Pub Date : 2022-06-29DOI: 10.1007/s40598-022-00205-2
Toshizumi Fukui, Takeki Tsuchiya
We discuss the notion of properness of a polynomial map (varvec{f}:mathbb {K}^mrightarrow mathbb {K}^n), (mathbb {K}=mathbb {C}) or (mathbb {R}), at a point of the target. We present a method to describe the set of non-proper points of (varvec{f}) with respect to Newton polyhedra of (varvec{f}). We obtain an explicit precise description of such a set of (varvec{f}) when (varvec{f}) satisfies certain condition (1.5). A relative version is also given in Sect. 3. Several tricks to describe the set of non-proper points of (varvec{f}) without the condition (1.5) is also given in Sect. 5.
{"title":"Properness of Polynomial Maps with Newton Polyhedra","authors":"Toshizumi Fukui, Takeki Tsuchiya","doi":"10.1007/s40598-022-00205-2","DOIUrl":"10.1007/s40598-022-00205-2","url":null,"abstract":"<div><p>We discuss the notion of properness of a polynomial map <span>(varvec{f}:mathbb {K}^mrightarrow mathbb {K}^n)</span>, <span>(mathbb {K}=mathbb {C})</span> or <span>(mathbb {R})</span>, at a point of the target. We present a method to describe the set of non-proper points of <span>(varvec{f})</span> with respect to Newton polyhedra of <span>(varvec{f})</span>. We obtain an explicit precise description of such a set of <span>(varvec{f})</span> when <span>(varvec{f})</span> satisfies certain condition (1.5). A relative version is also given in Sect. 3. Several tricks to describe the set of non-proper points of <span>(varvec{f})</span> without the condition (1.5) is also given in Sect. 5.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43371287","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-27DOI: 10.1007/s40598-022-00208-z
Avery St. Dizier, Alexander Yong
We connect generalized permutahedra with Schubert calculus. Thereby, we give sufficient vanishing criteria for Schubert intersection numbers of the flag variety. Our argument utilizes recent developments in the study of Schubitopes, which are Newton polytopes of Schubert polynomials. The resulting tableau test executes in polynomial time.
{"title":"Generalized Permutahedra and Schubert Calculus","authors":"Avery St. Dizier, Alexander Yong","doi":"10.1007/s40598-022-00208-z","DOIUrl":"10.1007/s40598-022-00208-z","url":null,"abstract":"<div><p>We connect generalized permutahedra with Schubert calculus. Thereby, we give sufficient vanishing criteria for Schubert intersection numbers of the flag variety. Our argument utilizes recent developments in the study of Schubitopes, which are Newton polytopes of Schubert polynomials. The resulting tableau test executes in polynomial time.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46821283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-27DOI: 10.1007/s40598-022-00200-7
Trevor Clark, Kostiantyn Drach, Oleg Kozlovski, Sebastian van Strien
In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen 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:
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.
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.
Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques. In this setting, some of these tools are:
the Enhanced Nest (a nest of puzzle pieces around critical points) from Kozlovski, Shen, van Strien (Ann Math 165:749–841, 2007), referred to below as KSS;
the Covering Lemma (which controls the moduli of pullbacks of annuli) from Kahn and Lyubich (Ann Math 169(2):561–593, 2009);
the QC-Criterion and the Spreading Principle from KSS.
The 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.
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:
puzzle pieces shrink to points,
(under some assumptions) topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate.
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 are the analogues of Lattès maps in this setting.
在全纯动力学中，复盒映射作为到精心选择的域的第一返回映射而出现。它们是类多项式映射的推广，其中返回映射的域可以具有无限多个分量。事实证明，它们在解决各种问题方面非常有用。本文的目的是：说明当复杂的盒映射不是由全局定义的映射引起时，以及当其域具有无限多个分量时，可能发生的一些病理，并给出避免这些问题的条件。为了表明，一旦有了有理映射的长方体映射，就可以假设这些条件在非常自然的环境中成立。因此，我们将这种复杂的盒映射称为动态自然映射。拥有这样的盒映射是解决一维动力学中许多问题的第一步。全纯动力学的许多结果依赖于组合和分析技术之间的相互作用。在这种情况下，其中一些工具是：Kozlovski，Shen，van Strien（Ann Math 165:749–8412007）的Enhanced Nest（一组围绕关键点的拼图），下文称为KSS；Kahn和Lyubich的覆盖引理（控制环空回撤模量）（Ann Math 169（2）:561–5932009）；质量控制准则和KSS的推广原理。本文的目的是让这些工具更容易访问，这样它们就可以用作“黑盒”，这样就不必在新的设置中重做校样。为了直观但相当详细地概述KSS和Kozlovski以及van Strien（Proc Lond Math Soc（3）99:275–2962009）对非重整化动态自然复盒映射的以下结果的证明：拼图收缩到点，（在某些假设下）拓扑共轭的非重整化多项式和盒映射是拟共形共轭的。我们证明了动态自然盒映射的基本遍历性质。这导致了一些必要的条件，当这样的盒子映射支持其填充的Julia集上的可测量不变线场时。这些映射是这种情况下Lattès映射的类似物。我们证明了复盒映射的Mañé定理的一个版本，该定理涉及沿避开临界点集邻域的点的轨道的展开。
