In this paper, we focus on constructing and refining geometric Markov partitions for pseudo-Anosov homeomorphisms that may contain spines. We introduce a systematic approach to constructing emph{adapted Markov partitions} for these homeomorphisms. Our primary result is an algorithmic construction of emph{adapted Markov partitions} for every generalized pseudo-Anosov map, starting from a single point. This algorithm is applied to the so-called emph{first intersection points} of the homeomorphism, producing emph{primitive Markov partitions} that behave well under iterations. We also prove that the set of emph{primitive geometric types} of a given order is finite, providing a canonical tool for classifying pseudo-Anosov homeomorphisms. We then construct new geometric Markov partitions from existing ones, maintaining control over their combinatorial properties and preserving their geometric types. The first geometric Markov partition we construct has a binary incidence matrix, which allows for the introduction of the sub-shift of finite type associated with any Markov partition's incidence matrix -- this is known as the emph{binary refinement}. We also describe a process that cuts any Markov partition along stable and unstable segments prescribed by a finite set of periodic codes, referred to as the $s$ and $U$-boundary refinements. Finally, we present an algorithmic construction of a Markov partition where all periodic boundary points are located at the corners of the rectangles in the partition, called the emph{corner refinement}. Each of these Markov partitions and their intrinsic combinatorial properties plays a crucial role in our algorithmic classification of pseudo-Anosov homeomorphisms up to topological conjugacy.
{"title":"Geometric Markov partitions for pseudo-Anosov homeomorphisms with prescribed combinatorics","authors":"Inti Cruz Diaz","doi":"arxiv-2409.03066","DOIUrl":"https://doi.org/arxiv-2409.03066","url":null,"abstract":"In this paper, we focus on constructing and refining geometric Markov\u0000partitions for pseudo-Anosov homeomorphisms that may contain spines. We\u0000introduce a systematic approach to constructing emph{adapted Markov\u0000partitions} for these homeomorphisms. Our primary result is an algorithmic\u0000construction of emph{adapted Markov partitions} for every generalized\u0000pseudo-Anosov map, starting from a single point. This algorithm is applied to\u0000the so-called emph{first intersection points} of the homeomorphism, producing\u0000emph{primitive Markov partitions} that behave well under iterations. We also\u0000prove that the set of emph{primitive geometric types} of a given order is\u0000finite, providing a canonical tool for classifying pseudo-Anosov\u0000homeomorphisms. We then construct new geometric Markov partitions from existing\u0000ones, maintaining control over their combinatorial properties and preserving\u0000their geometric types. The first geometric Markov partition we construct has a\u0000binary incidence matrix, which allows for the introduction of the sub-shift of\u0000finite type associated with any Markov partition's incidence matrix -- this is\u0000known as the emph{binary refinement}. We also describe a process that cuts any\u0000Markov partition along stable and unstable segments prescribed by a finite set\u0000of periodic codes, referred to as the $s$ and $U$-boundary refinements.\u0000Finally, we present an algorithmic construction of a Markov partition where all\u0000periodic boundary points are located at the corners of the rectangles in the\u0000partition, called the emph{corner refinement}. Each of these Markov partitions\u0000and their intrinsic combinatorial properties plays a crucial role in our\u0000algorithmic classification of pseudo-Anosov homeomorphisms up to topological\u0000conjugacy.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195905","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}
We proved that the normalized Ricci flow does not preserve the positivity of Ricci curvature of Riemannian metrics on every generalized Wallach space with $a_1+a_2+a_3le 1/2$, in particular on the spaces $operatorname{SU}(k+l+m)/operatorname{SU}(k)times operatorname{SU}(l) times operatorname{SU}(m)$ and $operatorname{Sp}(k+l+m)/operatorname{Sp}(k)times operatorname{Sp}(l) times operatorname{Sp}(m)$ independently on $k,l$ and $m$. The positivity of Ricci curvature is preserved for all original metrics with $operatorname{Ric}>0$ on generalized Wallach spaces $a_1+a_2+a_3> 1/2$ if the conditions $4left(a_j+a_kright)^2ge (1-2a_i)(1+2a_i)^{-1}$ hold for all ${i,j,k}={1,2,3}$. We also established that the spaces $operatorname{SO}(k+l+m)/operatorname{SO}(k)times operatorname{SO}(l)times operatorname{SO}(m)$ satisfy the above conditions for $max{k,l,m}le 11$, moreover, additional conditions were found to keep $operatorname{Ric}>0$ in cases when $max{k,l,m}le 11$ is violated. Similar questions have also been studied for all other generalized Wallach spaces given in the classification of Yuriui Nikonorov.
