prepoles, the same holds true. This is the first paper to consider this mixed case., Comment: 17 pages

We study transcendental meromorphic functions having two prepole asymptotic values and no critical values. We prove that these functions acting on their Julia sets are non-ergodic, which illustrates the antithesis of the Keen-Kotus result in [KK2] on the ergodicity of another subfamily of functions with two asymptotic values and no critical values., Comment: After we circulated this preprint, Professor Janina Kotus told us that her former Ph.D. student, Bart{\l}omiej Skorulski, has studied a more general family than the one in this paper. Therefore, we want to withdraw our paper. A reader who is interested in this interesting subject can refer to Skorulski's paper. We would like to thank Professor Janina Kotus for kindly pointing this to us

This paper is part of a general program in complex dynamics to understand parameter spaces of transcendental maps with finitely many singular values. The simplest families of such functions have two asymptotic values and no critical values. These families, up to affine conjugation, depend on two complex parameters. Understanding their parameter spaces is key to understanding families with more asymptotic values, just as understanding quadratic polynomials was for rational maps more generally. The first such families studied were the one-dimensional slices of the exponential family, $\exp(z) + a$, and the tangent family $\lambda \tan z$. The exponential case exhibited phenomena not seen for rational maps: Cantor bouquets in both the dynamic and parameter spaces, and no bounded hyperbolic components. The tangent case, with its two finite asymptotic values $\pm \lambda i$, is closer to the rational case, a kind of infinite degree version of the latter. In this paper, we consider a general family that interpolates between $\exp(z) + a$ and $\lambda \tan z$. Our new family has two asymptotic values and a one-dimensional slice for which one of the asymptotic values is constrained to be pole, the "polar asymptotic value" of the title. We show how the dynamic and parameter planes for this slice exhibit behavior that is a surprisingly delicate interplay between that of the $\exp(z) + a$ and $\lambda \tan z$ families., Comment: 13 figures

This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite. Here we look at a generalization of the family of polynomials $P_a(z)=z^{d-1}(z- \frac{da}{(d-1)})$, the family $f_{\lambda}=\lambda \tan^p z^q$. These functions have a super-attractive fixed point, and, depending on $p$, one or two asymptotic values. Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here. Adding transcendental methods to standard ones, we give a description of the dynamical properties; in particular we prove the Julia set of a hyperbolic map is either connected and locally connected or a Cantor set. We also give a description of the parameter plane of the family $f_{\lambda}$. Again there are similarities to and differences from the parameter plane of the family $P_a$ and again there are new techniques. In particular, we prove there is dense set of points on the boundaries of the hyperbolic components that are accessible along curves and we characterize these points., Comment: 42 pages, 6 figures

In this paper we continue the study, began in \cite{CJK2}, of the bifurcation locus of a family of meromorphic functions with two asymptotic values, no critical values and an attracting fixed point. If we fix the multiplier of the fixed point, either of the two asymptotic values determines a one-dimensional parameter slice for this family. We proved that the bifurcation locus divides this parameter slice into three regions, two of them analogous to the Mandelbrot set and one, the shift locus, analogous to the complement of the Mandelbrot set. In \cite{FK, CK} it was proved that the points in the bifurcation locus corresponding to functions with a parabolic cycle, or those for which some iterate of one of the asymptotic values lands on a pole are accessible boundary points of the hyperbolic components of the Mandelbrot-like sets. Here we prove these points, as well as the points where some iterate of the asymptotic value lands on a repelling periodic cycle, are also accessible from the shift locus., Comment: 20 pages, 5 figures

This paper is part of a program to understand the parameter spaces of dynamical systems generated by meromorphic functions with finitely many singular values. We give a full description of the parameter space for a specific family based on the exponential function that has precisely two finite asymptotic values and one attracting fixed point. It represents a step beyond the previous work in [GK] on degree 2 rational functions with analogous constraints: two critical values and an attracting fixed point. What is interesting and promising for pushing the general program even further, is that, despite the presence of the essential singularity, our new functions exhibit a dynamic structure as similar as one could hope to the rational case, and that the philosophy of the techniques used in the rational case could be adapted., Comment: 47 pages, 12 figures

