Positive topological entropy for monotone recurrence relations

2014 ◽  
Vol 35 (6) ◽  
pp. 1880-1901 ◽  
Author(s):  
LI GUO ◽  
XUE-QING MIAO ◽  
YA-NAN WANG ◽  
WEN-XIN QIN
Keyword(s):  
Recurrence Relation ◽  
Topological Entropy ◽  
Rotation Number ◽  
Recurrence Relations ◽  
High Dimensional ◽  
Positive Topological Entropy ◽  
Mather Theory

We associate the topological entropy of monotone recurrence relations with the Aubry–Mather theory. If there exists an interval$[{\it\rho}_{0},{\it\rho}_{1}]$such that, for each${\it\omega}\in ({\it\rho}_{0},{\it\rho}_{1})$, all Birkhoff minimizers with rotation number${\it\omega}$do not form a foliation, then the diffeomorphism on the high-dimensional cylinder defined via the monotone recurrence relation has positive topological entropy.

scholarly journals A Note on Tonelli Lagrangian Systems on $\mathbb{T}^2$ with Positive Topological Entropy on a High Energy Level

2020 ◽  
Vol 16 (4) ◽  
pp. 625-635
Author(s):  
J.G. Damasceno ◽  
◽  
J.A.G. Miranda ◽  
L.G. Perona ◽  
◽  
...  
Keyword(s):  
Energy Level ◽  
Topological Entropy ◽  
Dynamical Behavior ◽  
High Energy ◽  
Lagrangian Systems ◽  
High Energy Level ◽  
Positive Topological Entropy ◽  
Mather Theory ◽  
Closed Orbits ◽  
Lagrangian Flow

In this work we study the dynamical behavior of Tonelli Lagrangian systems defined on the tangent bundle of the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$. We prove that the Lagrangian flow restricted to a high energy level $E_{L}^{-1}(c)$ (i.e., $c > c_0(L)$) has positive topological entropy if the flow satisfies the Kupka-Smale property in $E_{L}^{-1}(c)$ (i.e., all closed orbits with energy c are hyperbolic or elliptic and all heteroclinic intersections are transverse on $E_{L}^{-1}(c)$). The proof requires the use of well-known results from Aubry – Mather theory.

scholarly journals Chaotic Motion in the Breathing Circle Billiard

2021 ◽  
Author(s):  
Claudio Bonanno ◽  
Stefano Marò
Keyword(s):  
Phase Space ◽  
Topological Entropy ◽  
Chaotic Motion ◽  
Moving Boundary ◽  
Point Particle ◽  
Invariant Curves ◽  
Positive Topological Entropy ◽  
Variational Techniques ◽  
Mather Theory ◽  
Class Of Functions

AbstractWe consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves in the phase space. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map has positive topological entropy. The proof relies on variational techniques based on the Aubry–Mather theory.

scholarly journals Dynamics of homeomorphisms of the torus homotopic to Dehn twists

2012 ◽  
Vol 34 (2) ◽  
pp. 409-422 ◽  
Author(s):  
SALVADOR ADDAS-ZANATA ◽  
FÁBIO A. TAL ◽  
BRÁULIO A. GARCIA
Keyword(s):  
Lebesgue Measure ◽  
Topological Entropy ◽  
Rotation Number ◽  
Vertical Velocity ◽  
Positive Topological Entropy ◽  
Dehn Twists ◽  
Bounded Motion ◽  
Rotation Set ◽  
Area Preserving ◽  
Uniformly Bounded

AbstractIn this paper, we consider torus homeomorphisms $f$ homotopic to Dehn twists. We prove that if the vertical rotation set of $f$ is reduced to zero, then there exists a compact connected essential ‘horizontal’ set $K$, invariant under $f$. In other words, if we consider the lift $\hat {f}$ of $f$ to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of $\hat {f}$. Also, we give a simple explicit condition which, when satisfied, implies that the vertical rotation set contains an interval and thus also implies positive topological entropy. As a corollary of the above results, we prove a version of Boyland’s conjecture to this setting: if $f$ is area preserving and has a lift $\hat {f}$ to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under $\hat {f}$, or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity.

scholarly journals Monotone recurrence relations, their Birkhoff orbits and topological entropy

1990 ◽  
Vol 10 (1) ◽  
pp. 15-41 ◽  
Author(s):  
Sigurd B. Angenent
Keyword(s):  
Large Class ◽  
Topological Entropy ◽  
Recurrence Relations ◽  
Symplectic Maps ◽  
Positive Topological Entropy ◽  
Twist Maps ◽  
Foregoing Discussion

