scholarly journals A spectral decomposition of the attractor of piecewise-contracting maps of the interval

2020 ◽  
pp. 1-21
Author(s):  
ALFREDO CALDERON ◽  
ELEONORA CATSIGERAS ◽  
PIERRE GUIRAUD
Keyword(s):  
Periodic Orbit ◽  
Finite Number ◽  
Spectral Decomposition ◽  
Discontinuity Point ◽  
Local Extremum ◽  
Cantor Set ◽  
Compact Interval ◽  
Limit Set ◽  
Local Extrema ◽  
Asymptotic Dynamics

We study the asymptotic dynamics of piecewise-contracting maps defined on a compact interval. For maps that are not necessarily injective, but have a finite number of local extrema and discontinuity points, we prove the existence of a decomposition of the support of the asymptotic dynamics into a finite number of minimal components. Each component is either a periodic orbit or a minimal Cantor set and such that the $\unicode[STIX]{x1D714}$ -limit set of (almost) every point in the interval is exactly one of these components. Moreover, we show that each component is the $\unicode[STIX]{x1D714}$ -limit set, or the closure of the orbit, of a one-sided limit of the map at a discontinuity point or at a local extremum.

On the finiteness of attractors for piecewise maps of the interval

2017 ◽  
Vol 39 (7) ◽  
pp. 1784-1804
Author(s):  
P. BRANDÃO ◽  
J. PALIS ◽  
V. PINHEIRO
Keyword(s):  
Periodic Orbit ◽  
Finite Number ◽  
Critical Points ◽  
Limit Set ◽  
Periodic Attractors ◽  
Flat Maps ◽  
Omega Limit Set

We consider piecewise $C^{2}$ non-flat maps of the interval and show that, for Lebesgue almost every point, its omega-limit set is either a periodic orbit, a cycle of intervals or the closure of the orbits of a subset of the critical points. In particular, every piecewise $C^{2}$ non-flat map of the interval displays only a finite number of non-periodic attractors.

scholarly journals On Limit Sets of Monotone Maps on Dendroids

2020 ◽  
Vol 5 (2) ◽  
pp. 311-316
Author(s):  
E.N. Makhrova
Keyword(s):  
Periodic Orbit ◽  
Periodic Points ◽  
Cantor Set ◽  
List Type ◽  
Limit Set ◽  
Limit Sets ◽  
Monotone Map ◽  
Monotone Maps

AbstractLet X be a dendrite, f : X → X be a monotone map. In the papers by I. Naghmouchi (2011, 2012) it is shown that ω-limit set ω(x, f ) of any point x ∈ X has the next properties: (1)\omega (x,f) \subseteq \overline {Per(f)} , where Per( f ) is the set of periodic points of f ;(2)ω(x, f ) is either a periodic orbit or a minimal Cantor set.In the paper by E. Makhrova, K. Vaniukova (2016 ) it is proved that (3)\Omega (f) = \overline {Per(f)} , where Ω( f ) is the set of non-wandering points of f.The aim of this note is to show that the above results (1) – (3) do not hold for monotone maps on dendroids.

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

Calculation of spectral determinants

1994 ◽  
Vol 447 (1930) ◽  
pp. 413-437 ◽  
Keyword(s):  
Periodic Orbit ◽  
Finite Number ◽  
Dirichlet Series ◽  
Energy Levels ◽  
Euler Product ◽  
Quantum Energy ◽  
Type Formula

A method for regularizing spectral determinants is developed which facilitates their computation from a finite number of eigenvalues. This is used to calcu­late the determinant ∆ for the hyperbola billiard over a range which includes 46 quantum energy levels. The result is compared with semiclassical periodic orbit evaluations of ∆ using the Dirichlet series, Euler product, and a Riemann-Siegel-type formula. It is found that the Riemann-Siegel-type expansion, which uses the least number of orbits, gives the closest approximation. This provides explicit numerical support for recent conjectures concerning the analytic proper­ties of semiclassical formulae, and in particular for the existence of resummation relations connecting long and short pseudo-orbits.