{"title":"The Dynamics of Complex Box Mappings","authors":"Trevor Clark, Kostiantyn Drach, Oleg Kozlovski, Sebastian van Strien","doi":"10.1007/s40598-022-00200-7","DOIUrl":"10.1007/s40598-022-00200-7","url":null,"abstract":"<div><p>In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen 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:</p><ul>\u0000 <li>\u0000 <p>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.</p>\u0000 </li>\u0000 <li>\u0000 <p>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 <i>dynamically natural</i>. Having such box mappings is the first step in tackling many problems in one-dimensional dynamics.</p>\u0000 </li>\u0000 <li>\u0000 <p>Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques. In this setting, some of these tools are:</p><ul>\u0000 <li>\u0000 <p>the Enhanced Nest (a nest of puzzle pieces around critical points) from Kozlovski, Shen, van Strien (Ann Math 165:749–841, 2007), referred to below as KSS;</p>\u0000 </li>\u0000 <li>\u0000 <p>the Covering Lemma (which controls the moduli of pullbacks of annuli) from Kahn and Lyubich (Ann Math 169(2):561–593, 2009);</p>\u0000 </li>\u0000 <li>\u0000 <p>the QC-Criterion and the Spreading Principle from KSS.</p>\u0000 </li>\u0000 </ul><p> The 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.</p>\u0000 </li>\u0000 <li>\u0000 <p>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:</p><ul>\u0000 <li>\u0000 <p>puzzle pieces shrink to points,</p>\u0000 </li>\u0000 <li>\u0000 <p>(under some assumptions) topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate.</p>\u0000 </li>\u0000 </ul>\u0000 </li>\u0000 <li>\u0000 <p>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 are the analogues of Lattès maps in this setting.</p>\u0000 </li>\u0000 <li>\u0000 ","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-022-00200-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50518555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-25DOI: 10.1007/s40598-022-00207-0
Ilya Karzhemanov
With any surjective rational map(f: mathbb {P}^n dashrightarrow mathbb {P}^n) of the projective space, we associate a numerical invariant (ML degree) and compute it in terms of a naturally defined vector bundle (E_f longrightarrow mathbb {P}^n).
{"title":"Maximum Likelihood Degree of Surjective Rational Maps","authors":"Ilya Karzhemanov","doi":"10.1007/s40598-022-00207-0","DOIUrl":"10.1007/s40598-022-00207-0","url":null,"abstract":"<div><p>With any <i>surjective rational map</i> <span>(f: mathbb {P}^n dashrightarrow mathbb {P}^n)</span> of the projective space, we associate a numerical invariant (<i>ML degree</i>) and compute it in terms of a naturally defined vector bundle <span>(E_f longrightarrow mathbb {P}^n)</span>.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45581553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-26DOI: 10.1007/s40598-022-00204-3
Andrei Gabrielov
Generic spherical quadrilaterals are classified up to isometry. Condition of genericity consists in the requirement that the images of the sides under the developing map belong to four distinct circles which have no triple intersections. Under this condition, it is shown that the space of quadrilaterals with prescribed angles consists of finitely many open curves. Degeneration at the endpoints of these curves is also determined.
{"title":"Classification of Generic Spherical Quadrilaterals","authors":"Andrei Gabrielov","doi":"10.1007/s40598-022-00204-3","DOIUrl":"10.1007/s40598-022-00204-3","url":null,"abstract":"<div><p>Generic spherical quadrilaterals are classified up to isometry. Condition of genericity consists in the requirement that the images of the sides under the developing map belong to four distinct circles which have no triple intersections. Under this condition, it is shown that the space of quadrilaterals with prescribed angles consists of finitely many open curves. Degeneration at the endpoints of these curves is also determined.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-022-00204-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42828660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-22DOI: 10.1007/s40598-022-00203-4
Grigory Solomadin
In this paper, for any Milnor hypersurface, we find the largest dimension of effective algebraic torus actions on it. The proof of the corresponding theorem is based on the computation of the automorphism group for any Milnor hypersurface. We find all generalized Buchstaber–Ray and Ray hypersurfaces that are toric varieties. We compute the Betti numbers of these hypersurfaces and describe their integral singular cohomology rings in terms of the cohomology of the corresponding ambient varieties.
{"title":"Cohomology Rings and Algebraic Torus Actions on Hypersurfaces in the Product of Projective Spaces and Bounded Flag Varieties","authors":"Grigory Solomadin","doi":"10.1007/s40598-022-00203-4","DOIUrl":"10.1007/s40598-022-00203-4","url":null,"abstract":"<div><p>In this paper, for any Milnor hypersurface, we find the largest dimension of effective algebraic torus actions on it. The proof of the corresponding theorem is based on the computation of the automorphism group for any Milnor hypersurface. We find all generalized Buchstaber–Ray and Ray hypersurfaces that are toric varieties. We compute the Betti numbers of these hypersurfaces and describe their integral singular cohomology rings in terms of the cohomology of the corresponding ambient varieties.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46354160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-16DOI: 10.1007/s40598-022-00202-5
A. Achak, A. Abouelaz, R. Daher, N. Safouane
We prove various mathematical aspects of the quantitative uncertainty principles, including Donoho–Stark’s uncertainty principle and a variant of Benedicks theorem for Lions transform.
