[{"publisher":"Springer Nature","intvolume":"         8","status":"public","department":[{"_id":"VaKa"}],"quality_controlled":"1","publication":"Arnold Mathematical Journal","page":"319-410","date_created":"2022-07-10T22:01:53Z","month":"06","acknowledgement":"We would also like to thank Dzmitry Dudko and Dierk Schleicher for many stimulating discussions and encouragement during our work on this project, and Weixiao Shen, Mikhail Hlushchanka and the referee for helpful comments. We are grateful to Leon Staresinic who carefully read the revised version of the manuscript and provided many helpful suggestions.","related_material":{"link":[{"relation":"erratum","url":"https://doi.org/10.1007/s40598-022-00209-y"},{"relation":"erratum","url":"https://doi.org/10.1007/s40598-022-00218-x"}]},"project":[{"_id":"9B8B92DE-BA93-11EA-9121-9846C619BF3A","call_identifier":"H2020","name":"Spectral rigidity and integrability for billiards and geodesic flows","grant_number":"885707"}],"doi":"10.1007/s40598-022-00200-7","ddc":["500"],"language":[{"iso":"eng"}],"title":"The dynamics of complex box mappings","ec_funded":1,"citation":{"ieee":"T. Clark, K. Drach, O. Kozlovski, and S. V. Strien, “The dynamics of complex box mappings,” <i>Arnold Mathematical Journal</i>, vol. 8, no. 2. Springer Nature, pp. 319–410, 2022.","ista":"Clark T, Drach K, Kozlovski O, Strien SV. 2022. The dynamics of complex box mappings. Arnold Mathematical Journal. 8(2), 319–410.","chicago":"Clark, Trevor, Kostiantyn Drach, Oleg Kozlovski, and Sebastian Van Strien. “The Dynamics of Complex Box Mappings.” <i>Arnold Mathematical Journal</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s40598-022-00200-7\">https://doi.org/10.1007/s40598-022-00200-7</a>.","mla":"Clark, Trevor, et al. “The Dynamics of Complex Box Mappings.” <i>Arnold Mathematical Journal</i>, vol. 8, no. 2, Springer Nature, 2022, pp. 319–410, doi:<a href=\"https://doi.org/10.1007/s40598-022-00200-7\">10.1007/s40598-022-00200-7</a>.","short":"T. Clark, K. Drach, O. Kozlovski, S.V. Strien, Arnold Mathematical Journal 8 (2022) 319–410.","apa":"Clark, T., Drach, K., Kozlovski, O., &#38; Strien, S. V. (2022). The dynamics of complex box mappings. <i>Arnold Mathematical Journal</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40598-022-00200-7\">https://doi.org/10.1007/s40598-022-00200-7</a>","ama":"Clark T, Drach K, Kozlovski O, Strien SV. The dynamics of complex box mappings. <i>Arnold Mathematical Journal</i>. 2022;8(2):319-410. doi:<a href=\"https://doi.org/10.1007/s40598-022-00200-7\">10.1007/s40598-022-00200-7</a>"},"author":[{"last_name":"Clark","first_name":"Trevor","full_name":"Clark, Trevor"},{"last_name":"Drach","full_name":"Drach, Kostiantyn","first_name":"Kostiantyn","orcid":"0000-0002-9156-8616","id":"fe8209e2-906f-11eb-847d-950f8fc09115"},{"last_name":"Kozlovski","first_name":"Oleg","full_name":"Kozlovski, Oleg"},{"full_name":"Strien, Sebastian Van","first_name":"Sebastian Van","last_name":"Strien"}],"type":"journal_article","day":"01","oa":1,"publication_status":"published","volume":8,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"file_date_updated":"2022-07-12T10:04:55Z","issue":"2","article_processing_charge":"No","abstract":[{"text":"In holomorphic dynamics, complex box mappings arise as first return maps to wellchosen domains. They are a generalization of polynomial-like mapping, where the domain of the return map can have infinitely many components. They turned out to be extremely useful in tackling diverse problems. The purpose of this paper is:\r\n• To illustrate some pathologies that can occur when a complex box mapping is not induced by a globally defined map and when its domain has infinitely many components, and to give conditions to avoid these issues.