Mathematical Sciences: The Method of Induced Hyperbolicity for One-Dimensional Maps and Two-Dimensional Diffeomorphisms

利用主要研究者早期发展的诱导双曲性方法，研究一族非线性动力系统。特别令人感兴趣的将是它们的随机行为。我们将考察费根鲍姆不动点不稳定流形附近的单参数微分同胚族。Sullivan的方法将被用来研究具有绝对连续不变测度的映射的拓扑共轭何时蕴含拟共形共轭。动力系统的不变度量在吸引或排斥的系统的不变集上是正的。主要研究人员将研究在某些参数空间中接近费根鲍姆极限的这类系统。这种倍增周期分叉的极限引起了各种物理和生物科学研究人员的浓厚兴趣。刚刚超过这一极限，就会出现混乱。

Using the method of induced hyperbolicity developed earlier by the principal investigator, families of nonlinear dynamical systems will be studied. Of particular interest will be their stochastic behavior. One-parameter families of diffeomorphisms close to the unstable manifold of the Feigenbaum fixed point will be inspected. And Sullivan's approach will be used to investigate when topological conjugacy implies quasiconformal conjugacy for maps with absolutely continuous invariant measures. An invariant measure of a dynamical system is positive on invariant sets of the system which attract or repel. The principal investigator will study such systems which are close in some parameter space to the Feigenbaum limit. This limit of period-doubling bifurcations has raised intense interest from investigators in a wide variety of physical and biological sciences. Just past this limit lies the onset of chaos.