论文标题
在单位间隔上扩展和收缩图的迭代功能系统
Iterated function systems of affine expanding and contracting maps on the unit interval
论文作者
论文摘要
我们通过扩展和收缩仿射图产生的单位间隔分析了迭代功能系统的两点动作，在这种间隔中，膨胀和收缩率由一对$（m，n）整数确定。 该动态取决于Lyapunov指数。对于负Lyapunov指数，我们建立同步，这意味着具有不同初始点的轨道的收敛。对于消失的Lyapunov指数，我们建立了间歇性，其中一组全密度迭代的轨道很近，但间歇性地分开了。对于积极的Lyapunov指数，我们显示了对两点动态的绝对连续固定度量的存在，并讨论了其后果。 对于非依赖整数的非负Lyapunov指数$（m，n）$，我们为两点动作的绝对连续的固定度量提供明确的表达式。在零lyapunov指数的情况下，这些固定措施是无限的$σ$ finite措施。对于不同的Lyapunov指数，我们在这里找到了两点动作系统的相变，在该系统中，固定度量的支持在间歇性动力学和在过渡点处的无限固定度量爆炸。
We analyze the two-point motions of iterated function systems on the unit interval generated by expanding and contracting affine maps, where the expansion and contraction rates are determined by a pair $(M,N)$ of integers. This dynamics depends on the Lyapunov exponent. For a negative Lyapunov exponent we establish synchronization, meaning convergence of orbits with different initial points. For a vanishing Lyapunov exponent we establish intermittency, where orbits are close for a set of iterates of full density, but are intermittently apart. For a positive Lyapunov exponent we show the existence of an absolutely continuous stationary measure for the two-point dynamics and discuss its consequences. For nonnegative Lyapunov exponent and pairs $(M,N)$ that are multiplicatively dependent integers, we provide explicit expressions for absolutely continuous stationary measures of the two-point motions. These stationary measures are infinite $σ$-finite measures in the case of zero Lyapunov exponent. For varying Lyapunov exponent we find here a phase transition for the system of two-point motions, in which the support of the stationary measure explodes with intermittent dynamics and an infinite stationary measure at the transition point.
