scholarly journals Zero-dimensional isomorphic dynamical models

2018 ◽  
Vol 40 (8) ◽  
pp. 2116-2130
Author(s):  
TOMASZ DOWNAROWICZ ◽  
LEI JIN ◽  
WOLFGANG LUSKY ◽  
YIXIAO QIAO
Keyword(s):  
Dynamical System ◽  
Probability Measure ◽  
Extreme Points ◽  
Dimensional System ◽  
Choquet Simplex ◽  
Countable Union ◽  
Topological Dynamical System ◽  
Dynamical Models ◽  
Bauer Simplex

By an assignment we mean a mapping from a Choquet simplex $K$ to probability measure-preserving systems obeying some natural restrictions. We prove that if $\unicode[STIX]{x1D6F7}$ is an aperiodic assignment on a Choquet simplex $K$ such that the set of extreme points $\mathsf{ex}K$ is a countable union $\bigcup _{n}E_{n}$, where each set $E_{n}$ is compact, zero-dimensional and the restriction of $\unicode[STIX]{x1D6F7}$ to the Bauer simplex $K_{n}$ spanned by $E_{n}$ can be ‘embedded’ in some topological dynamical system, then $\unicode[STIX]{x1D6F7}$ can be ‘realized’ in a zero-dimensional system.

A minimal distal map on the torus with sub-exponential measure complexity

2018 ◽  
Vol 40 (4) ◽  
pp. 953-974 ◽  
Author(s):  
WEN HUANG ◽  
LEIYE XU ◽  
XIANGDONG YE
Keyword(s):  
Dynamical System ◽  
Probability Measure ◽  
Borel Probability Measure ◽  
Skew Product ◽  
Topological Dynamical System ◽  
Borel Probability ◽  
Product Map

In this paper the notion of sub-exponential measure complexity for an invariant Borel probability measure of a topological dynamical system is introduced. Then a minimal distal skew product map on the torus with sub-exponential measure complexity is constructed.

scholarly journals Characteristic measures of symbolic dynamical systems

2021 ◽  
pp. 1-7
Author(s):  
JOSHUA FRISCH ◽  
OMER TAMUZ
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Automorphism Group ◽  
Probability Measure ◽  
Automorphism Groups ◽  
Topological Dynamical System ◽  
Zero Entropy ◽  
Characteristic Measure

Abstract A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

scholarly journals Inverse-square law between time and amplitude for crossing tipping thresholds

2019 ◽  
Vol 475 (2222) ◽  
pp. 20180504 ◽  
Author(s):  
Paul Ritchie ◽  
Özkan Karabacak ◽  
Jan Sieber
Keyword(s):  
Dynamical System ◽  
Feedback Loop ◽  
Tipping Point ◽  
Dimensional System ◽  
Stochastic Forcing ◽  
Inverse Square Law ◽  
System A ◽  
Asymptotic Expressions ◽  
Higher Dimensional ◽  
Random Disturbances

A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive feedback loop. We study what happens if one turns around after one has crossed the threshold. We derive a simple criterion that relates how far the parameter exceeds the tipping threshold maximally and how long the parameter stays above the threshold to avoid tipping in an inverse-square law to observable properties of the dynamical system near the fold. For the case when the dynamical system is subject to stochastic forcing we give an approximation to the probability of tipping if a parameter changing in time reverses near the tipping point. The derived approximations are valid if the parameter change in time is sufficiently slow. We demonstrate for a higher-dimensional system, a model for the Indian summer monsoon, how numerically observed escape from the equilibrium converge to our asymptotic expressions. The inverse-square law between peak of the parameter forcing and the time the parameter spends above a given threshold is also visible in the level curves of equal probability when the system is subject to random disturbances.

scholarly journals On quantum theory in terms of white noise

1977 ◽  
Vol 68 ◽  
pp. 21-34 ◽  
Author(s):  
T. Hida ◽  
L. Streit
Keyword(s):  
Quantum Theory ◽  
Probability Measure ◽  
Heat Equation ◽  
Correlation Functions ◽  
Time Parameter ◽  
Imaginary Time ◽  
Dynamical Models ◽  
Quantum Dynamical ◽  
Reverse Transition ◽  
Generalized Heat Equation

It has often been pointed out that a much more manageable structure is obtained from quantum theory if the time parameter t is chosen imaginary instead of real. Under a replacement of t by i·t the Schrödinger equation turns into a generalized heat equation, time ordered correlation functions transform into the moments of a probability measure, etc. More recently this observation has become extremely important for the construction of quantum dynamical models, where criteria were developed by E. Nelson, by K. Osterwalder and R. Schrader and others [8] which would permit the reverse transition to real time after one has constructed an imaginary time (“Euclidean”) model.