{"title":"Ricci curvature and normalized Ricci flow on generalized Wallach spaces","authors":"Nurlan Abiev","doi":"arxiv-2409.02570","DOIUrl":"https://doi.org/arxiv-2409.02570","url":null,"abstract":"We proved that the normalized Ricci flow does not preserve the positivity of\u0000Ricci curvature of Riemannian metrics on every generalized Wallach space with\u0000$a_1+a_2+a_3le 1/2$, in particular on the spaces\u0000$operatorname{SU}(k+l+m)/operatorname{SU}(k)times operatorname{SU}(l)\u0000times operatorname{SU}(m)$ and\u0000$operatorname{Sp}(k+l+m)/operatorname{Sp}(k)times operatorname{Sp}(l)\u0000times operatorname{Sp}(m)$ independently on $k,l$ and $m$. The positivity of\u0000Ricci curvature is preserved for all original metrics with\u0000$operatorname{Ric}>0$ on generalized Wallach spaces $a_1+a_2+a_3> 1/2$ if the\u0000conditions $4left(a_j+a_kright)^2ge (1-2a_i)(1+2a_i)^{-1}$ hold for all\u0000${i,j,k}={1,2,3}$. We also established that the spaces\u0000$operatorname{SO}(k+l+m)/operatorname{SO}(k)times operatorname{SO}(l)times\u0000operatorname{SO}(m)$ satisfy the above conditions for $max{k,l,m}le 11$,\u0000moreover, additional conditions were found to keep $operatorname{Ric}>0$ in\u0000cases when $max{k,l,m}le 11$ is violated. Similar questions have also been\u0000studied for all other generalized Wallach spaces given in the classification of\u0000Yuriui Nikonorov.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195912","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}
This article establishes estimates on the dimension of the global attractor of the two-dimensional rotating Navier-Stokes equation for viscous, incompressible fluids on the $beta$-plane. Previous results in this setting by M.A.H. Al-Jaboori and D. Wirosoetisno (2011) had proved that the global attractor collapses to a single point that depends only the longitudinal coordinate, i.e., zonal flow, when the rotation is sufficiently fast. However, an explicit quantification of the complexity of the global attractor in terms of $beta$ had remained open. In this paper, such estimates are established which are valid across a wide regime of rotation rates and are consistent with the dynamically degenerate regime previously identified. Additionally, a decomposition of solutions is established detailing the asymptotic behavior of the solutions in the limit of large rotation.