FOX-ARTIN ARC AND CONSTRUCTING KLEINIAN GROUPS ACTING ON S3 WITH LIMIT SET A WILD CANTOR SET

1995 ◽  
Vol 06 (01) ◽  
pp. 19-32 ◽  
Author(s):  
NIKOLAY GUSEVSKII ◽  
HELEN KLIMENKO
Keyword(s):  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
Limit Set ◽  
Limit Sets ◽  
Geometrically Finite ◽  
Wild Cantor Set ◽  
Wild Cantor Sets

We construct purely loxodromic, geometrically finite, free Kleinian groups acting on S3 whose limit sets are wild Cantor sets. Our construction is closely related to the construction of the wild Fox–Artin arc.

scholarly journals Lyapunov optimization for non-generic one-dimensional expanding Markov maps

2019 ◽  
Vol 40 (9) ◽  
pp. 2571-2592 ◽  
Author(s):  
MAO SHINODA ◽  
HIROKI TAKAHASI
Keyword(s):  
Periodic Orbit ◽  
Finite Number ◽  
Dense Subset ◽  
Equilibrium States ◽  
Positive Entropy ◽  
Hölder Continuous ◽  
Perturbation Theorem ◽  
One Dimensional ◽  
Lyapunov Optimization ◽  
Shift Map

For a non-generic, yet dense subset of$C^{1}$expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new$C^{1}$perturbation theorem which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

Adding machines and wild attractors

1997 ◽  
Vol 17 (6) ◽  
pp. 1267-1287 ◽  
Author(s):  
HENK BRUIN ◽  
GERHARD KELLER ◽  
MATTHIAS ST. PIERRE
Keyword(s):  
Dynamical System ◽  
Critical Point ◽  
Cantor Set ◽  
Limit Set ◽  
Unimodal Maps ◽  
Irrational Rotations ◽  
Circle Rotation ◽  
Omega Limit Set ◽  
Adding Machines

We investigate the dynamics of unimodal maps $f$ of the interval restricted to the omega limit set $X$ of the critical point for cases where $X$ is a Cantor set. In particular, many cases where $X$ is a measure attractor of $f$ are included. We give two classes of examples of such maps, both generalizing unimodal Fibonacci maps [LM, BKNS]. In all cases $f_{|X}$ is a continuous factor of a generalized odometer (an adding machine-like dynamical system), and at the same time $f_{|X}$ factors onto an irrational circle rotation. In some of the examples we obtain irrational rotations on more complicated groups as factors.

Generalized Specification Property and Distributional Chaos

2003 ◽  
Vol 13 (07) ◽  
pp. 1683-1694 ◽  
Author(s):  
F. Balibrea ◽  
B. Schweizer ◽  
A. Sklar ◽  
J. Smítal
Keyword(s):  
Compact Interval ◽  
Limit Set ◽  
Distributional Chaos ◽  
Specification Property ◽  
Continuous Map ◽  
Basic Set ◽  
Made In

Let f be a continuous map from a compact interval into itself. Continuing the work begun by Schweizer and Smítal [1994], we prove that the restriction of f to any basic set (i.e. any nonsolenoidal, infinite, maximal ω-limit set) satisfies a generalization of the specification property. We apply this generalization to establish several conjectures made in the abovementioned paper, e.g. the fact that distributional chaos is stable.

scholarly journals On holomorphic flows on Stein surfaces: Transversality, dicriticalness and stability

2014 ◽  
Vol 25 (10) ◽  
pp. 1450093
Author(s):  
T. Ito ◽  
B. Scárdua ◽  
Y. Yamagishi
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Finite Number ◽  
Sufficient Conditions ◽  
First Integral ◽  
Holomorphic Vector Field ◽  
Necessary And Sufficient ◽  
Holonomy Map

We study the classification of the pairs (N, X) where N is a Stein surface and X is a ℂ-complete holomorphic vector field with isolated singularities on N. We describe the role of transverse sections in the classification of X and give necessary and sufficient conditions on X in order to have N biholomorphic to ℂ2. As a sample of our results, we prove that N is biholomorphic to ℂ2 if H2(N, ℤ) = 0, X has a finite number of singularities and exhibits a singularity with three separatrices or, equivalently, a singularity with first jet of the form [Formula: see text] where λ1/λ2 ∈ ℚ+. We also study flows with many periodic orbits (i.e. orbits diffeomorphic to ℂ*), in a sense we will make clear, proving they admit a meromorphic first integral or they exhibit some special periodic orbit, whose holonomy map is a non-resonant nonlinearizable diffeomorphism map.

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