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Random perturbations of predominantly expanding 1D maps Blumenthal, Alex
Description
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.
Item Metadata
Title |
Random perturbations of predominantly expanding 1D maps
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2018-03-22T19:31
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Description |
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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Extent |
49 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Maryland
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Series | |
Date Available |
2018-09-19
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0372087
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Postdoctoral
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International