Complex Horseshoes and the Dynamics of Mappings of Two Complex Variables

Horseshoes for $\mathcal{C}^{1+α}$ mappings with hyperbolic measures

We present here a construction of horseshoes for any $\mathcal{C}^{1+α}$ mapping $f$ preserving an ergodic hyperbolic measure $μ$ with $h_μ(f)>0$ and then deduce that the exponential growth rate of the number of periodic points for any $\mathcal{C}^{1+α}$ mapping $f$ is greater than or equal to $h_μ(f)$. We also prove that the exponential growth rate of the number of hyperbolic periodic points is equal to the hyperbolic entropy. The hyperbolic entropy means the entropy resulting from hyperbolic measures.

Horseshoes for singly thermostated hamiltonians

This note studies 1 and 2 degree of freedom hamiltonian systems that are thermostated by a single-variable thermostat. Under certain conditions on the hamiltonian and thermostat, the existence of a horseshoe in the flow of the thermostated system is proven.

Real Polynomial Diffeomorphisms with Maximal Entropy: II. Small Jacobian

This paper concerns the Henon family of diffeomorphisms of the plane for which the absolute value of the jacobian parameter is bounded by .08. We describe the parameter locus for which the mapping is a horseshoe.

Dynamics of the Morse Oscillator: Analytical Expressions for Trajectories, Action-Angle Variables, and Chaotic Dynamics

We consider the one degree-of-freedom Hamiltonian system defined by the Morse potential energy function (the "Morse oscillator"). We use the geometry of the level sets to construct explicit expressions for the trajectories as a function of time, their period for the bounded trajectories, and action-angle variables. We use these trajectories to prove sufficient conditions for chaotic dynamics, in the sense of Smale horseshoes, for the time-periodically perturbed Morse oscillator using a Melnikov type approach.

Local analytic conjugacy of resonant analytic mappings in two variables, in the non-archimedean setting

In this note, we consider locally invertible analytic mappings in two dimensions, with coefficients in a non-archimedean field. Suppose such a map has a Jacobian with eigenvalues $λ_1$ and $λ_2$ so that $|λ_1|>1$ and $λ_2$ is a positive power of $λ_1$, or that $λ_1=1$ and $|λ_2|\neq 1$. We prove that two formal maps with eigenvalues satisfying either of these conditions are analytically equivalent if and only if they are formally equivalent.

拟周期可逆映射的不变曲线及其应用

Invariant curves of quasi-periodic reversible mappings and its application

本文研究拟周期可逆映射的不变曲线存在性，通过正规形定理和修正同调方程保证可逆性，并应用于非线性共振振荡器，探讨Birkhoff常数的影响。