Dynamical Systems
New submissions
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New submissions for Fri, 29 Mar 24
- [1] arXiv:2403.18940 [pdf, other]
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Title: On the discontinuities of Hausdorff dimension in generic dynamical Lagrange spectrumAuthors: Christian Camilo Silva VillamilSubjects: Dynamical Systems (math.DS)
Let $\varphi_0$ be a $C^2$-conservative diffeomorphism of a compact surface $S$ and let $\Lambda_0$ be a mixing horseshoe of $\varphi_0$. Given a smooth real function $f$ defined in $S$ and some diffeomorphism $\varphi$, close to $\varphi_0$, let $\mathcal{L}_{\varphi, f}$ be the Lagrange spectrum associated to the hyperbolic continuation $\Lambda(\varphi)$ of the horseshoe $\Lambda_0$ and $f$. We show that, for generic choices of $\varphi$ and $f$, if $L_{\varphi, f}$ is the map that gives the Hausdorff dimension of the set $\mathcal{L}_{\varphi, f}\cap (-\infty, t)$ for $t\in \mathbb{R}$, then there are at most two points that can be limit of a infinite sequence of discontinuities of $L_{\varphi, f}$.
- [2] arXiv:2403.19003 [pdf, other]
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Title: Finding Birkhoff Averages via Adaptive FilteringSubjects: Dynamical Systems (math.DS); Numerical Analysis (math.NA)
In many applications, one is interested in classifying trajectories of Hamiltonian systems as invariant tori, islands, or chaos. The convergence rate of ergodic Birkhoff averages can be used to categorize these regions, but many iterations of the return map are needed to implement this directly. Recently, it has been shown that a weighted Birkhoff average can be used to accelerate the convergence, resulting in a useful method for categorizing trajectories.
In this paper, we show how a modified version the reduced rank extrapolation method (named Birkhoff RRE) can also be used to find optimal weights for the weighted average with a single linear least-squares solve.Using these, we classify trajectories with fewer iterations of the map than the standard weighted Birkhoff average. Furthermore, for the islands and invariant circles, a subsequent eigenvalue problem gives the number of islands and the rotation number. Using these numbers, we find Fourier parameterizations of invariant circles and islands. We show examples of Birkhoff RRE on the standard map and on magnetic field line dynamics. - [3] arXiv:2403.19274 [pdf, other]
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Title: Extracting coherent sets in aperiodically driven flows from generators of Mather semigroupsSubjects: Dynamical Systems (math.DS); Numerical Analysis (math.NA)
Coherent sets are time-dependent regions in the physical space of nonautonomous flows that exhibit little mixing with their neighborhoods, robustly under small random perturbations of the flow. They thus characterize the global long-term transport behavior of the system. We propose a framework to extract such time-dependent families of coherent sets for nonautonomous systems with an ergodic driving dynamics and (small) Brownian noise in physical space. Our construction involves the assembly and analysis of an operator on functions over the augmented space of the associated skew product that, for each fixed state of the driving, propagates distributions on the corresponding physical-space fibre according to the dynamics. This time-dependent operator has the structure of a semigroup (it is called the Mather semigroup), and we show that a spectral analysis of its generator allows for a trajectory-free computation of coherent families, simultaneously for all states of the driving. Additionally, for quasi-periodically driven torus flows, we propose a tailored Fourier discretization scheme for this generator and demonstrate our method by means of three examples of two-dimensional flows.
- [4] arXiv:2403.19566 [pdf, ps, other]
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Title: Level-2 IFS Thermodynamic Formalism: Gibbs probabilities in the space of probabilities and the push-forward mapSubjects: Dynamical Systems (math.DS); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Probability (math.PR)
We will denote by $\mathcal{M}$ the space of Borel probabilities on the symbolic space $\Omega=\{1,2...,m\}^\mathbb{N}$. $\mathcal{M}$ is equipped Monge-Kantorovich metric. We consider here the push-forward map $\mathfrak{T}:\mathcal{M} \to \mathcal{M}$ as a dynamical system. The space of Borel probabilities on $\mathcal{M}$ is denoted by $\mathfrak{M}$. Given a continuous function $A: \mathcal{M}\to \mathbb{R}$, an {\it a priori} probability $\Pi_0$ on $\mathcal{M}$, and a certain convolution operation acting on pairs of probabilities on $\mathcal{M}$, we define an associated Level-2 IFS Ruelle operator. We show the existence of an eigenfunction and an eigenprobability $\hat{\Pi}\in\mathfrak{M}$ for such an operator. Under a normalization condition for $A$, we show the existence of some $\mathfrak{T}$-invariant probabilities $\hat{\Pi}\in\mathfrak{M}.$ We are able to define the variational entropy of such $\hat{\Pi}$ and a related maximization pressure problem associated to $A$. In some particular examples, we show how to get eigenprobabilities solutions on $\mathfrak{M}$ for the Level-2 Thermodynamic Formalism problem from eigenprobabilities on $\mathcal{M}$ for the classical (Level-1) Thermodynamic Formalism. These examples highlight the fact that our approach is a natural generalization of the classic case.