In the early 1980's, computers made it possible to observe that in complex dynamics, one often sees dynamical behavior reflected in parameter space and vice versa. This duality was first exploited by Douady, Hubbard and their students in early work on rational maps. See \cite{DH,BH} for example. Here, we continue to study these ideas in the realm of transcendental functions. In \cite{KK1}, it was shown that for the tangent family, $\lambda \tan z$, the way the hyperbolic components meet at a point where the asymptotic value eventually lands on infinity reflects the dynamic behavior of the functions at infinity. In the first part of this paper we show that this duality extends to a much more general class of transcendental meromorphic functions that we call {\em generalized Nevanlinna functions} with the additional property that infinity is not an asymptotic value. In particular, we show that in "dynamically natural" one dimensional slices of parameter space, there are "hyperbolic-like" components with a unique distinguished boundary point whose dynamics reflect the behavior inside an asymptotic tract at infinity. Our main result is that {\em every} parameter point in such a slice for which the asymptotic value eventually lands on a pole is such a distinguished boundary point. In the second part of the paper, we apply this result to the families $\lambda \tan^p z^q$, $p,q \in \mathbb Z^+$, to prove that all hyperbolic components of period greater than $1$ are bounded., Comment: 31 pages, 3 figures

In this paper we study the transition to chaos for the restriction to the real and imaginary axes of the tangent family $\{ T_t(z)=i t\tan z\}_{0< t\leq \pi}$. Because tangent maps have no critical points but have an essential singularity at infinity and two symmetric asymptotic values, there are new phenomena: as $t$ increases we find single instances of "period quadrupling", "period splitting" and standard "period doubling"; there follows a general pattern of "period merging" where two attracting cycles of period $2^n$ "merge" into one attracting cycle of period $2^{n+1}$, and "cycle doubling" where an attracting cycle of period $2^{n+1}$ "becomes" two attracting cycles of the same period. We use renormalization to prove the existence of these bifurcation parameters. The uniqueness of the cycle doubling and cycle merging parameters is quite subtle and requires a new approach. To prove the cycle doubling and merging parameters are, indeed, unique, we apply the concept of "holomorphic motions" to our context. In addition, we prove that there is an "infinitely renormalizable" tangent map $T_{t_\infty}$. It has no attracting or parabolic cycles. Instead, it has a strange attractor contained in the real and imaginary axes which is forward invariant and minimal under $T^2_{t_\infty}$. The intersection of this strange attractor with the real line consists of two binary Cantor sets and the intersection with the imaginary line is totally disconnected, perfect and unbounded., Comment: 52 pages, 15 figures

We study the parameter planes of certain one-dimensional, dynamically-defined slices of holomorphic families of entire and meromorphic transcendental maps of finite type. Our planes are defined by constraining the orbits of all but one of the singular values, and leaving free one asymptotic value. We study the structure of the regions of parameters, which we call {\em shell components}, for which the free asymptotic value tends to an attracting cycle of non-constant multiplier. The exponential and the tangent families are examples that have been studied in detail, and the hyperbolic components in those parameter planes are shell components. Our results apply to slices of both entire and meromorphic maps. We prove that shell components are simply connected, have a locally connected boundary and have no center, i.e., no parameter value for which the cycle is superattracting. Instead, there is a unique parameter in the boundary, the {\em virtual center}, which plays the same role. For entire slices, the virtual center is always at infinity, while for meromorphic ones it maybe finite or infinite. In the dynamical plane we prove, among other results, that the basins of attraction which contain only one asymptotic value and no critical points are simply connected. Our dynamical plane results apply without the restriction of finite type., Comment: 41 pages, 13 figures

We define two classes of topological infinite degree covering maps modeled on two families of transcendental holomorphic maps. The first, which we call exponential maps of type $(p,q)$, are branched covers and is modeled on transcendental entire maps of the form $P e^{Q}$, where $P$ and $Q$ are polynomials of degrees $p$ and $q$. The second is the class of universal covering maps from the plane to the sphere with two removed points modeled on transcendental meromorphic maps with two asymptotic values. The problem we address is to give a combinatorial characterization of the holomorphic maps contained in these classes whose post-singular sets are finite. The main results in this paper are that a post-singularly finite topological exponential map of type $(0,1)$ or a certain post-singularly finite topological exponential map of type $(p,1)$ or a post-singularly finite universal covering map from the plane to the sphere with two points removed is combinatorially equivalent to a holomorphic same type map if and only if this map has bounded geometry., Comment: arXiv admin note: substantial text overlap with arXiv:1209.6044, arXiv:1112.2557

Let $G = \langle A,B \rangle$ be a non-elementary two generator subgroup of the isometry group of $\mathbb{H}^2$, the hyperbolic plane. If $G$ is discrete and free and geometrically finite, its quotient is a pair of pants and in prior work we produced a formula for the number of essential self intersections (ESIs) of any primitive geodesic on the quotient. An ESI is a point where the geodesic has a self-intersection on a seam. Self-intersections of geodesics on arbitrary hyperbolic surfaces have recently been studied by Basmajian and Chas. Here we extend our results to two generator subgroups $G$ of isometries of $\mathbb{H}^3$, hyperbolic three-space, which are discrete, free and geometrically finite. We generalize our definition of ESIs and give a geometric interpretation of them in the quotient manifold. We show that they satisfy the same formulas.

