Julia sets for polynomial diffeomorphisms of C2{\Bbb C}^2 are not semianalytic

For any polynomial diffeomorphism $f$ of ${\Bbb C}^2$ with positive entropy, neither the Julia set of $f$ nor of its inverse $f^{-1}$ is semi-analytic

Dynamics of Rational Surface Automorphisms: Rotation Domains

We consider rational surface automorphisms with positive entropy. A Fatou component is said to be a rotation domain if the automorphism induces a torus action on it. Here we construct a rational surface automorphism with positive entropy with the following property: it has a rotation domain which contains both a curve of fixed points and isolated fixed points. This Fatou component cannot be imbedded into complex euclidean space, so we introduce a global linear model space and show that it can be globally linearized in this model

No smooth Julia sets for polynomial diffeomorphisms of C2\mathbb{C}^2 with positive entropy

For any polynomial diffeomorphism $f$ of $\mathbb{C}^2$ with positive entropy, neither the Julia set of $f$ nor of its inverse $f^{-1}$ is $C^1$ smooth as a manifold-with-boundary

Exponential Martingales and Time integrals of Brownian Motion

We find a simple expression for the probability density of $\int \exp (B_s - s/2) ds$ in terms of its distribution function and the distribution function for the time integral of $\exp (B_s + s/2)$. The relation is obtained with a change of measure argument where expectations over events determined by the time integral are replaced by expectations over the entire probability space. We develop precise information concerning the lower tail probabilities for these random variables as well as for time integrals of geometric Brownian motion with arbitrary constant drift. In particular, $E[ \exp\big(\theta / \int \exp (B_s)ds\big) ]$ is finite iff $\theta < 2$. We present a new formula for the price of an Asian call option

One-Factor Term Structure without Forward Rates

We construct a no-arbitrage model of bond prices where the long bond is used as a numeraire. We develop bond prices and their dynamics without developing any model for the spot rate or forward rates. The model is arbitrage free and all nominal interest rates remain positive in the model. We give examples where our model does not have a spot rate; other examples include both spot and forward rates

Linear Recurrences in the Degree Sequences of Monomial Mappings

Let $A$ be an integer matrix, and let $f_A$ be the associated monomial map. We give a connection between the eigenvalues of $A$ and existence of a linear recurrence relation in the sequence of degrees

Entropy of real rational surface automorphisms

We compare real and complex dynamics for automorphisms of rational surfaces that are obtained by lifting \chg{some} quadratic birational maps of the plane. In particular, we show how to exploit the existence of an invariant cubic curve to understand how the real part of an automorphism acts on homology. We apply this understanding to give examples where the entropy of the full (complex) automorphism is the same as its real restriction. Conversely and by different methods, we exhibit different examples where the entropy is strictly decreased by restricting to the real part of the surface. Finally, we give an example of a rational surface automorphism with positive entropy whose periodic cycles are all real

Degree growth of matrix inversion: birational maps of symmetric, cyclic matrices

We consider two (densely defined) involutions on the space of $q\times q$ matrices; $I(x_{ij})$ is the matrix inverse of $(x_{ij})$, and $J(x_{ij})$ is the matrix whose $ij$th entry is the reciprocal $x_{ij}^{-1}$. Let $K=I\circ J$. The set ${\cal SC}_q$ of symmetric, cyclic matrices is invariant under $K$. In this paper, we determine the degrees of the iterates $K^n=K\circ...\circ K$ restricted to ${\cal SC}_q$

Linear Fractional Recurrences: Periodicities and Integrability

We consider k-step recurrences of the form $z_{n+k} = A(z)/B(z)$, where A and B are linear functions of $z_n, z_{n+1}, ..., z_{n+k-1}$, which we call k-step linear fractional recurrences. The first Theorem in this paper shows that for each k there are k-step linear fractional recurrences which are periodic of period 4k. Among this class of recurrences, there is also the so-called Lyness process, which has the form $A(z)/B(z) = (a +z_{n+1} + z_{n+2} + ... + z_{n+k-1})/z_n$. The second Theorem shows that the Lyness process has quadratic degree growth. The Lyness process is integrable, and we discuss its known integrals

On the Degree Growth of Birational Mappings in Higher Dimension

Let $f$ be a birational map of ${\bf C}^d$, and consider the degree complexity, or asymptotic degree growth rate $\delta(f)=\lim_{n\to\infty}({\rm deg}(f^n))^{1/n}$. We introduce a family of elementary maps, which have the form $f=L\circ J$, where $L$ is (invertible) linear, and $J(x_1,...,x_d)=(x_1^{-1},...,x_d^{-1})$. We develop a method of regularization and show how it can be used to compute $\delta$ for an elementary map