scholarly journals Contractibility of a persistence map preimage

2020 ◽  
Vol 4 (4) ◽  
pp. 509-523
Author(s):  
Jacek Cyranka ◽  
Konstantin Mischaikow ◽  
Charles Weibel
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Time Series ◽  
Data Driven ◽  
Dimensional System ◽  
Multiple Time ◽  
Multiple Time Series ◽  
Topological Features ◽  
Persistence Diagram ◽  
Persistence Diagrams

Abstract This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of snapshots of solutions, what conclusions can be drawn about solutions of the original dynamical system? We address this challenge in the context of an N dimensional system of ordinary differential equation defined in $${\mathbb {R}}^N$$ R N . To each point in $${\mathbb {R}}^N$$ R N (e.g. an initial condition) we associate a persistence diagram. The main result of this paper is that under this association the preimage of every persistence diagram is contractible. As an application we provide conditions under which multiple time series of persistence diagrams can be used to conclude the existence of a fixed point of the differential equation that generates the time series.

A Remark on Invariant Scrambled Sets

2012 ◽  
Vol 204-208 ◽  
pp. 4776-4779
Author(s):  
Lin Huang ◽  
Huo Yun Wang ◽  
Hong Ying Wu
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Periodic Point ◽  
Metric Space ◽  
Cantor Sets ◽  
Countable Union ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Isolated Points ◽  
Weakly Mixing

By a dynamical system we mean a compact metric space together with a continuous map . This article is devoted to study of invariant scrambled sets. A dynamical system is a periodically adsorbing system if there exists a fixed point and a periodic point such that and are dense in . We show that every topological weakly mixing and periodically adsorbing system contains an invariant and dense Mycielski scrambled set for some , where has no isolated points. A subset is a Myceilski set if it is a countable union of Cantor sets.

scholarly journals When is a dynamical system mean sensitive?

2017 ◽  
Vol 39 (06) ◽  
pp. 1608-1636 ◽  
Author(s):  
FELIPE GARCÍA-RAMOS ◽  
JIE LI ◽  
RUIFENG ZHANG
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
The Other ◽  
Positive Entropy ◽  
Topological Dynamical System ◽  
Other Hand ◽  
Open Question ◽  
Uniquely Ergodic ◽  
Mean Sensitivity ◽  
Transitive System

This article is devoted to studying which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things, we show that every uniquely ergodic, mixing system with positive entropy is mean sensitive. On the other hand, we provide an example of a transitive system which is cofinitely sensitive or Devaney chaotic with positive entropy but fails to be mean sensitive. As applications of our theory and examples, we negatively answer an open question regarding equicontinuity/sensitivity dichotomies raised by Tu, we introduce and present results of locally mean equicontinuous systems and we show that mean sensitivity of the induced hyperspace does not imply that of the phase space.

When are all closed subsets recurrent?

2016 ◽  
Vol 37 (7) ◽  
pp. 2223-2254 ◽  
Author(s):  
JIE LI ◽  
PIOTR OPROCHA ◽  
XIANGDONG YE ◽  
RUIFENG ZHANG
Keyword(s):  
Dynamical System ◽  
Periodic Points ◽  
Topological Dynamical System ◽  
Open Questions ◽  
Weakly Mixing ◽  
Uniformly Rigid ◽  
Compact Subsets

In the paper we study relations of rigidity, equicontinuity and pointwise recurrence between an invertible topological dynamical system (t.d.s.) $(X,T)$ and the t.d.s. $(K(X),T_{K})$ induced on the hyperspace $K(X)$ of all compact subsets of $X$, and provide some characterizations. Among other examples, we construct a minimal, non-equicontinuous, distal and uniformly rigid t.d.s. and a weakly mixing t.d.s. which induces dense periodic points on the hyperspace $K(X)$ but itself does not have dense distal points, solving in that way a few open questions from earlier articles by Dong, and Li, Yan and Ye.

GALTON'S QUINCUNX: RANDOM WALK OR CHAOS?

2007 ◽  
Vol 17 (12) ◽  
pp. 4463-4469 ◽  
Author(s):  
KEVIN JUDD
Keyword(s):  
Dynamical System ◽  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Mechanical Device ◽  
Dynamical Models ◽  
The Central Limit Theorem ◽  
Random Events ◽  
Low Dimensional ◽  
Deterministic Dynamical System

In 1873 Francis Galton had constructed a simple mechanical device where a ball is dropped vertically through a harrow of pins that deflect the ball sideways as it falls. Galton called the device a quincunx, although today it is usually referred to as a Galton board. Statisticians often employ (conceptually, if not physically) the quincunx to illustrate random walks and the central limit theorem. In particular, how a Binomial or Gaussian distribution results from the accumulation of independent random events, that is, the collisions in the case of the quincunx. But how valid is the assumption of "independent random events" made by Galton and countless subsequent statisticians? This paper presents evidence that this assumption is almost certainly not valid and that the quincunx has the richer, more predictable qualities of a low-dimensional deterministic dynamical system. To put this observation into a wider context, the result illustrates that statistical modeling assumptions can obscure more informative dynamics. When such dynamical models are employed they will yield better predictions and forecasts.

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