The discreteness problem, that is, the problem of determining whether or not a given finitely generated group G of orientation preserving isometries of hyperbolic three-space is discrete as a subgroup of the whole isometry group of hyperbolic three space, is a challenging problem that has been investigated for more than a century and is still open. It is known that G is discrete if, and only if, every non-elementary two generator subgroup is. Several sufficient conditions for discreteness are also known as are some necessary conditions, though no single necessary and sufficient condition is known. There is a finite discreteness algorithm for the two generator subgroups of the isometry group of hyperbolic two-space. But the situation in three dimensions is more delicate because there are geometrically infinite groups. We present a semi-algorithm, that is, a procedure that terminates sometimes but not always. There is no standard way to find an infinite sequence of distinct elements that converges to the identity to show that a group is not discrete. Our semi-algorithm either produces such an infinite sequence or finds a finite sequence that produces a right angled hexagon in hyperbolic three-space which has a special property that is a generalization of the notion of convexity. We call it a canonical hexagon. If the group is discrete, free and geometrically finite, it always has an essentially unique canonical hexagon which the procedure finds in a finite number of steps., Comment: The result is false as stated. The article is withdrawn

In this paper we define a topological class of branched covering maps of the plane called {\em topological exponential maps of type $(p,q)$} and denoted by $\TE_{p,q}$, where $p\geq 0$ and $q\geq 1$. We follow the framework given in \cite{Ji} to study the problem of combinatorially characterizing an entire map $P e^{Q}$, where $P$ is a polynomial of degree $p$ and $Q$ is a polynomial of degree $q$ using an {\em iteration scheme defined by Thurston} and a {\em bounded geometry condition}. We first show that an element $f \in {\TE}_{p,q}$ with finite post-singular set is combinatorially equivalent to an entire map $P e^{Q}$ if and only if it has bounded geometry with compactness. Thus to complete the characterization, we only need to check that the bounded geometry actually implies compactness. We show this for some $f\in \TE_{p,1}$, $p\geq 1$. Our main result in this paper is that a post-singularly finite map $f$ in $\TE_{0,1}$ or a post-singularly finite map $f$ in $\TE_{p,1}$, $p\geq 1$, with only one non-zero simple breanch point $c$ such that either $c$ is periodic or $c$ and $f(c)$ are both not periodic, is combinatorially equivalent to a post-singularly finite entire map of either the form $e^{\lambda z}$ or the form $ \alpha z^{p}e^{\lambda z}$, where $\alpha=(-\lambda/p)^{p}e^{- \lambda (-p/\lambda)^{p}}$, respectively, if and only if it has bounded geometry. This is the first result in this direction for a family of transcendental holomorphic maps with critical points., Comment: arXiv admin note: text overlap with arXiv:1112.2557

In this paper we consider the topological class, denoted by AV2, of universal covering maps from the plane to the sphere with two removed points. We prove that an element f in AV2 with finite post-singular set is combinatorially equivalent to a meromorphic transcendental map g with constant Schwarzian derivative if any only if f satisfies a Bounded Geometry condition., Comment: The topological constrain condition has been modified and corrected

We consider non-elementary representations of two generator free groups in $PSL(2,\mathbb{C})$, not necessarily discrete or free, $G = < A, B >$. A word in $A$ and $B$, $W(A,B)$, is a palindrome if it reads the same forwards and backwards. A word in a free group is {\sl primitive} if it is part of a minimal generating set. Primitive elements of the free group on two generators can be identified with the positive rational numbers. We study the geometry of palindromes and the action of $G$ in $\HH^3$ whether or not $G$ is discrete. We show that there is a {\sl core geodesic} $\L$ in the convex hull of the limit set of $G$ and use it to prove three results: the first is that there are well defined maps from the non-negative rationals and from the primitive elements to $\L$; the second is that $G$ is geometrically finite if and only if the axis of every non-parabolic palindromic word in $G$ intersects $\L$ in a compact interval; the third is a description of the relation of the pleating locus of the convex hull boundary to the core geodesic and to palindromic elements., Comment: 19 pages

We give a unified geometric approach to some theorems about primitive elements and palindromes in free groups of rank 2. The geometric treatment gives new proofs of the theorems. Dedicated to Bill Harvey on his 65th birthday., Comment: 15 The current version replaces the first version; we have corrected the statement of theorem 2.1 and added a new theorem 2.4 and included some additional references

