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Given an end-periodic homeomorphism $f:S\to <\#comment/>S$we give a lower bound on the Handel–Miller stretch factor of $f$in terms of thecore characteristic of $f$, which is a measure of topological complexity for an end-periodic homeomorphism. We also show that the growth rate of this bound is sharp. 

Abstract We prove an explicit characterization of the points in Thurston’s Master Teapot, which can be implemented algorithmically to test whether a point in $$\mathbb {C}\times \mathbb {R}$$ belongs to the complement of the Master Teapot. As an application, we show that the intersection of the Master Teapot with the unit cylinder is not symmetrical under reflection through the plane that is the product of the imaginary axis of $$\mathbb {C}$$ and $$\mathbb {R}$$ . 

Abstract Alpine regions are changing rapidly due to loss of snow and ice in response to ongoing climate change. While studies have documented ecological responses in alpine lakes and streams to these changes, our ability to predict such outcomes is limited. We propose that the application of fundamental rules of life can help develop necessary predictive frameworks. We focus on four key rules of life and their interactions: the temperature dependence of biotic processes from enzymes to evolution; the wavelength dependence of the effects of solar radiation on biological and ecological processes; the ramifications of the non‐arbitrary elemental stoichiometry of life; and maximization of limiting resource use efficiency across scales. As the cryosphere melts and thaws, alpine lakes and streams will experience major changes in temperature regimes, absolute and relative inputs of solar radiation in ultraviolet and photosynthetically active radiation, and relative supplies of resources (e.g., carbon, nitrogen, and phosphorus), leading to nonlinear and interactive effects on particular biota, as well as on community and ecosystem properties. We propose that applying these key rules of life to cryosphere‐influenced ecosystems will reduce uncertainties about the impacts of global change and help develop an integrated global view of rapidly changing alpine environments. However, doing so will require intensive interdisciplinary collaboration and international cooperation. More broadly, the alpine cryosphere is an example of a system where improving our understanding of mechanistic underpinnings of living systems might transform our ability to predict and mitigate the impacts of ongoing global change across the daunting scope of diversity in Earth's biota and environments.