本文建立了对$beta$平面上粘性不可压缩流体的二维旋转纳维-斯托克斯方程全局吸引子维度的估计。此前，M.A.H. Al-Jaboori 和 D. Wirosoetisno（2011 年）在此背景下的结果证明，当旋转速度足够快时，全局吸引子会坍缩为一个仅取决于纵坐标的单点，即纵向流。然而，用 $beta$ 来明确量化全局吸引子的复杂性仍然是个未知数。本文建立的这种估计值在很宽的旋转速率范围内都是有效的，并且与之前确定的动力学退化机制是一致的。此外，本文还建立了解的分解，详细说明了大旋转极限下解的渐近行为。
{"title":"Upper bounds on the dimension of the global attractor of the 2D Navier-Stokes equations on the $β-$plane","authors":"Aseel Farhat, Anuj Kumar, Vincent R. Martinez","doi":"arxiv-2409.02868","DOIUrl":"https://doi.org/arxiv-2409.02868","url":null,"abstract":"This article establishes estimates on the dimension of the global attractor\u0000of the two-dimensional rotating Navier-Stokes equation for viscous,\u0000incompressible fluids on the $beta$-plane. Previous results in this setting by\u0000M.A.H. Al-Jaboori and D. Wirosoetisno (2011) had proved that the global\u0000attractor collapses to a single point that depends only the longitudinal\u0000coordinate, i.e., zonal flow, when the rotation is sufficiently fast. However,\u0000an explicit quantification of the complexity of the global attractor in terms\u0000of $beta$ had remained open. In this paper, such estimates are established\u0000which are valid across a wide regime of rotation rates and are consistent with\u0000the dynamically degenerate regime previously identified. Additionally, a\u0000decomposition of solutions is established detailing the asymptotic behavior of\u0000the solutions in the limit of large rotation.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195911","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}
A discrete-time deterministic dynamical system is governed at every step by a predetermined law. However the dynamics can lead to many complexities in the phase space and in the domain of observables that makes it comparable to a stochastic process. This article presents two different ways of representing a dynamical system by stochastic processes. The first is a step-skew product system, in which a finite state Markov process drives a dynamics on Euclidean space. The second is a skew-product system, in which a deterministic, mixing flow intermittently drives a deterministic flow through a topological space created by gluing cylinders. This system is called a perturbed pipe-flow. We show how these three representations are interchangeable. The inter-connections also reveal how a deterministic chaotic system partitions the phase space at a local level, and also mixes the phase space at a global level.
{"title":"Discrete-time dynamics, step-skew products, and pipe-flows","authors":"Suddhasattwa Das","doi":"arxiv-2409.02318","DOIUrl":"https://doi.org/arxiv-2409.02318","url":null,"abstract":"A discrete-time deterministic dynamical system is governed at every step by a\u0000predetermined law. However the dynamics can lead to many complexities in the\u0000phase space and in the domain of observables that makes it comparable to a\u0000stochastic process. This article presents two different ways of representing a\u0000dynamical system by stochastic processes. The first is a step-skew product\u0000system, in which a finite state Markov process drives a dynamics on Euclidean\u0000space. The second is a skew-product system, in which a deterministic, mixing\u0000flow intermittently drives a deterministic flow through a topological space\u0000created by gluing cylinders. This system is called a perturbed pipe-flow. We\u0000show how these three representations are interchangeable. The inter-connections\u0000also reveal how a deterministic chaotic system partitions the phase space at a\u0000local level, and also mixes the phase space at a global level.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195951","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}
Reaction networks are a general framework widely used in modelling diverse phenomena in different science disciplines. The dynamical process of a reaction network endowed with mass-action kinetics is a mass-action system. In this paper we study dynamics of first order mass-action systems. We prove that every first order endotactic mass-action system has a weakly reversible deficiency zero realization, and has a unique equilibrium which is exponentially globally asymptotically stable (and is positive) in each (positive) stoichiometric compatibility class. In particular, we prove that global attractivity conjecture holds for every linear complex balanced mass-action system. In this way, we exclude the possibility of first order endotactic mass-action systems to admit multistationarity or multistability. The result indicates that the importance of binding molecules in reactants is crucial for (endotactic) reaction networks to have complicated dynamics like limit cycles. The proof relies on the fact that $mathcal{A}$-endotacticity of first order reaction networks implies endotacticity for a finite set $mathcal{A}$, which is also proved in this paper. Out of independent interest, we provide a sufficient condition for endotacticity of reaction networks which are not necessarily of first order.
