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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. 

In this note we make progress toward a conjecture of Durham–Fanoni–Vlamis, showing that every infinite-type surface with fi­ni­te-invariance index \(1\) and no nondisplaceable compact subsurfaces fails to have a good graph of curves, that is, a connected graph where vertices represent homotopy classes of essential simple closed curves and with a natural mapping class group action having infinite diameter orbits. Our arguments use tools developed by Mann–Rafi in their study of the coarse geometry of big mapping class groups. 

Abstract Given an irreducible, end‐periodic homeomorphism of a surface with finitely many ends, all accumulated by genus, the mapping torus, , is the interior of a compact, irreducible, atoroidal 3‐manifold with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of in terms of the translation length of on the pants graph of . This builds on work of Brock and Agol in the finite‐type setting. We also construct a broad class of examples of irreducible, end‐periodic homeomorphisms and use them to show that our bound is asymptotically sharp. 

Classical parking functions are defined as the parking preferences for $$n$$ cars driving (from west to east) down a one-way street containing parking spaces labeled from $$1$$ to $$n$$ (from west to east). Cars drive down the street toward their preferred spot and park there if the spot is available. Otherwise, the car continues driving down the street and takes the first available parking space, if such a space exists. If all cars can park using this parking rule, we call the $$n$$-tuple containing the cars' parking preferences a parking function.   In this paper, we introduce a generalization of the parking rule allowing cars whose preferred space is taken to first proceed up to $$k$$ spaces west of their preferred spot to park before proceeding east if all of those $$k$$ spaces are occupied. We call parking preferences which allow all cars to park under this new parking rule $$k$$-Naples parking functions of length $$n$$. This generalization gives a natural interpolation between classical parking functions, the case when $k=0$, and all $$n$$-tuples of positive integers $$1$$ to $$n$$, the case when $$k\geq n-1$$. Our main result provides a recursive formula for counting $$k$$-Naples parking functions of length $$n$$. We also give a characterization for the $k=1$ case by introducing a new function that maps $$1$$-Naples parking functions to classical parking functions, i.e. $$0$$-Naples parking functions. Lastly, we present a bijection between $$k$$-Naples parking functions of length $$n$$ whose entries are in weakly decreasing order and a family of signature Dyck paths.