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Random perturbations of predominantly expanding 1D maps Blumenthal, Alex


We consider a model of 1D multimodal circle maps with strong expansion on most of phase space, including, e.g., the one-parameter family $f_a(x) = L \sin x + a$ for $a \in [0,1)$ with fixed $L >> 1$. Even when L is quite large, the problem of deciding the asymptotic regime (stochastic versus regular) of $f_a$ for a given $a$ involves infinite-precision knowledge of infinite trajectories: outside special cases, this problem is typically impossible to resolve from any checkable finite-time conditions on the dynamics of $f_a$. We contend that the corresponding problem for (possibly quite small) IID random perturbations of the $f_a$ is far more tractable. In our model, we perturb $f_a$ at each timestep by an IID uniformly distributed random variable in the interval $[- \epsilon, \epsilon]$ for a fixed (yet arbitrarily small) $\epsilon > 0$. We obtain a checkable condition, involving finite trajectories of $f_a$ of length ~ $\log(\epsilon^{-1})$, for this random composition to admit (1) a unique, absolutely continuous stationary ergodic measure and (2) a Lyapunov exponent of size approximately $\log L$. Joint with Yun Yang.

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