LEADER 06547nam 22006373 450 001 9910965973803321 005 20231110214826.0 010 $a9781470470241 010 $a1470470241 035 $a(MiAaPQ)EBC6939726 035 $a(Au-PeEL)EBL6939726 035 $a(CKB)21420568900041 035 $a(OCoLC)1309058109 035 $a(EXLCZ)9921420568900041 100 $a20220327d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aAbelian Networks IV. Dynamics of Nonhalting Networks 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2022. 210 4$dİ2022. 215 $a1 online resource (104 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.276 311 08$aPrint version: Chan, Swee Hong Abelian Networks IV. Dynamics of Nonhalting Networks Providence : American Mathematical Society,c2022 9781470451417 327 $aCover -- Title page -- Chapter 1. Introduction -- 1.1. Flashback -- 1.2. Atemporal dynamics -- 1.3. Relating atemporal dynamics to traditional dynamics -- 1.4. Computational questions -- 1.5. The torsion group of a nonhalting abelian network -- 1.6. Critical networks -- 1.7. Example: Rotor networks and abelian mobile agents -- 1.8. Proof ideas -- 1.9. Summary of notation -- Chapter 2. Commutative Monoid Actions -- 2.1. Injective actions and Grothendieck group -- 2.2. The case of finite commutative monoids -- Chapter 3. Review of Abelian Networks -- 3.1. Definition of abelian networks -- 3.2. Legal and complete executions -- 3.3. Locally recurrent states -- 3.4. The production matrix -- 3.5. Subcritical, critical, and supercritical abelian networks -- 3.6. Examples: sandpiles, rotor-routing, toppling, etc -- Chapter 4. The Torsion Group of an Abelian Network -- 4.1. The removal lemma -- 4.2. Recurrent components -- 4.3. Construction of the torsion group -- 4.4. Relations to the critical group in the halting case -- Chapter 5. Critical Networks: Recurrence -- 5.1. Recurrent configurations and the burning test -- 5.2. Thief networks of a critical network -- 5.3. The capacity and the level of a configuration -- 5.4. Stoppable levels: When does the torsion group act transitively? -- Chapter 6. Critical Networks: Dynamics -- 6.1. Activity as a component invariant -- 6.2. Near uniqueness of legal executions -- Chapter 7. Rotor and Agent Networks -- 7.1. The cycle test for recurrence -- 7.2. Counting recurrent components -- 7.3. Determinantal generating functions for recurrent configurations -- Chapter 8. Concluding Remarks -- 8.1. A unified notion of recurrence and burning test -- 8.2. Forbidden subconfiguration test for recurrence -- 8.3. Number of recurrent configurations in a recurrent component -- Acknowledgement -- Bibliography -- Back Cover. 330 $a"An abelian network is a collection of communicating automata whose state transitions and message passing each satisfy a local commutativity condition. This paper is a continuation of the abelian networks series of Bond and Levine (2016), for which we extend the theory of abelian networks that halt on all inputs to networks that can run forever. A nonhalting abelian network can be realized as a discrete dynamical system in many different ways, depending on the update order. We show that certain features of the dynamics, such as minimal period length, have intrinsic definitions that do not require specifying an update order. We give an intrinsic definition of the torsion group of a finite irreducible (halting or nonhalting) abelian network, and show that it coincides with the critical group of Bond and Levine (2016) if the network is halting. We show that the torsion group acts freely on the set of invertible recurrent components of the trajectory digraph, and identify when this action is transitive. This perspective leads to new results even in the classical case of sinkless rotor networks (deterministic analogues of random walks). In Holroyd et. al (2008) it was shown that the recurrent configurations of a sinkless rotor network with just one chip are precisely the unicycles (spanning subgraphs with a unique oriented cycle, with the chip on the cycle). We generalize this result to abelian mobile agent networks with any number of chips. We give formulas for generating series such as where n is the number of recurrent chip-and-rotor configurations with n chips; D is the diagonal matrix of outdegrees, and A is the adjacency matrix. A consequence is that the sequence (n)n1 completely determines the spectrum of the simple random walk on the network"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aAbelian groups 606 $aCombinatorics -- Graph theory -- Graphs and abstract algebra (groups, rings, fields, etc.)$2msc 606 $aGroup theory and generalizations -- Abelian groups -- Finite abelian groups$2msc 606 $aGroup theory and generalizations -- Semigroups -- Commutative semigroups$2msc 606 $aGroup theory and generalizations -- Semigroups -- Semigroups in automata theory, linguistics, etc.$2msc 606 $aDynamical systems and ergodic theory -- Topological dynamics -- Cellular automata$2msc 606 $aDynamical systems and ergodic theory -- Low-dimensional dynamical systems -- Combinatorial dynamics (types of periodic orbits)$2msc 615 0$aAbelian groups. 615 7$aCombinatorics -- Graph theory -- Graphs and abstract algebra (groups, rings, fields, etc.). 615 7$aGroup theory and generalizations -- Abelian groups -- Finite abelian groups. 615 7$aGroup theory and generalizations -- Semigroups -- Commutative semigroups. 615 7$aGroup theory and generalizations -- Semigroups -- Semigroups in automata theory, linguistics, etc.. 615 7$aDynamical systems and ergodic theory -- Topological dynamics -- Cellular automata. 615 7$aDynamical systems and ergodic theory -- Low-dimensional dynamical systems -- Combinatorial dynamics (types of periodic orbits). 676 $a512/.25 676 $a512.25 686 $a05C25$a20K01$a20M14$a20M35$a37B15$a37E15$2msc 700 $aChan$b Swee Hong$01800465 701 $aLevine$b Lionel$01800466 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910965973803321 996 $aAbelian Networks IV. Dynamics of Nonhalting Networks$94345289 997 $aUNINA