Let $F= < a,b>$ be a rank two free group. A word $W(a,b)$ in $F$ is {\sl primitive} if it, along with another group element, generates the group. It is a {\sl palindrome} (with respect to $a$ and $b$) if it reads the same forwards and backwards. It is known that in a rank two free group any primitive element is conjugate either to a palindrome or to the product of two palindromes, but known iteration schemes for all primitive words give only a representative for the conjugacy class. Here we derive a new iteration scheme that gives either the unique palindrome in the conjugacy class or expresses the word as a unique product of two unique palindromes. We denote these words by $E_{p/q}$ where $p/q$ is rational number expressed in lowest terms. We prove that $E_{p/q}$ is a palindrome if $pq$ is even and the unique product of two unique palindromes if $pq$ is odd. We prove that the pairs $(E_{p/q},E_{r/s})$ generate the group when $|ps-rq|=1$. This improves the previously known result that held only for $pq$ and $rs$ both even. The derivation of the enumeration scheme also gives a new proof of the known results about primitives., Comment: Final revisions, to appear J Algebra

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 function system} coming from the sequence $f_1,f_2,f_3,....$ We prove that a sufficient condition on the domain $X$ for all limit functions of any $\{F_n\}$ to be constant is also necessary. We prove the condition is a quasiconformal invariant. Finally, we address the question of uniqueness of limit functions., Comment: 8 Pages To appear PAMS

Determining the space of free discrete two generator groups of M\"obius transformations is an old and difficult problem. In this paper we show how to construct large balls of full dimension in this space. To do this, we begin with a marked discrete group of non-separating disjoint circle type. Such a group determines three disjoint or tangent planes. We prove that there is a whole family of discrete groups that share these planes. We find a set of six real numbers that serve as parameters for this family and construct an embedding of our parameters into a classical representation of the full space of free discrete groups. We see that each planar family fills out a ball of full dimension in the classical embedding., Comment: 10 pages

For two generator free Fuchsian groups, the quotient three manifold is a genus two solid handlebody and its boundary is a hyperelliptic Riemann surface. The convex core is also a hyperelliptic Riemann surface. We find the Weierstrass points of both of these surfaces. We then generalize the notion of a hyperelliptic Riemann surface to a ``hyperelliptic'' three manifold. We show that the handlebody has a unique order two isometry fixing six unique geodesic line segments, which we call the {\sl Weierstrass lines} of the handlebody. The Weierstrass lines are, of course, the analogue of the Weierstrass points on the boundary surface. Further, we show that the manifold is foliated by surfaces equidistant from the convex core, each fixed by the isometry of order two. The restriction of this involution to the equidistant surface fixes six {\sl generalized Weierstrass points} on the surface., Comment: 43 pages 11 figures to appear Geometria Dedicata

We study the dynamics of the tangent family z -> lambda tan z for lambda complex and give a complete classification of their stable behavior. We also characterize the the hyperbolic components and give a combinatorial description their deployment in the parameter plane., Comment: revised May 1997

Consider the two-parameter family of real analytic maps $F_{a,b}:x \mapsto x+ a+{b\over 2\pi} \sin(2\pi x)$ which are lifts of degree one endomorphisms of the circle. The purpose of this paper is to provide a proof that for any closed interval $I$, the set of maps $F_{a,b}$ whose rotation interval is $I$, form a contractible set.

We construct new coordinates for the Teichm\"uller space Teich of a punctured torus into $\bold{R} \times\bold{R}^+$. The coordinates depend on the representation of Teich as a space of marked Kleinian groups $G_\mu$ that depend holomorphically on a parameter $\mu$ varying in a simply connected domain in $\bold{C}$. They describe the geometry of the hyperbolic manifold $\bold{H}^3/G_\mu$; they reflect exactly the visual patterns one sees in the limit sets of the groups $G_\mu$; and they are directly computable from the generators of $G_\mu$., Comment: 6 pages

In this paper, we assume that $G$ is a finitely generated torsion free non-elementary Kleinian group with $\Omega(G)$ nonempty. We show that the maximal number of elements of $G$ that can be pinched is precisely the maximal number of rank 1 parabolic subgroups that any group isomorphic to $G$ may contain. A group with this largest number of rank 1 maximal parabolic subgroups is called {\it maximally parabolic}. We show such groups exist. We state our main theorems concisely here. Theorem I. The limit set of a maximally parabolic group is a circle packing; that is, every component of its regular set is a round disc. Theorem II. A maximally parabolic group is geometrically finite. Theorem III. A maximally parabolic pinched function group is determined up to conjugacy in $PSL(2,{\bf C})$ by its abstract isomorphism class and its parabolic elements.