Existence of the Lyapunov exponent for S-unimodal maps
Abstract
In this paper, we show that for any S-unimodal map T on [0,1] with a non-flat critical point the Lyapunov exponent exists for Lebesgue almost every point and is equal to a constant λT. Moreover, λT=0 if and only if T admits neither an absolutely continuous T-invariant probability measure with positive entropy nor a strictly stable periodic orbit. Consequently, if an S-unimodal map with a non-flat critical point is infinitely renormalizable or non-statistical then for Lebesgue almost every x∈ [0,1] the Lyapunov exponent along the orbit of x exists and is equal to $0$. A key ingredient is the following result of independent interest. If an S-unimodal map with a non-flat critical point has no periodic attractor then for Lebesgue almost every x∈ [0,1] the lower Lyapunov exponent along the orbit of x is non-negative. This shows that, in the absence of periodic attractors, exponential contraction cannot occur along the orbit of Lebesgue almost every point.
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BibTeX
@misc{arima2026existence,
title = {{Existence of the Lyapunov exponent for $S$-unimodal maps}},
author = {Yuya Arima},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.04187},
eprint = {2607.04187},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04187},
}
RIS
TY - UNPB
TI - Existence of the Lyapunov exponent for S-unimodal maps
AU - Yuya Arima
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2607.04187
UR - https://arxiv.org/abs/2607.04187
AB - In this paper, we show that for any S-unimodal map T on [0,1] with a non-flat critical point the Lyapunov exponent exists for Lebesgue almost every point and is equal to a constant λ_T. Moreover, λ_T=0 if and only if T admits neither an absolutely continuous T-invariant probability measure with positive entropy nor a strictly stable periodic orbit. Consequently, if an S-unimodal map with a non-flat critical point is infinitely renormalizable or non-statistical then for Lebesgue almost every x∈ [0,1] the Lyapunov exponent along the orbit of x exists and is equal to $0$. A key ingredient is the following result of independent interest. If an S-unimodal map with a non-flat critical point has no periodic attractor then for Lebesgue almost every x∈ [0,1] the lower Lyapunov exponent along the orbit of x is non-negative. This shows that, in the absence of periodic attractors, exponential contraction cannot occur along the orbit of Lebesgue almost every point.
ER -