Extreme Value Laws for Dynamical Systems with Countable Extremal Sets
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We consider stationary stochastic processes arising from dynamical systems by evaluating a given observable along the orbits of the system. We focus on the extremal behaviour of the process, which is related to the entrance in certain regions of the phase space, which correspond to neighbourhoods of the maximal set \(\mathcal M\), i.e.,the set of points where the observable is maximised. The main novelty here is the fact that we consider that the set \(\mathcal M\) may have a countable number of points, which are associated by belonging to the orbit of a certain point, and may have accumulation points. In order to prove the existence of distributional limits and study the intensity of clustering, given by the Extremal Index, we generalise the conditions previously introduced in Freitas (Adv Math 231(5): 2626–2665, 2012, Stoch Process Appl 125(4): 1653–1687, 2015).
KeywordsDynamical systems Extreme value laws Extremal index Clustering Countable maximal set
Mathematics Subject Classification37A50 60G70 37B20 60G10 37C25
DA was partially supported by the Grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. ACMF and JMF were partially supported by FCT projects FAPESP/19805/2014 and PTDC/MAT-CAL/3884/2014. FBR was supported by BREUDS, a Brazilian-European partnership of the FP7-PEOPLE-2012-IRSES program, with project Number 318999, which is supported by an FP7 International International Research Staff Exchange Scheme (IRSES) grant of the European Union. All authors were partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020. JMF would like to thank Mike Todd for careful reading and useful suggestions.
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