{"title":"Global stability of first order endotactic reaction systems","authors":"Chuang Xu","doi":"arxiv-2409.01598","DOIUrl":"https://doi.org/arxiv-2409.01598","url":null,"abstract":"Reaction networks are a general framework widely used in modelling diverse\u0000phenomena in different science disciplines. The dynamical process of a reaction\u0000network endowed with mass-action kinetics is a mass-action system. In this\u0000paper we study dynamics of first order mass-action systems. We prove that every\u0000first order endotactic mass-action system has a weakly reversible deficiency\u0000zero realization, and has a unique equilibrium which is exponentially globally\u0000asymptotically stable (and is positive) in each (positive) stoichiometric\u0000compatibility class. In particular, we prove that global attractivity\u0000conjecture holds for every linear complex balanced mass-action system. In this\u0000way, we exclude the possibility of first order endotactic mass-action systems\u0000to admit multistationarity or multistability. The result indicates that the\u0000importance of binding molecules in reactants is crucial for (endotactic)\u0000reaction networks to have complicated dynamics like limit cycles. The proof\u0000relies on the fact that $mathcal{A}$-endotacticity of first order reaction\u0000networks implies endotacticity for a finite set $mathcal{A}$, which is also\u0000proved in this paper. Out of independent interest, we provide a sufficient condition for\u0000endotacticity of reaction networks which are not necessarily of first order.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195946","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}
Given a continuous linear cocycle $mathcal{A}$ over a homeomorphism $f$ of a compact metric space $X$, we investigate its set $mathcal{R}$ of Lyapunov-Perron regular points, that is, the collection of trajectories of $f$ that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain results roughly saying that the set $mathcal{R}$ is of first Baire category (i.e., meager) in $X$, unless some rigid structure is present. In some settings, this rigid structure forces the Lyapunov exponents to be defined everywhere and to be independent of the point; that is what we call complete regularity.
{"title":"Complete regularity of linear cocycles and the Baire category of the set of Lyapunov-Perron regular points","authors":"Jairo Bochi, Yakov Pesin, Omri Sarig","doi":"arxiv-2409.01798","DOIUrl":"https://doi.org/arxiv-2409.01798","url":null,"abstract":"Given a continuous linear cocycle $mathcal{A}$ over a homeomorphism $f$ of a\u0000compact metric space $X$, we investigate its set $mathcal{R}$ of\u0000Lyapunov-Perron regular points, that is, the collection of trajectories of $f$\u0000that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain\u0000results roughly saying that the set $mathcal{R}$ is of first Baire category\u0000(i.e., meager) in $X$, unless some rigid structure is present. In some\u0000settings, this rigid structure forces the Lyapunov exponents to be defined\u0000everywhere and to be independent of the point; that is what we call complete\u0000regularity.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195920","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}
Let $omega$ be a plane autonomous system and C its configuration of algebraic integral curves. If the singularities of C are quasi homogeneous we give new conditions for existence of a Darboux integrating factor or a Darboux first integral. This is used to construct new components of the center variety in degree 3.
让 $omega$ 是一个平面自治系统，C 是其代数积分曲线的配置。如果 C 的奇点是准均质的，我们就给出了达尔布积分因子或达尔布第一积分存在的新条件。这将用于构造 3 度中心变的新分量。
{"title":"New solutions of the Poincaré Center Problem in degree 3","authors":"Hans-Christian von Bothmer","doi":"arxiv-2409.01751","DOIUrl":"https://doi.org/arxiv-2409.01751","url":null,"abstract":"Let $omega$ be a plane autonomous system and C its configuration of\u0000algebraic integral curves. If the singularities of C are quasi homogeneous we\u0000give new conditions for existence of a Darboux integrating factor or a Darboux\u0000first integral. This is used to construct new components of the center variety\u0000in degree 3.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195922","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}
Let $f$ be a polynomial-like map with dominant topological degree $d_tgeq 2$ and let $d_{k-1}
让 $f$ 是一个拓扑阶数为 $d_tgeq 2$ 的类多项式映射，让 $d_{k-1}
