scholarly journals Complexity of injective piecewise contracting interval maps

2018 ◽  
Vol 40 (1) ◽  
pp. 64-88 ◽  
Author(s):  
E. CATSIGERAS ◽  
P. GUIRAUD ◽  
A. MEYRONEINC
Keyword(s):  
Growth Rate ◽  
Cantor Set ◽  
Interval Maps ◽  
Piecewise Affine ◽  
Complexity Function ◽  
Asymptotic Dynamics

We study the complexity of the itineraries of injective piecewise contracting maps on the interval. We prove that for any such map the complexity function of any itinerary is eventually affine. We also prove that the growth rate of the complexity is bounded from above by the number, $N-1$, of discontinuities of the map. To show that this bound is optimal, we construct piecewise affine contracting maps whose itineraries all have the complexity $(N-1)n+1$. In these examples, the asymptotic dynamics take place in a minimal Cantor set containing all the discontinuities.

scholarly journals Subshifts on Infinite Alphabets and Their Entropy

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1293
Author(s):  
Sharwin Rezagholi
Keyword(s):  
Growth Rate ◽  
Markov Chains ◽  
Topological Entropy ◽  
Spectral Radius ◽  
Symbolic Dynamics ◽  
Infinite Graphs ◽  
Complexity Function ◽  
Countably Infinite

We analyze symbolic dynamics to infinite alphabets by endowing the alphabet with the cofinite topology. The topological entropy is shown to be equal to the supremum of the growth rate of the complexity function with respect to finite subalphabets. For the case of topological Markov chains induced by countably infinite graphs, our approach yields the same entropy as the approach of Gurevich We give formulae for the entropy of countable topological Markov chains in terms of the spectral radius in l2.

scholarly journals Cut and project sets with polytopal window I: Complexity

2020 ◽  
pp. 1-33
Author(s):  
HENNA KOIVUSALO ◽  
JAMES J. WALTON
Keyword(s):  
Growth Rate ◽  
Natural Condition ◽  
Algebraic Topology ◽  
Complexity Function

We calculate the growth rate of the complexity function for polytopal cut and project sets. This generalizes work of Julien where the almost canonical condition is assumed. The analysis of polytopal cut and project sets has often relied on being able to replace acceptance domains of patterns by so-called cut regions. Our results correct mistakes in the literature where these two notions are incorrectly identified. One may only relate acceptance domains and cut regions when additional conditions on the cut and project set hold. We find a natural condition, called the quasicanonical condition, guaranteeing this property and demonstrate by counterexample that the almost canonical condition is not sufficient for this. We also discuss the relevance of this condition for the current techniques used to study the algebraic topology of polytopal cut and project sets.

scholarly journals On piecewise affine interval maps with countably many laps

2011 ◽  
Vol 31 (3) ◽  
pp. 753-762 ◽  
Author(s):  
Jozef Bobok ◽  
◽  
Martin Soukenka
Keyword(s):  
Interval Maps ◽  
Piecewise Affine

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.

scholarly journals Almost minimal systems and periodicity in hyperspaces

2017 ◽  
Vol 38 (6) ◽  
pp. 2158-2179 ◽  
Author(s):  
LEOBARDO FERNÁNDEZ ◽  
CHRIS GOOD ◽  
MATE PULJIZ
Keyword(s):  
Metric Space ◽  
Necessary Conditions ◽  
Periodic Points ◽  
Cantor Set ◽  
Full Description ◽  
Compact Metric Space ◽  
Symmetric Products ◽  
Interval Maps ◽  
Admissible Sets ◽  
Induced Maps

Given a self-map of a compact metric space $X$, we study periodic points of the map induced on the hyperspace of closed non-empty subsets of $X$. We give some necessary conditions on admissible sets of periods for these maps. Seemingly unrelated to this, we construct an almost totally minimal homeomorphism of the Cantor set. We also apply our theory to give a full description of admissible period sets for induced maps of the interval maps. The description of admissible periods is also given for maps induced on symmetric products.

The threshold energy for atom displacement in fcc metals

1990 ◽  
Vol 48 (4) ◽  
pp. 530-531
Author(s):  
Wilfried Sigle ◽  
Matthias Hohenstein ◽  
Alfred Seeger
Keyword(s):  
Growth Rate ◽  
Elevated Temperatures ◽  
Threshold Energy ◽  
Dislocation Loops ◽  
Absolute Minimum ◽  
Voltage Electron ◽  
Atom Displacement ◽  
Electron Microscopes ◽  
Fcc Metal

Prolonged electron irradiation of metals at elevated temperatures usually leads to the formation of large interstitial-type dislocation loops. The growth rate of the loops is proportional to the total cross-section for atom displacement,which is implicitly connected with the threshold energy for atom displacement, Ed . Thus, by measuring the growth rate as a function of the electron energy and the orientation of the specimen with respect to the electron beam, the anisotropy of Ed can be determined rather precisely. We have performed such experiments in situ in high-voltage electron microscopes on Ag and Au at 473K as a function of the orientation and on Au as a function of temperature at several fixed orientations.Whereas in Ag minima of Ed are found close to <100>,<110>, and <210> (13-18eV), (Fig.1) atom displacement in Au requires least energy along <100>(15-19eV) (Fig.2). Au is thus the first fcc metal in which the absolute minimum of the threshold energy has been established not to lie in or close to the <110> direction.

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