On local aspects of topological weak mixing in dimension one and beyond

2011 ◽  
Vol 202 (3) ◽  
pp. 261-288 ◽  
Author(s):  
Piotr Oprocha ◽  
Guohua Zhang
Keyword(s):  
Weak Mixing ◽  
Dimension One ◽  
Topological Weak Mixing

scholarly journals On weak mixing, minimality and weak disjointness of all iterates

2011 ◽  
Vol 32 (5) ◽  
pp. 1661-1672 ◽  
Author(s):  
DOMINIK KWIETNIAK ◽  
PIOTR OPROCHA
Keyword(s):  
Topological Dynamical System ◽  
Multiple Recurrence ◽  
Weak Mixing ◽  
Compact Metric Space ◽  
Open Questions ◽  
Weakly Mixing ◽  
Recurrence Theorem ◽  
Topological Weak Mixing ◽  
Mixing Property ◽  
Transitive System

AbstractThis article addresses some open questions about the relations between the topological weak mixing property and the transitivity of the map f×f2×⋯×fm, where f:X→X is a topological dynamical system on a compact metric space. The theorem stating that a weakly mixing and strongly transitive system is Δ-transitive is extended to a non-invertible case with a simple proof. Two examples are constructed, answering the questions posed by Moothathu [Diagonal points having dense orbit. Colloq. Math. 120(1) (2010), 127–138]. The first one is a multi-transitive non-weakly mixing system, and the second one is a weakly mixing non-multi-transitive system. The examples are special spacing shifts. The latter shows that the assumption of minimality in the multiple recurrence theorem cannot be replaced by weak mixing.

Topological weak-mixing of interval exchange maps

1997 ◽  
Vol 17 (5) ◽  
pp. 1183-1209 ◽  
Author(s):  
A. NOGUEIRA ◽  
D. RUDOLPH
Keyword(s):  
Measure Zero ◽  
Interval Exchange ◽  
Weak Mixing ◽  
Interval Map ◽  
Constant Height ◽  
Interval Lengths ◽  
Topological Weak Mixing ◽  
Interval Exchange Maps

An interval map with only one discontinuity is isomorphic to a rotation of the circle, and has continuous eigenfunctions. What we show here is that for almost every choice of lengths of the intervals, this is the only way an irreducible interval exchange can have a somewhere continuous eigenfunction. We show slightly more, considering certain towers over the interval exchange, showing that outside of a set of choices for interval lengths of measure zero these have a somewhere continuous eigenfunction only if they are isomorphic to either a rotation, or a tower of constant height over an interval exchange.

scholarly journals Topological weak mixing and diffusion at all times for a class of Hamiltonian systems

2021 ◽  
pp. 1-15
Author(s):  
BASSAM FAYAD ◽  
MARIA SAPRYKINA
Keyword(s):  
Hamiltonian Systems ◽  
Weak Mixing ◽  
Topological Weak Mixing ◽  
And Diffusion ◽  
Strong Diffusion ◽  
Diffusion Properties

Abstract We present examples of nearly integrable analytic Hamiltonian systems with several strong diffusion properties: topological weak mixing and diffusion at all times. These examples are obtained by AbC constructions with several frequencies.

On local aspects of topological weak mixing, sequence entropy and chaos

2013 ◽  
Vol 34 (5) ◽  
pp. 1615-1639 ◽  
Author(s):  
PIOTR OPROCHA ◽  
GUOHUA ZHANG
Keyword(s):  
Dynamical Systems ◽  
Qualitative Theory ◽  
Broad Sense ◽  
Weak Mixing ◽  
Sequence Entropy ◽  
Weakly Mixing ◽  
Topological Weak Mixing ◽  
Topological Sequence Entropy ◽  
Mixing Sequence

AbstractIn this paper we show that for every$n\geq 2$there are minimal systems with perfect weakly mixing sets of order$n$and all weakly mixing sets of order$n+ 1$trivial. We present some relations between weakly mixing sets and topological sequence entropy; in particular, we prove that invertible minimal systems with non-trivial weakly mixing sets of order three always have positive topological sequence entropy. We also study relations between weak mixing of sets and other well-established notions from qualitative theory of dynamical systems like (regional) proximality, chaos and equicontinuity in a broad sense.

New proofs that weak mixing is generic

1976 ◽  
Vol 32 (3) ◽  
pp. 263-278 ◽  
Author(s):  
Steven Alpern
Keyword(s):  
Weak Mixing

scholarly journals Exact solutions for the total variation denoising problem of piecewise constant images in dimension one

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Riccardo Cristoferi
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Total Variation ◽  
Piecewise Constant ◽  
Geometrical Properties ◽  
Total Variation Denoising ◽  
Fidelity Term ◽  
Dimension One

AbstractA method for obtaining the exact solution for the total variation denoising problem of piecewise constant images in dimension one is presented. The validity of the algorithm relies on some results concerning the behavior of the solution when the parameter λ in front of the fidelity term varies. Albeit some of them are well-known in the community, here they are proved with simple techniques based on qualitative geometrical properties of the solutions.

scholarly journals Nonlinear Conditions for Ultradifferentiability

2021 ◽  
Author(s):  
David Nicolas Nenning ◽  
Armin Rainer ◽  
Gerhard Schindl
Keyword(s):  
General Validity ◽  
Dimension One ◽  
Analogous Theorem

AbstractA remarkable theorem of Joris states that a function f is $$C^\infty $$ C ∞ if two relatively prime powers of f are $$C^\infty $$ C ∞ . Recently, Thilliez showed that an analogous theorem holds in Denjoy–Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris’s result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.

Is there large weak mixing in heavy nuclei?

1994 ◽  
Vol 49 (6) ◽  
pp. 3297-3300 ◽  
Author(s):  
J. J. Szymanski ◽  
J. D. Bowman ◽  
M. Leuschner ◽  
B. A. Brown ◽  
I. C. Girit
Keyword(s):  
Heavy Nuclei ◽  
Weak Mixing
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