Topological conjugacy can be found in number of problems like
(1) Topological conjugacy for sofic systems and extensions of automorphisms of finite subsystems of topological Markov shifts .
(2) Topological Conjugacy of Relative Topological Dynamics
(3) Topological conjugacy of singularities of vector fields
(4) Topological Conjugacy of Affine Transformations of Tori
Thank you Romeo PG(Just as indicated it's about relevance and places to relate it to), Thank you too Maged Bin-Saad.
Like stated topological conjugacy arises in various areas. In dynamical systems (discrete and continuous ) topological conjugate systems have equivalent flows. Hence conjugacy is an important in the classification of dynamic systems. For example or dynamic systems arising from differential equations, the Hartman-Grobman theorem establishes when the resulting dynamic systems is conjugate to the the linearized equation at an equilibrium point.
http://www.math.wsu.edu/math/faculty/schumaker/Math512/512F10Ch4B.pdf
Conjugacy and equivalence of flows arose in the early 1900's as the subject of topological dynamics developed, mostly intially developed by G. D. Birkhoff based on the work of Poincare, Bendixion and others. Later, Hartman, Grobman, Smale, Pugh, Anosov, Arnold and others furthered developed the subject and showed that such systems give rise to very topologically complicated invariant sets, i.e., limit cycles, wandering points, strange attractors ( the basis of what is know today as chaos) with the concept of stable and unstable manifolds and the and prototypical strange attractor being the Smale Horseshoe.
A typical approach introduced by Poincare (known as the Poincare map) was to reformulate the problem of continuous dynamical systems into one of iterated maps on the underlying topological space and study the limit sets as the number of iterations go to infinity.
This is how Hartman proved the Hartman-Grobman theorem.
http://www.pavleck.net/bookinfo/ordinary-differential-equations_28.djvu/
Smale developed the Horseshoe map in the studying the orbits of the van der Pol oscillator. Interesting topological dynamic also and chaotic behavior (strange attractors) arise in the classical Lorentz system better known as the Lorentz butterfly.
https://en.wikipedia.org/wiki/Topological_dynamics
https://en.wikipedia.org/wiki/Horseshoe_map
https://en.wikipedia.org/wiki/Van_der_Pol_oscillator
https://en.wikipedia.org/wiki/Lorenz_system
Thank you once more Prof. Prevatt and Prof. Breuer for the added answers.
No Peter it is not "applied mathematics" although some of the pervious of chaos theory are from the applied sciences are. Topological dynamics is concerned with the topology of the invariant sets of dynamical systems, be they continuous, discrete (e.g., fractals). It arose out of the asymptotic properties of differential equations x_dot=f(x) around the stationary points ( (f(x0) =0). However, the subject is quite broad, from flows on manifolds evolution equations in Banach spaces.
http://math.sci.ccny.cuny.edu/docs?name=Top%20Dynamics%20Survey.pdf
https://www.amazon.com/Ergodic-Topological-Dynamics-Homogeneous-Mathematical-ebook/dp/B01DM26UWO/ref=sr_1_2?s=books&ie=UTF8&qid=1531514887&sr=1-2&keywords=topological+dynamics
https://www.amazon.com/Dynamics-Evolutionary-Equations-Mathematical-Sciences/dp/0387983473/ref=sr_1_10?s=books&ie=UTF8&qid=1531516291&sr=1-10
It encompasses a large area and it is just not about R^n but it has been expanded to infinite dimensions. The concept of exponential dichotomy is contained in the theory and is applicable in the tangent bundles of manifolds to describe general evolution equations on Banach spaces. Topological dynamics is the study of the topology of invariant sets. Does it have applications - sure but that was not what Birkhoff, Hartmon Ansov, Smale, Pugh, Sacker, Sell , etc. were after etc. were after - they were after the classification of the invariant sets which can be quite complicated, e.g., measure zero but everywhere dense. It has been hoped that the structure of these sets would lead to information of the structural stability of the dynamic systems producing them.
https://en.wikipedia.org/wiki/Structural_stability
The closely thing we have to that is the Pugh closing lemma
https://en.wikipedia.org/wiki/Pugh's_closing_lemma
but that requires the diffeomorphism be continuous with continuous first derivative. Expanding Pugh's closing lemma is one of Smale's problems.