- [5] arXiv:2403.19567 [pdf, ps, other]
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Title: Poissonian Actions of Polish GroupsComments: 43 pagesSubjects: Dynamical Systems (math.DS); Probability (math.PR)
We define and study Poissonian actions of Polish groups as a framework to Poisson suspensions, characterize them spectrally, and provide a complete characterization of their ergodicity. We further construct 'spatial' Poissonian actions, answering partially a question of Glasner, Tsirelson & Weiss about L\'evy groups. We also construct for every diffeomorphism group an ergodic free spatial probability preserving actions. This constitutes a new class of Polish groups admitting non-essentially countable orbit equivalence relations, obtaining progress on a problem of Kechris.
- [6] arXiv:2403.19621 [pdf, ps, other]
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Title: Holomorphically conjugate polynomial automorphisms of C^2 are polynomially conjugateSubjects: Dynamical Systems (math.DS); Complex Variables (math.CV)
We confirm a conjecture of Friedland and Milnor: if two polynomial automorphisms f and g in Aut(C^2) with dynamical degree >1 are conjugate by some holomorphic diffeomorphism \phi of C^2, then \phi is a polynomial automorphism; thus, f and g are conjugate inside Aut(C^2). We also discuss a number of variations on this result.
Cross-lists for Fri, 29 Mar 24
- [7] arXiv:2403.18858 (cross-list from math.OA) [pdf, ps, other]
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Title: Corrigendum to "A New Uniqueness Theorem for the Tight C*-algebra of an Inverse Semigroup" [C. R. Math. Acad. Sci. Soc. R. Can. 44 (2022), no. 4, 88--112]Comments: 4 pagesJournal-ref: C. R. Math. Rep. Acad. Sci. Canada Vol. 46 (1) 2024, pp. 11-15Subjects: Operator Algebras (math.OA); Dynamical Systems (math.DS)
We correct the proof of Theorem 4.1 from [C. R. Math. Acad. Sci. Soc. R. Can. \textbf{44} (2022), no. 4, 88--112].
- [8] arXiv:2403.19017 (cross-list from math.OC) [pdf, ps, other]
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Title: Pole Placement and Feedback Stabilization for Discrete Linear Ensemble SystemsAuthors: Xudong ChenSubjects: Optimization and Control (math.OC); Dynamical Systems (math.DS)
We consider discrete ensembles of linear, scalar control systems with single-inputs. Assuming that all the individual systems are unstable, we investigate whether there exist linear feedback control laws that can asymptotically stabilize the ensemble system. We provide necessary/sufficient conditions for feasibility of pole placement in the left half plane and for feedback stabilizability of the ensemble systems.
- [9] arXiv:2403.19281 (cross-list from math.DG) [pdf, other]
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Title: On potentials whose level sets are orbitsComments: 21 pages, 1 figureSubjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Dynamical Systems (math.DS)
A level orbit of a mechanical Hamiltonian system is a solution of Newton equation that is contained in a level set of the potential energy. In 2003, Mark Levi asked for a characterization of the smooth potential energy functions on the plane with the property that any point on the plane lies on a level orbit; we call such functions Levi potentials. The basic examples are the radial monotone increasing smooth functions. In this paper we show that any Levi potential that is analytic or has totally path-disconnected critical set must be radial. Nevertheless, we show that every compact convex subset of the plane is the critical set of a Levi potential. A crucial observation for these theorems is that, outside the critical set, the family of level sets of a Levi potential forms a solution of the inverse curvature flow.
- [10] arXiv:2403.19393 (cross-list from cond-mat.soft) [pdf, ps, other]
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Title: Memory signatures in path curvature of self-avoidant model particles are revealed by time delayed self mutual informationSubjects: Soft Condensed Matter (cond-mat.soft); Dynamical Systems (math.DS)
Emergent behavior in active systems is a complex byproduct of local, often pairwise, interactions. One such interaction is self-avoidance, which experimentally can arise as a response to self-generated environmental signals; such experiments have inspired non-Markovian mathematical models. In previous work, we set out to find ``hallmarks of self-avoidant memory" in a particle model for environmentally responsive swimming droplets. In our analysis, we found that transient self-trapping was a spatial hallmark of the particle's self-avoidant memory response. The self-trapping results from the combined effects of behaviors at multiple scales: random reorientations, which occur on the diffusion scale, and the self-avoidant memory response, which occurs on the ballistic (and longer) timescales. In this work, we use the path curvature as it encodes the self-trapping response to estimate an ``effective memory lifetime" by analyzing the decay of its time-delayed mutual information and subsequently determining the longevity of significant nonlinear correlations. This effective memory lifetime (EML) is longer in systems where the curvature is a product of both self-avoidance and random reorientations as compared to systems without self-avoidance.