\r\n• To show that once one has a box mapping for a rational map, these conditions can be assumed to hold in a very natural setting. Thus, we call such complex box mappings dynamically natural. Having such box mappings is the first step in tackling many problems in one-dimensional dynamics.\r\n• Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques. In this setting, some of these tools are:\r\n  • the Enhanced Nest (a nest of puzzle pieces around critical points) from Kozlovski, Shen, van Strien (AnnMath 165:749–841, 2007), referred to below as KSS;\r\n  • the Covering Lemma (which controls the moduli of pullbacks of annuli) from Kahn and Lyubich (Ann Math 169(2):561–593, 2009);\r\n   • the QC-Criterion and the Spreading Principle from KSS.\r\nThe purpose of this paper is to make these tools more accessible so that they can be used as a ‘black box’, so one does not have to redo the proofs in new settings.\r\n• To give an intuitive, but also rather detailed, outline of the proof from KSS and Kozlovski and van Strien (Proc Lond Math Soc (3) 99:275–296, 2009) of the following results for non-renormalizable dynamically natural complex box mappings:\r\n   • puzzle pieces shrink to points,\r\n   • (under some assumptions) topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate.\r\n• We prove the fundamental ergodic properties for dynamically natural box mappings. This leads to some necessary conditions for when such a box mapping supports a measurable invariant line field on its filled Julia set. These mappings\r\nare the analogues of Lattès maps in this setting.\r\n• We prove a version of Mañé’s Theorem for complex box mappings concerning expansion along orbits of points that avoid a neighborhood of the set of critical points.","lang":"eng"}],"_id":"11553","date_published":"2022-06-01T00:00:00Z","file":[{"date_updated":"2022-07-12T10:04:55Z","access_level":"open_access","checksum":"16e7c659dee9073c6c8aeb87316ef201","file_name":"2022_ArnoldMathematicalJournal_Clark.pdf","creator":"kschuh","file_size":2509915,"success":1,"date_created":"2022-07-12T10:04:55Z","content_type":"application/pdf","file_id":"11559","relation":"main_file"}],"publication_identifier":{"issn":["2199-6792"],"eissn":["2199-6806"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2023-02-16T10:02:12Z","scopus_import":"1","oa_version":"None","has_accepted_license":"1","year":"2022","article_type":"original"},{"intvolume":"       408","status":"public","publication":"Advances in Mathematics","department":[{"_id":"VaKa"}],"quality_controlled":"1","isi":1,"publisher":"Elsevier","date_created":"2022-08-01T17:08:16Z","month":"10","doi":"10.1016/j.aim.2022.108591","ddc":["510"],"keyword":["General Mathematics"],"language":[{"iso":"eng"}],"project":[{"name":"Spectral rigidity and integrability for billiards and geodesic flows","grant_number":"885707","_id":"9B8B92DE-BA93-11EA-9121-9846C619BF3A","call_identifier":"H2020"}],"acknowledgement":"We are grateful to a number of colleagues for helpful and inspiring discussions during the time when we worked on this project, in particular Dima Dudko, Misha Hlushchanka, John Hubbard, Misha Lyubich, Oleg Kozlovski, and Sebastian van Strien. Finally, we would like to thank our dynamics research group for numerous helpful and enjoyable discussions: Konstantin Bogdanov, Roman Chernov, Russell Lodge, Steffen Maaß, David Pfrang, Bernhard Reinke, Sergey Shemyakov, and Maik Sowinski. We gratefully acknowledge support by the Advanced Grant “HOLOGRAM” (#695 621) of the European Research Council (ERC), as well as hospitality of Cornell University in