{"title":"On the support of measures of large entropy for polynomial-like maps","authors":"Sardor Bazarbaev, Fabrizio Bianchi, Karim Rakhimov","doi":"arxiv-2409.02039","DOIUrl":"https://doi.org/arxiv-2409.02039","url":null,"abstract":"Let $f$ be a polynomial-like map with dominant topological degree $d_tgeq 2$\u0000and let $d_{k-1}<d_t$ be its dynamical degree of order $k-1$. We show that the\u0000support of every ergodic measure whose measure-theoretic entropy is strictly\u0000larger than $log sqrt{d_{k-1} d_t}$ is supported on the Julia set, i.e., the\u0000support of the unique measure of maximal entropy $mu$. The proof is based on\u0000the exponential speed of convergence of the measures $d_t^{-n}(f^n)^*delta_a$\u0000towards $mu$, which is valid for a generic point $a$ and with a controlled\u0000error bound depending on $a$. Our proof also gives a new proof of the same\u0000statement in the setting of endomorphisms of $mathbb P^k(mathbb C)$ - a\u0000result due to de Th'elin and Dinh - which does not rely on the existence of a\u0000Green current.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195915","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}
This paper addresses the perturbation of higher-dimensional non-smooth autonomous differential systems characterized by two zones separated by a codimension-one manifold, with an integral manifold foliated by crossing periodic solutions. Our primary focus is on developing the Melnikov method to analyze the emergence of limit cycles originating from the periodic integral manifold. While previous studies have explored the Melnikov method for autonomous perturbations of non-smooth differential systems with a linear switching manifold and with a periodic integral manifold, either open or of codimension 1, our work extends to non-smooth differential systems with a non-linear switching manifold and more general periodic integral manifolds, where the persistence of periodic orbits is of interest. We illustrate our findings through several examples, highlighting the applicability and significance of our main result.
{"title":"Limit cycles bifurcating from periodic integral manifold in non-smooth differential systems","authors":"Oscar A. R. Cespedes, Douglas D. Novaes","doi":"arxiv-2409.01851","DOIUrl":"https://doi.org/arxiv-2409.01851","url":null,"abstract":"This paper addresses the perturbation of higher-dimensional non-smooth\u0000autonomous differential systems characterized by two zones separated by a\u0000codimension-one manifold, with an integral manifold foliated by crossing\u0000periodic solutions. Our primary focus is on developing the Melnikov method to\u0000analyze the emergence of limit cycles originating from the periodic integral\u0000manifold. While previous studies have explored the Melnikov method for\u0000autonomous perturbations of non-smooth differential systems with a linear\u0000switching manifold and with a periodic integral manifold, either open or of\u0000codimension 1, our work extends to non-smooth differential systems with a\u0000non-linear switching manifold and more general periodic integral manifolds,\u0000where the persistence of periodic orbits is of interest. We illustrate our\u0000findings through several examples, highlighting the applicability and\u0000significance of our main result.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195916","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}
In the study of properties within one-dimensional dynamics, the assumption of a negative Schwarzian derivative has been shown to be very useful. However, this condition may appear somewhat arbitrary, as it is not a dynamical condition in any sense other than that it is preserved for its iterates. In this brief work, we show that the assumption of a negative Schwarzian derivative it is not entirely arbitrary but rather strictly related to the fulfillment of the Minimum Principle for the derivative of the map and its iterates, which is the key point in the proof of Singer's Theorem.
{"title":"On the Use of the Schwarzian derivative in Real One-Dimensional Dynamics","authors":"Felipe Correa, Bernardo San Martín","doi":"arxiv-2409.00959","DOIUrl":"https://doi.org/arxiv-2409.00959","url":null,"abstract":"In the study of properties within one-dimensional dynamics, the assumption of\u0000a negative Schwarzian derivative has been shown to be very useful. However,\u0000this condition may appear somewhat arbitrary, as it is not a dynamical\u0000condition in any sense other than that it is preserved for its iterates. In\u0000this brief work, we show that the assumption of a negative Schwarzian\u0000derivative it is not entirely arbitrary but rather strictly related to the\u0000fulfillment of the Minimum Principle for the derivative of the map and its\u0000iterates, which is the key point in the proof of Singer's Theorem.","PeriodicalId":501035,"journal":{"name":"arXiv - MATH - Dynamical Systems","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142195919","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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