AbstractA generalization of the class of monotone twistmaps to maps ofs1×RNis proposed. The existence of Birkhoff orbits is studied, and a criterion for positive topological entropy is given. These results are then specialized to the case of monotone twist maps. Finally it is shown that there is a large class of symplectic maps to which the foregoing discussion applies.

scholarly journals ℤd-actions with prescribed topological and ergodic properties

2011 ◽  
Vol 32 (1) ◽  
pp. 191-209 ◽  
Author(s):  
YURI LIMA
Keyword(s):  
Topological Entropy ◽  
Topological Dynamics ◽  
Ergodic Transformations ◽  
Positive Topological Entropy ◽  
Ergodic Properties ◽  
Uniquely Ergodic

AbstractWe extend constructions of Hahn and Katznelson [On the entropy of uniquely ergodic transformations. Trans. Amer. Math. Soc.126 (1967), 335–360] and Pavlov [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28 (2008), 1291–1322] to ℤd-actions on symbolic dynamical spaces with prescribed topological and ergodic properties. More specifically, we describe a method to build ℤd-actions which are (totally) minimal, (totally) strictly ergodic and have positive topological entropy.

scholarly journals ELEMENTARY OSCILLATION CRITERIA FOR A THREE TERM RECURRENCE RELATION WITH OSCILLATORY COEFFICIENT SEQUENCE

1998 ◽  
Vol 29 (3) ◽  
pp. 227-232
Author(s):  
GUANG ZHANG ◽  
SUI-SUN CHENG
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Qualitative Properties ◽  
Oscillation Criteria ◽  
Three Term Recurrence Relation

Qualitative properties of recurrence relations with coefficients taking on both positive and negative values are difficult to obtain since mathematical tools are scarce. In this note we start from scratch and obtain a number of oscillation criteria for one such relation : $x_{n+1}-x_n+p_nx_{n-r}\le 0$.

scholarly journals On the quotient moment of lower generalized order statistics and characterization

2015 ◽  
Vol 11 (1) ◽  
pp. 73-89
Author(s):  
Devendra Kumar
Keyword(s):  
Order Statistics ◽  
Recurrence Relation ◽  
Conditional Expectation ◽  
General Class ◽  
Recurrence Relations ◽  
Generalized Order Statistics ◽  
Lower Records ◽  
Moments Of Order Statistics ◽  
Generalized Order

Abstract In this paper we consider general class of distribution. Recurrence relations satisfied by the quotient moments and conditional quotient moments of lower generalized order statistics for a general class of distribution are derived. Further the results are deduced for quotient moments of order statistics and lower records and characterization of this distribution by considering the recurrence relation of conditional expectation for general class of distribution satisfied by the quotient moment of the lower generalized order statistics.

scholarly journals Distinct partitions and overpartitions

2021 ◽  
Vol 38 (1) ◽  
pp. 149-158
Author(s):  
MIRCEA MERCA ◽  
Keyword(s):  
Partition Function ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Convergent Series ◽  
The Other ◽  
Large Integer ◽  
Linear Recurrence ◽  
Fast Computation ◽  
Real Numbers ◽  
Very High

In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function $Q(n)$. One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of $Q(n)$. This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.

A Necessary Condition of Nonminimal Aubry–Mather Sets for Monotone Recurrence Relations

2014 ◽  
Vol 24 (01) ◽  
pp. 1450012 ◽  
Author(s):  
Ya-Nan Wang ◽  
Wen-Xin Qin
Keyword(s):  
Rotation Number ◽  
Recurrence Relations ◽  
Irrational Rotation ◽  
Necessary Condition ◽  
Irrational Rotation Number

In this paper, we show that a necessary condition for nonminimal Aubry–Mather sets of monotone recurrence relations is that the set of all Birkhoff minimizers with some irrational rotation number does not constitute a foliation, i.e. the gaps of the minimal Aubry–Mather set are not filled up with Birkhoff minimizers.

scholarly journals Some Recurrence Relations and Hilbert Series of Right-Angled Affine Artin Monoid M(D~n∞)

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Young Chel Kuwn ◽  
Zaffar Iqbal ◽  
Abdul Rauf Nizami ◽  
Mobeen Munir ◽  
Sana Riaz ◽  
...  
Keyword(s):  
Growth Rate ◽  
Recurrence Relation ◽  
Recurrence Relations ◽  
Hilbert Series ◽  
The Right

We find the Hilbert series of the right-angled affine Artin monoid M(D~n∞). We also discuss its recurrence relation and the growth rate.

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