the spring of 2018 while much of this work was prepared. The first-named author also acknowledges the support of the ERC Advanced Grant “SPERIG” (#885 707).","type":"journal_article","author":[{"id":"fe8209e2-906f-11eb-847d-950f8fc09115","orcid":"0000-0002-9156-8616","last_name":"Drach","first_name":"Kostiantyn","full_name":"Drach, Kostiantyn"},{"full_name":"Schleicher, Dierk","first_name":"Dierk","last_name":"Schleicher"}],"day":"29","title":"Rigidity of Newton dynamics","citation":{"chicago":"Drach, Kostiantyn, and Dierk Schleicher. “Rigidity of Newton Dynamics.” <i>Advances in Mathematics</i>. Elsevier, 2022. <a href=\"https://doi.org/10.1016/j.aim.2022.108591\">https://doi.org/10.1016/j.aim.2022.108591</a>.","ieee":"K. Drach and D. Schleicher, “Rigidity of Newton dynamics,” <i>Advances in Mathematics</i>, vol. 408, no. Part A. Elsevier, 2022.","ista":"Drach K, Schleicher D. 2022. Rigidity of Newton dynamics. Advances in Mathematics. 408(Part A), 108591.","mla":"Drach, Kostiantyn, and Dierk Schleicher. “Rigidity of Newton Dynamics.” <i>Advances in Mathematics</i>, vol. 408, no. Part A, 108591, Elsevier, 2022, doi:<a href=\"https://doi.org/10.1016/j.aim.2022.108591\">10.1016/j.aim.2022.108591</a>.","short":"K. Drach, D. Schleicher, Advances in Mathematics 408 (2022).","ama":"Drach K, Schleicher D. Rigidity of Newton dynamics. <i>Advances in Mathematics</i>. 2022;408(Part A). doi:<a href=\"https://doi.org/10.1016/j.aim.2022.108591\">10.1016/j.aim.2022.108591</a>","apa":"Drach, K., &#38; Schleicher, D. (2022). Rigidity of Newton dynamics. <i>Advances in Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.aim.2022.108591\">https://doi.org/10.1016/j.aim.2022.108591</a>"},"ec_funded":1,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)"},"volume":408,"file_date_updated":"2023-02-02T07:39:09Z","oa":1,"publication_status":"published","abstract":[{"text":"We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit can be distinguished in combinatorial terms from all other orbits), or the orbit of this point eventually lands in the filled-in Julia set of a polynomial-like restriction of the original map. As a corollary, we show that the Julia sets of Newton maps in many non-trivial cases are locally connected; in particular, every cubic Newton map without Siegel points has locally connected Julia set.\r\nIn the parameter space of Newton maps of arbitrary degree we obtain the following rigidity result: any two combinatorially equivalent Newton maps are quasiconformally conjugate in a neighborhood of their Julia sets provided that they either non-renormalizable, or they are both renormalizable “in the same way”.\r\nOur main tool is a generalized renormalization concept called “complex box mappings” for which we extend a dynamical rigidity result by Kozlovski and van Strien so as to include irrationally indifferent and renormalizable situations.","lang":"eng"}],"_id":"11717","date_published":"2022-10-29T00:00:00Z","file":[{"file_size":2164036,"creator":"dernst","file_name":"2022_AdvancesMathematics_Drach.pdf","access_level":"open_access","date_updated":"2023-02-02T07:39:09Z","checksum":"2710e6f5820f8c20a676ddcbb30f0e8d","file_id":"12474","relation":"main_file","content_type":"application/pdf","success":1,"date_created":"2023-02-02T07:39:09Z"}],"article_number":"108591","article_processing_charge":"Yes (via OA deal)","issue":"Part A","publication_identifier":{"issn":["0001-8708"]},"scopus_import":"1","external_id":{"isi":["000860924200005"]},"date_updated":"2023-08-03T12:36:07Z","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","has_accepted_license":"1","year":"2022","oa_version":"Published Version","article_type":"original"}]