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The Bollobás set pairs inequality is a fundamental result in extremal set theory with many applications. In this paper, for $$n \geqslant k \geqslant t \geqslant 2$$, we consider a collection of $$k$$ families $$\mathcal{A}_i: 1 \leq i \leqslant k$$ where $$\mathcal{A}_i = \{ A_{i,j} \subset [n] : j \in [n] \}$$ so that $$A_{1, i_1} \cap \cdots \cap A_{k,i_k} \neq \varnothing$$ if and only if there are at least $$t$$ distinct indices $$i_1,i_2,\dots,i_k$$. Via a natural connection to a hypergraph covering problem, we give bounds on the maximum size $$\beta_{k,t}(n)$$ of the families with ground set $[n]$.
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Extremes and extremal indices for level set observables on hyperbolic systems *
Abstract Consider an ergodic measure preserving dynamical system ( T , X , μ ), and an observable ϕ : X → R . For the time series X n ( x ) = ϕ ( T n ( x )), we establish limit laws for the maximum process M n = max k ⩽ n X k in the case where ϕ is an observable maximized on a line segment, and ( T , X , μ ) is a hyperbolic dynamical system. Such observables arise naturally in weather and climate applications. We consider the extreme value laws and extremal indices for these observables on hyperbolic toral automorphisms, Sinai dispersing billiards and coupled expanding maps. In particular we obtain clustering and nontrivial extremal indices due to self intersection of submanifolds under iteration by the dynamics, not arising from any periodicity.
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- Award ID(s):
- 2009923
- PAR ID:
- 10427546
- Date Published:
- Journal Name:
- Nonlinearity
- Volume:
- 34
- Issue:
- 2
- ISSN:
- 0951-7715
- Page Range / eLocation ID:
- 1136 to 1167
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1088/1361-6544/abd85f
