Dimensions of stable sets and scrambled sets in positive finite entropy systems

2011 ◽  
Vol 32 (2) ◽  
pp. 599-628 ◽  
Author(s):  
CHUN FANG ◽  
WEN HUANG ◽  
YINGFEI YI ◽  
PENGFEI ZHANG
Keyword(s):  
Dynamical System ◽  
Riemannian Manifold ◽  
Continuum Hypothesis ◽  
Large Set ◽  
Metric Entropy ◽  
Positive Entropy ◽  
Stable Sets ◽  
Stable And Unstable Sets ◽  
The Continuum ◽  
Hyperbolic Points

AbstractWe study the dimensions of stable sets and scrambled sets of a dynamical system with positive finite entropy. We show that there is a measure-theoretically ‘large’ set containing points whose sets of ‘hyperbolic points’ (i.e. points lying in the intersections of the closures of the stable and unstable sets) admit positive Bowen dimension entropies; under the continuum hypothesis, this set also contains a scrambled set with positive Bowen dimension entropies. For several kinds of specific invertible dynamical systems, the lower bounds of the Hausdorff dimension of these sets are estimated. In particular, for a diffeomorphism on a smooth Riemannian manifold with positive entropy, such a lower bound is given in terms of the metric entropy and Lyapunov exponent.

scholarly journals Several results on compact metrizable spaces in $$\mathbf {ZF}$$

2021 ◽  
Author(s):  
Kyriakos Keremedis ◽  
Eleftherios Tachtsis ◽  
Eliza Wajch
Keyword(s):  
Continuum Hypothesis ◽  
Compact Spaces ◽  
Power Set ◽  
Compact Metrizable Space ◽  
Permutation Models ◽  
Independence Results ◽  
The Axiom Of Choice ◽  
Metrizable Spaces ◽  
Transfer Theorems ◽  
The Continuum

AbstractIn the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investigated. Some of the statements are provable in $$\mathbf {ZF}$$ ZF , some are shown to be independent of $$\mathbf {ZF}$$ ZF . For independence results, distinct models of $$\mathbf {ZF}$$ ZF and permutation models of $$\mathbf {ZFA}$$ ZFA with transfer theorems of Pincus are applied. New symmetric models of $$\mathbf {ZF}$$ ZF are constructed in each of which the power set of $$\mathbb {R}$$ R is well-orderable, the Continuum Hypothesis is satisfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube $$[0, 1]^{\mathbb {R}}$$ [ 0 , 1 ] R .

scholarly journals Static stability predicts the continuum of interleg coordination patterns in Drosophila

2018 ◽  
Author(s):  
Nicholas S. Szczecinski ◽  
Till Bockemühl ◽  
Alexander S. Chockley ◽  
Ansgar Büschges
Keyword(s):  
Experimental Data ◽  
Static Stability ◽  
Fruit Flies ◽  
Coordination Pattern ◽  
Large Set ◽  
Summary Statement ◽  
Walking Speeds ◽  
Coordination Patterns ◽  
Phase Relationships ◽  
The Continuum

AbstractDuring walking, insects must coordinate the movements of their six legs for efficient locomotion. This interleg coordination is speed-dependent; fast walking in insects is associated with tripod coordination patterns, while slow walking is associated with more variable, tetrapod-like patterns. To date, however, there has been no comprehensive explanation as to why these speed-dependent shifts in interleg coordination should occur in insects. Tripod coordination would be sufficient at low walking speeds. The fact that insects use a different interleg coordination pattern at lower speeds suggests that it is more optimal or advantageous at these speeds. Furthermore, previous studies focused on discrete tripod and tetrapod coordination patterns. Experimental data, however, suggest that changes observed in interleg coordination are part of a speed-dependent spectrum. Here, we explore these issues in relation to static stability as an important aspect of interleg coordination in Drosophila. We created a model that uses basic experimentally measured parameters in fruit flies to find the interleg phase relationships that maximize stability for a given walking speed. Based on this measure, the model predicted a continuum of interleg coordination patterns spanning the complete range of walking speeds. Furthermore, for low walking speeds the model predicted tetrapod-like patterns to be most stable, while at high walking speeds tripod coordination emerged as most optimal. Finally, we validated the basic assumption of a continuum of interleg coordination patterns in a large set of experimental data from walking fruit flies and compared these data with the model-based predictions.Summary statementA simple stability-based modelling approach can explain why walking insects use different leg coordination patterns in a speed-dependent way.

Some remarks on the partition calculus of ordinals

1999 ◽  
Vol 64 (2) ◽  
pp. 436-442 ◽  
Author(s):  
Péter Komjáth
Keyword(s):  
Regular Cardinal ◽  
Continuum Hypothesis ◽  
Alternative Proof ◽  
Partition Relation ◽  
Negative Relation ◽  
Partition Calculus ◽  
Counter Example ◽  
The Continuum

One of the early partition relation theorems which include ordinals was the observation of Erdös and Rado [7] that if κ = cf(κ) > ω then the Dushnik–Miller theorem can be sharpened to κ→(κ, ω + 1)2. The question on the possible further extension of this result was answered by Hajnal who in [8] proved that the continuum hypothesis implies ω1 ↛ (ω1, ω + 2)2. He actually proved the stronger result ω1 ↛ (ω: 2))2. The consistency of the relation κ↛(κ, (ω: 2))2 was later extensively studied. Baumgartner [1] proved it for every κ which is the successor of a regular cardinal. Laver [9] showed that if κ is Mahlo there is a forcing notion which adds a witness for κ↛ (κ, (ω: 2))2 and preserves Mahloness, ω-Mahloness of κ, etc. We notice in connection with these results that λ→(λ, (ω: 2))2 holds if λ is singular, in fact λ→(λ, (μ: n))2 for n < ω, μ < λ (Theorem 4).In [11] Todorčević proved that if cf(λ) > ω then a ccc forcing can add a counter-example to λ→(λ, ω + 2)2. We give an alternative proof of this (Theorem 5) and extend it to larger cardinals: if GCH holds, cf (λ) > κ = cf (κ) then < κ-closed, κ+-c.c. forcing adds a counter-example to λ→(λ, κ + 2)2 (Theorem 6).Erdös and Hajnal remarked in their problem paper [5] that Galvin had proved ω2→(ω1, ω + 2)2 and he had also asked if ω2→(ω1, ω + 3)2 is true. We show in Theorem 1 that the negative relation is consistent.

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