{"title":"Quantitative Uncertainty Principles Related to Lions Transform","authors":"A. Achak, A. Abouelaz, R. Daher, N. Safouane","doi":"10.1007/s40598-022-00202-5","DOIUrl":"10.1007/s40598-022-00202-5","url":null,"abstract":"<div><p>We prove various mathematical aspects of the quantitative uncertainty principles, including Donoho–Stark’s uncertainty principle and a variant of Benedicks theorem for Lions transform.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43801122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-24DOI: 10.1007/s40598-022-00201-6
Alexey Gorinov
The fundamental theorem of affine geometry says that if a self-bijection f of an affine space of dimenion n over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then f of the expected type, namely f is a composition of an affine map and an automorphism of the field. We prove a two-sided analogue of this: namely, we consider self-bijections as above which take affine subspaces to affine subspaces but which are allowed to take left subspaces to right ones and vice versa. We show that under some conditions these maps again are of the expected type.
{"title":"Two-Sided Fundamental Theorem of Affine Geometry","authors":"Alexey Gorinov","doi":"10.1007/s40598-022-00201-6","DOIUrl":"10.1007/s40598-022-00201-6","url":null,"abstract":"<div><p>The fundamental theorem of affine geometry says that if a self-bijection <i>f</i> of an affine space of dimenion <i>n</i> over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then <i>f</i> of the expected type, namely <i>f</i> is a composition of an affine map and an automorphism of the field. We prove a two-sided analogue of this: namely, we consider self-bijections as above which take affine subspaces to affine subspaces but which are allowed to take left subspaces to right ones and vice versa. We show that under some conditions these maps again are of the expected type.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41287513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-03DOI: 10.1007/s40598-022-00199-x
Gabriela Estevez, Pablo Guarino
A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper, we establish this principle for a large class of bicritical circle maps, which are (C^3) circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo in (J Eur Math Soc 1:339–392, 1999) for the case of a single critical point. When combined with the recent papers (Estevez et al. in Complex bounds for multicritical circle maps with bounded type rotation number, arXiv:2005.02377, 2020; Yampolsky in C R Math Rep Acad Sci Can 41:57–83, 2019), our main theorem implies (C^{1+alpha }) rigidity for real-analytic bicritical circle maps with rotation number of bounded type (Corollary 1.1).
重整化理论中已经在许多重要情况下建立了一个一般的假设，即重整化轨道的指数收敛意味着拓扑共轭实际上是光滑的（当限制于原始系统的吸引子时）。本文建立了一大类双临界圆映射的这一原理，这些映射是具有无理旋转数和恰好两个（非平坦）临界点的（C^3）圆同胚。这里给出的证明是对de Faria和de Melo在（《欧洲数学会杂志》1999年第1卷第139-392页）中给出的单临界点情况下的双临界设置的改编。当与最近的论文相结合时（Estevez等人在《具有有界型旋转数的多临界圆映射的复界》中，arXiv:2005.023772020；Yampolsky在《C R Math Rep Acad Sci Can 41:57–832019》中），我们的主要定理暗示了具有有界类型旋转数的实解析双临界圆映射（推论1.1）的（C^｛1+alpha｝）刚性。
{"title":"Renormalization of Bicritical Circle Maps","authors":"Gabriela Estevez, Pablo Guarino","doi":"10.1007/s40598-022-00199-x","DOIUrl":"10.1007/s40598-022-00199-x","url":null,"abstract":"<div><p>A general <i>ansatz</i> in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper, we establish this principle for a large class of <i>bicritical circle maps</i>, which are <span>(C^3)</span> circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo in (J Eur Math Soc 1:339–392, 1999) for the case of a single critical point. When combined with the recent papers (Estevez et al. in Complex bounds for multicritical circle maps with bounded type rotation number, arXiv:2005.02377, 2020; Yampolsky in C R Math Rep Acad Sci Can 41:57–83, 2019), our main theorem implies <span>(C^{1+alpha })</span> rigidity for real-analytic bicritical circle maps with rotation number of <i>bounded type</i> (Corollary 1.1).</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49563940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-22DOI: 10.1007/s40598-022-00197-z
Talia Blum, John R. Doyle, Trevor Hyde, Colby Kelln, Henry Talbott, Max Weinreich
A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.
{"title":"Dynamical Moduli Spaces and Polynomial Endomorphisms of Configurations","authors":"Talia Blum, John R. Doyle, Trevor Hyde, Colby Kelln, Henry Talbott, Max Weinreich","doi":"10.1007/s40598-022-00197-z","DOIUrl":"10.1007/s40598-022-00197-z","url":null,"abstract":"<div><p>A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47885513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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