Diophantine approximation by orbits of expanding Markov maps

2012 ◽  
Vol 33 (2) ◽  
pp. 585-608 ◽  
Author(s):  
LINGMIN LIAO ◽  
STÉPHANE SEURET
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Gibbs Measures ◽  
Finite Partition ◽  
Markov Map

AbstractIn 1995, Hill and Velani introduced the ‘shrinking targets’ theory. Given a dynamical system ([0,1],T), they investigated the Hausdorff dimension of sets of points whose orbits are close to some fixed point. In this paper, we study the sets of points well approximated by orbits {Tnx}n≥0, where Tis an expanding Markov map with a finite partition supported by [0,1]. The dimensions of these sets are described using the multifractal properties of invariant Gibbs measures.

scholarly journals SHRINKING TARGETS FOR NONAUTONOMOUS DYNAMICAL SYSTEMS CORRESPONDING TO CANTOR SERIES EXPANSIONS

2015 ◽  
Vol 92 (2) ◽  
pp. 205-213 ◽  
Author(s):  
LIOR FISHMAN ◽  
BILL MANCE ◽  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Wide Class ◽  
Series Expansions ◽  
Closed Formula ◽  
Nonautonomous Dynamical Systems ◽  
Nonautonomous Dynamical System ◽  
Cantor Series

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the interval$[0,1)$, viewed as$\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilising our theorem. In particular, we give a Diophantine approximation interpretation of our scheme.

A NOTE ON INHOMOGENEOUS DIOPHANTINE APPROXIMATION IN BETA-DYNAMICAL SYSTEM

2014 ◽  
Vol 91 (1) ◽  
pp. 34-40 ◽  
Author(s):  
YUEHUA GE ◽  
FAN LÜ
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Real Number ◽  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Positive Function ◽  
Real Numbers ◽  
Image Position

AbstractWe study the distribution of the orbits of real numbers under the beta-transformation$T_{{\it\beta}}$for any${\it\beta}>1$. More precisely, for any real number${\it\beta}>1$and a positive function${\it\varphi}:\mathbb{N}\rightarrow \mathbb{R}^{+}$, we determine the Lebesgue measure and the Hausdorff dimension of the following set:$$\begin{eqnarray}E(T_{{\it\beta}},{\it\varphi})=\{(x,y)\in [0,1]\times [0,1]:|T_{{\it\beta}}^{n}x-y|<{\it\varphi}(n)\text{ for infinitely many }n\in \mathbb{N}\}.\end{eqnarray}$$

scholarly journals BACH FLOWS OF PRODUCT MANIFOLDS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250039 ◽  
Author(s):  
SANJIT DAS ◽  
SAYAN KAR
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Ricci Flow ◽  
Flow Characteristics ◽  
Geometric Flow ◽  
Flow Equations ◽  
First Order ◽  
Bach Tensor ◽  
Point Condition ◽  
Future Work

We investigate various aspects of a geometric flow defined using the Bach tensor. First, using a well-known split of the Bach tensor components for (2, 2) unwarped product manifolds, we solve the Bach flow equations for typical examples of product manifolds like S2 × S2, R2 × S2. In addition, we obtain the fixed-point condition for general (2, 2) manifolds and solve it for a restricted case. Next, we consider warped manifolds. For Bach flows on a special class of asymmetrically warped 4-manifolds, we reduce the flow equations to a first-order dynamical system, which is solved exactly to find the flow characteristics. We compare our results for Bach flow with those for Ricci flow and discuss the differences qualitatively. Finally, we conclude by mentioning possible directions for future work.

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 Gaps problems and frequencies of patches in cut and project sets

2016 ◽  
Vol 161 (1) ◽  
pp. 65-85 ◽  
Author(s):  
ALAN HAYNES ◽  
HENNA KOIVUSALO ◽  
JAMES WALTON ◽  
LORENZO SADUN
Keyword(s):  
Hausdorff Dimension ◽  
Frequency Spectrum ◽  
Diophantine Approximation

AbstractWe establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets of full Hausdorff dimension.

scholarly journals Vertical Shift and Simultaneous Diophantine Approximation on Polynomial Curves

2014 ◽  
Vol 58 (1) ◽  
pp. 1-26
Author(s):  
Faustin Adiceam
Keyword(s):  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Rational Approximations ◽  
Vertical Shift ◽  
Polynomial Curves ◽  
Integer Coefficients ◽  
Simultaneous Diophantine Approximation ◽  
First Results

AbstractThe Hausdorff dimension of the set of simultaneously τ-well-approximable points lying on a curve defined by a polynomial P(X) + α, where P(X) ∈ ℤ[X] and α ∈ ℝ, is studied when τ is larger than the degree of P(X). This provides the first results related to the computation of the Hausdorff dimension of the set of well-approximable points lying on a curve that is not defined by a polynomial with integer coefficients. The proofs of the results also include the study of problems in Diophantine approximation in the case where the numerators and the denominators of the rational approximations are related by some congruential constraint.

scholarly journals Badly approximable points on self-affine sponges and the lower Assouad dimension

2017 ◽  
Vol 39 (3) ◽  
pp. 638-657 ◽  
Author(s):  
TUSHAR DAS ◽  
LIOR FISHMAN ◽  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Assouad Dimension ◽  
Badly Approximable Points ◽  
Bounded Below ◽  
Badly Approximable ◽  
Sierpinski Sponges

We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. For self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpiński sponges/carpets (also known as Bedford–McMullen sponges/carpets) and the case of Barański carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.

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