scholarly journals Equidistribution problems in complex dynamics of higher dimension

2017 ◽  
Vol 28 (07) ◽  
pp. 1750057 ◽  
Author(s):  
Tien-Cuong Dinh ◽  
Nessim Sibony
Keyword(s):  
Dynamical System ◽  
Complex Dynamics ◽  
Fundamental Problem ◽  
Arbitrary Dimension ◽  
Deep Understanding ◽  
Periodic Points ◽  
Higher Dimension ◽  
Open Problems ◽  
New Techniques ◽  
Ergodic Properties

Equidistribution of the orbits of points, subvarieties or of periodic points in complex dynamics is a fundamental problem. It is often related to strong ergodic properties of the dynamical system and to a deep understanding of analytic cycles, or more generally positive closed currents, of arbitrary dimension and degree. The later topic includes the study of the potentials and super-potentials of positive closed currents, their intersection with or without dimension excess. In this paper, we will survey some results and tools developed during the last two decades. Related concepts, new techniques and open problems will be presented.

SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1351-1355
Author(s):  
VLADIMIR FEDORENKO
Keyword(s):  
Topological Entropy ◽  
Complex Dynamics ◽  
Periodic Points ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
Circle Maps ◽  
Interval Maps ◽  
Chain Recurrent Set ◽  
Recurrent Set

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

scholarly journals Almost all $S$-integer dynamical systems have many periodic points

1998 ◽  
Vol 18 (2) ◽  
pp. 471-486 ◽  
Author(s):  
T. B. WARD
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Zeta Function ◽  
Periodic Points ◽  
Radius Of Convergence ◽  
Dynamical Zeta Function ◽  
Hyperbolic Dynamical Systems ◽  
Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

scholarly journals Generalizations of the Poincaré Birkhoff fixed point theorem

1977 ◽  
Vol 17 (3) ◽  
pp. 375-389 ◽  
Author(s):  
Walter D. Neumann
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Periodic Points ◽  
Open Problems ◽  
Number Of Components ◽  
Birkhoff Theorem

It is shown how George D. Birkhoff's proof of the Poincaré Birkhoff theorem can be modified using ideas of H. Poincaré to give a rather precise lower bound on the number of components of the set of periodic points of the annulus. Some open problems related to this theorem are discussed.

scholarly journals Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

2018 ◽  
Vol 28 (04) ◽  
pp. 1830011
Author(s):  
Mio Kobayashi ◽  
Tetsuya Yoshinaga
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Chaotic Behavior ◽  
Periodic Points ◽  
Two Dimensional ◽  
Stable Fixed Point ◽  
Bifurcation Structure ◽  
Unstable Fixed Point ◽  
Gaussian Map ◽  
Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

scholarly journals Complex Canard Explosion in a Fractional-Order FitzHugh–Nagumo Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950111 ◽  
Author(s):  
Mohammed-Salah Abdelouahab ◽  
René Lozi ◽  
Guanrong Chen
Keyword(s):  
Dynamical System ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Search Algorithm ◽  
Two Dimensional ◽  
Mixed Mode Oscillations ◽  
The Stability ◽  
Two Parameters ◽  
Complex Phenomena ◽  
Autonomous Dynamical System

This article investigates the complex phenomena of canard explosion with mixed-mode oscillations, observed from a fractional-order FitzHugh–Nagumo (FFHN) model. To rigorously analyze the dynamics of the FFHN model, a new mathematical notion, referred to as Hopf-like bifurcation (HLB), is introduced. HLB provides a precise definition for the change between a fixed point and an [Formula: see text]-asymptotically [Formula: see text]-periodic solution of the fractional-order dynamical system, as well as the stability of the FFHN model and the appearance of the HLB. The existence of canard oscillations in the neighborhoods of such HLB points are numerically investigated. Using a new algorithm, referred to as the global-local canard explosion search algorithm, the appearance of various patterns of solutions is revealed, with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied. The numbers of such oscillations versus the two parameters, respectively, are perfectly fitted using exponential functions. Finally, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional order close to one. After all, the article demonstrates that the FFHN model is a very simple two-dimensional model with an incredible ability to present the complex dynamics of neurons.

Remarks on Livšic' theory for nonabelian cocycles

1999 ◽  
Vol 19 (3) ◽  
pp. 703-721 ◽  
Author(s):  
KLAUS SCHMIDT
Keyword(s):  
Dynamical System ◽  
Homomorphic Image ◽  
Periodic Points ◽  
Skew Product ◽  
Topologically Transitive ◽  
Polish Group ◽  
Bounded Distortion

Let $(X,\phi)$ be a hyperbolic dynamical system and let $(G,\delta)$ be a Polish group. Motivated by Nicol and Pollicott, and then by Parry we study conditions under which two Hölder maps $f,g: X\longrightarrow G$ are Hölder cohomologous.In the context of Nicol and Pollicott we show that if $f$ and $g$ are measurably cohomologous and the distortion of the metric $\delta $ by the cocycles defined by $f$ and $g$ is bounded in an appropriate sense, then $f$ and $g$ are Hölder cohomologous.Two further results extend the main theorems recently presented by Parry. Under the hypothesis of bounded distortion we show that, if $f$ and $g$ give equal weight to all periodic points of $\phi $, then $f$ and $g$ are Hölder cohomologous. If the metric $\delta $ is bi-invariant, and if the skew-product $\phi _f$ defined by $f$ is topologically transitive, then conjugacy of weights implies that $g$ is Hölder conjugate to $\alpha \cdot f$ for some isometric automorphism $\alpha $ of $G$. The weaker condition that $g$-weights of periodic points are close to the identity whenever their $f$-weights are close to the identity implies that $g$ is continuously cohomologous to a homomorphic image of $f$.

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.

scholarly journals General Asymptotic Supnorm Estimates for Solutions of One-Dimensional Advection-Diffusion Equations in Heterogeneous Media

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
José A. Barrionuevo ◽  
Lucas S. Oliveira ◽  
Paulo R. Zingano
Keyword(s):  
Initial Data ◽  
Large Time ◽  
Heterogeneous Media ◽  
Diffusion Equations ◽  
Open Problems ◽  
New Techniques ◽  
One Dimensional ◽  
Advection Diffusion

We derive general bounds for the large time size of supnorm values ∥u(·,t)∥L∞(ℝ) of solutions to one-dimensional advection-diffusion equations ut+(b(x,t)u)x=uxx,x∈ℝ,t>0 with initial data u(·,0)∈Lp0(ℝ)∩L∞(ℝ) for some 1≤p0<∞ and arbitrary bounded advection speeds b(x,t), introducing new techniques based on suitable energy arguments. Some open problems and related results are also given.

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