- [11] arXiv:2403.19453 (cross-list from math.NT) [pdf, ps, other]
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Title: The submodularity of the covolume function in global function fieldsAuthors: Gukyeong BangComments: 18 pagesSubjects: Number Theory (math.NT); Dynamical Systems (math.DS)
In this paper, we study the submodularity of the covolume function in global function fields. The submodular property is often needed in the study of homogeneous dynamics, especially to define a Margulis function. We proved that the covolume function is submodular when the class group of the global function field is trivial.
- [12] arXiv:2403.19550 (cross-list from math.SP) [pdf, ps, other]
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Title: Spectral gap for surfaces of infinite volume with negative curvatureAuthors: Zhongkai TaoSubjects: Spectral Theory (math.SP); Analysis of PDEs (math.AP); Dynamical Systems (math.DS)
We prove that the imaginary parts of scattering resonances for negatively curved asymptotically hyperbolic surfaces are uniformly bounded away from zero and provide a resolvent bound in the resulting resonance-free strip. This provides an essential spectral gap without the pressure condition. This is done by adapting the methods of [arXiv:1004.3361], [arXiv:1012.4391] and [arXiv:2201.08259] and answers a question posed in [arXiv:1504.06589].
- [13] arXiv:2403.19582 (cross-list from math.PR) [pdf, ps, other]
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Title: Generalized law of iterated logarithm for the Lorentz gas with infinite horizonSubjects: Probability (math.PR); Dynamical Systems (math.DS)
We obtain a generalized law of iterated logarithm for a class of dependent processes with superdiffusive behaviour. Our results apply in particular to the Lorentz gas with infinite horizon.
- [14] arXiv:2403.19642 (cross-list from math.NT) [pdf, ps, other]
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Title: Square patterns in dynamical orbitsComments: 17 pagesSubjects: Number Theory (math.NT); Dynamical Systems (math.DS)
Let $q$ be an odd prime power. Let $f\in \mathbb{F}_q[x]$ be a polynomial having degree at least $2$, $a\in \mathbb{F}_q$, and denote by $f^n$ the $n$-th iteration of $f$. Let $\chi$ be the quadratic character of $\mathbb{F}_q$, and $\mathcal{O}_f(a)$ the forward orbit of $a$ under iteration by $f$. Suppose that the sequence $(\chi(f^n(a)))_{n\geq 1}$ is periodic, and $m$ is its period. Assuming a mild and generic condition on $f$, we show that, up to a constant, $m$ can be bounded from below by $|\mathcal{O}_f(a)|/q^\frac{2\log_{2}(d)+1}{2\log_2(d)+2}$. More informally, we prove that the period of the appearance of squares in an orbit of an element provides an upper bound for the size of the orbit itself. Using a similar method, we can also prove that, up to a constant, we cannot have more than $q^\frac{2\log_2(d)+1}{2\log_2(d)+2}$ consecutive squares or non-squares in the forward orbit of $a$. In addition, we provide a classification of all polynomials for which our generic condition does not hold.
Replacements for Fri, 29 Mar 24
- [15] arXiv:2207.08085 (replaced) [pdf, ps, other]
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Title: Quasi-compactness of transfer operators for topological Markov shifts with holesAuthors: Haruyoshi TanakaComments: To appear in Discrete and Continuous Dynamical SystemsSubjects: Dynamical Systems (math.DS); Probability (math.PR)
- [16] arXiv:2309.14646 (replaced) [pdf, ps, other]
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Title: Concentration of dimension in extremal points of left-half lines in the Lagrange spectrumComments: 24 pagesSubjects: Dynamical Systems (math.DS)
- [17] arXiv:2310.18931 (replaced) [pdf, ps, other]
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Title: Equilibria decomposition-based comparison of reaction networks of Wnt signalingSubjects: Dynamical Systems (math.DS)
- [18] arXiv:2401.00654 (replaced) [pdf, ps, other]
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Title: Global rigidity for some partially hyperbolic abelian actions with 1-dimensional centerAuthors: Sven SandfeldtComments: 61 pages, 1 figure. Simplified proof of Theorem 1.3. Removed an assumption from Theorem ASubjects: Dynamical Systems (math.DS)
- [19] arXiv:2403.14585 (replaced) [pdf, ps, other]
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Title: Slow decay of correlations for generic mixing automorphismsAuthors: Valery V. RyzhikovComments: in Russian languageSubjects: Dynamical Systems (math.DS)
- [20] arXiv:2210.16446 (replaced) [pdf, ps, other]
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Title: Measurable Imbeddings, Free Products, and Graph ProductsAuthors: Özkan DemirComments: v2, submitted to Confluentes MathematiciSubjects: Group Theory (math.GR); Dynamical Systems (math.DS)
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