The investigates arise of course often out of studying specific equations of mathematical physics or even mathematical biology (predator - prey equation). However, at the end of the day much of mathematics arises that way. However, the goal is to get a deeper understanding of the invariant sets (equilibrium if you will) be it statistical (ergodic) or density (everywhere dense, zero measure) etc.
Prof. Breuer, just as you indicated I wanted to dwell more on pure aspects of the question posed and so "what is topological conjugacy relevant to" will be okay but its usage in topological dynamics, topology in terms of homeomorphisms and iterated functions will be appreciated. Upon the whole, the contributions above have added colour to what I am looking for. Remember these days we have pure and applied topology and one can work with the two as well.
Prof. Truman explained importantly which is also appreciated that topological dynamics is concerned with the topology of the invariant sets of (topological) dynamical systems.
We can even talk of minimal sets, distal dynamical systems, poisson stability and chain recurrence.
As many properties like proximality, distality, behaviour of fixed points (if they are attracting or repelling) etc are invariant under topological conjugacy, sometimes one can use it to make the calculation easier. My student Alok Kumar Yadav has used topological conjugacy to check behaviour of fixed points of `affine' maps on the unit circle. He took the conjugation by a suitable isometry and it made the calculation easier.
That is an very important application of conjugacy. For example in Hartman's proof of the Hartman-Grobman Theorem published in 1960, he converts the question concerning solutions of the differential equation x'=f(x) around a fixed point x0 (f(x0)=0) when the fixed point of the hyperbolic (the eigenvalues of Df(x0) have non zero real part) and showed that a conjugacy between that and the linearized equation (x'=Df(x0)x) and contrasts mapping form R^n to R^n where the solution is conjugate to the solution of the xn=T^nx and establishes the existence of the stable and unstable manifolds for the original equation. By using topological conjunct - the qualitative behavior ( structure and topology) in the neighborhood fixed points, attractors, the simplest conjugate system - which is a linear system in the case of hyperbolic fixed points. A truly foundational result in dynamic systems which has been extended to dynamical systems and differential equation on manifolds and Banach spaces. In fact Phil's original theorem converted it to a problem of a diffeomorphism on R^n.
https://en.wikipedia.org/wiki/Hartman–Grobman_theorem
Grobman independently discovered the result using a similar approach independently about a year later. However, because the scientist in the US and the former Soviet Union didn't talk much in the 1960's - neither was aware either's result until later.
In the late 1960's, one of Hartman's students, Charles Pugh, proved the the result for iterated maps on a manifold and Palis extended it to diffeomorphisms on Banach spaces. In 1984 Quandt established a similar result for endomorphisms on a Banach spaces.
https://core.ac.uk/download/pdf/82304960.pdf
Because of its potential important to applications, even today work is ongoing to simplify the proof so that new understanding can be attained and the results become more available to a wider audience. Hartman's original theory was very functional analytic based and could be somewhat difficult to follow. And of course since it is really a geometric and topological result at the end of the day, more geometric proofs are being developed.
https://arxiv.org/pdf/1405.6733.pdf
Many fixed point theorems can be stated using iterations of a homeomorphism and the fixed point structure of the phase space near the fixed point is homeomorphic to the phase space of any conjugate system. Smale showed the utility of this in his papers in numerical analysis in finding critical points and in his work in mathematical economics looking at equilibrium of the price model.
Article Stephen Smale and the Economic Theory of General Equilibrium
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Example of two differential systems which are topologically equivalent but. not topologically conjugate
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