Topological conjugacy

 ( Topological Dynamics ) 

In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially.Arnold V. I. Geometric Methods in the Theory of Ordinary Differential Equations (Springer, 2020) [1]

To illustrate this directly: suppose that $f$ and $g$ are iterated functions, and there exists a homeomorphism $h$ such that

$g\; =\; h^\{-1\}\; \backslash circ\; f\; \backslash circ\; h,$

so that $f$ and $g$ are topologically conjugate. Then one must have

$g^n\; =\; h^\{-1\}\; \backslash circ\; f^n\; \backslash circ\; h,$

and so the iterated systems are topologically conjugate as well. Here, $\backslash circ$ denotes function composition.

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Definition $f\backslash colon\; X\; \backslash to\; X,\; g\backslash colon\; Y\; \backslash to\; Y$, and $h\backslash colon\; Y\; \backslash to\; X$ are continuous functions on topological spaces, $X$ and $Y$.

$f$ being topologically semiconjugate to $g$ means, by definition, that $h$ is a surjection such that $f\; \backslash circ\; h\; =\; h\; \backslash circ\; g$.

$f$ and $g$ being topologically conjugate means, by definition, that they are topologically semiconjugate and $h$ is furthermore injective, then Bijection, and its Inverse function is continuous too; i.e. $h$ is a homeomorphism; further, $h$ is termed a topological conjugation between $f$ and $g$.

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Flows Similarly, $\backslash phi$ on $X$, and $\backslash psi$ on $Y$ are flows, with $X,\; Y$, and $h\backslash colon\; Y\backslash to\; X$ as above.

$\backslash phi$ being topologically semiconjugate to $\backslash psi$ means, by definition, that $h$ is a surjection such that $\backslash phi(h(y),\; t)\; =\; h\; \backslash circ\; \backslash psi(y,\; t)$, for each $y\backslash in\; Y$, $t\backslash in\; \backslash mathbb\{R\}$.

$\backslash phi$ and $\backslash psi$ being topologically conjugate means, by definition, that they are topologically semiconjugate and is a homeomorphism. Arnold V. I. Geometric Methods in the Theory of Ordinary Differential Equations (Springer, 2020) [2]

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Examples

• The logistic map and the tent map are topologically conjugate.
  Alligood, K. T., Sauer, T., and Yorke, J.A. (1997). 9780387946771, Springer. ISBN 9780387946771
• The logistic map of unit height and the Bernoulli map are topologically conjugate.
• For certain values in the parameter space, the Hénon map when restricted to its Julia set is topologically conjugate or semi-conjugate to the shift map on the space of two-sided sequences in two symbols.

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Discussion Topological conjugation – unlike semiconjugation – defines an equivalence relation in the space of all continuous surjections of a topological space to itself, by declaring $f$ and $g$ to be related if they are topologically conjugate. This equivalence relation is very useful in the theory of , since each class contains all functions which share the same dynamics from the topological viewpoint. For example, periodic point of $g$ are mapped to homeomorphic orbits of $f$ through the conjugation. Writing $g\; =\; h^\{-1\}\; \backslash circ\; f\; \backslash circ\; h$ makes this fact evident: $g^n\; =\; h^\{-1\}\; \backslash circ\; f^n\; \backslash circ\; h$. Speaking informally, topological conjugation is a "change of coordinates" in the topological sense.

However, the analogous definition for flows is somewhat restrictive. In fact, we are requiring the maps $\backslash phi(\backslash cdot,\; t)$ and $\backslash psi(\backslash cdot,\; t)$ to be topologically conjugate for each $t$, which is requiring more than simply that orbits of $\backslash phi$ be mapped to orbits of $\backslash psi$ homeomorphically. This motivates the definition of topological equivalence, which also partitions the set of all flows in $X$ into classes of flows sharing the same dynamics, again from the topological viewpoint.

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Topological equivalence We say that two flows $\backslash phi$ and $\backslash psi$ are topologically equivalent, if there is a homeomorphism $h:Y\; \backslash to\; X$, mapping orbits of $\backslash psi$ to orbits of $\backslash phi$ homeomorphically, and preserving orientation of the orbits. In other words, letting $\backslash mathcal\{O\}$ denote an orbit, one has

$h(\backslash mathcal\{O\}(y,\; \backslash psi))\; =\; \backslash \{h\; \backslash circ\; \backslash psi(y,\; t):\; t\; \backslash in\; \backslash mathbb\{R\}\backslash \}\; =\; \backslash \{\backslash phi(h(y),\; t):\; t\; \backslash in\; \backslash mathbb\{R\}\backslash \}\; =\; \backslash mathcal\{O\}(h(y),\; \backslash phi)$

for each $y\; \backslash in\; Y$. In addition, one must line up the flow of time: for each $y\; \backslash in\; Y$, there exists a $\backslash delta\; >\; 0$ such that, if , and if is such that $\backslash phi(h(y),\; s)\; =\; h\; \backslash circ\; \backslash psi(y,\; t)$, then $s\; >\; 0$.

Overall, topological equivalence is a weaker equivalence criterion than topological conjugacy, as it does not require that the time term is mapped along with the orbits and their orientation. An example of a topologically equivalent but not topologically conjugate system would be the non-hyperbolic class of two dimensional systems of differential equations that have closed orbits. While the orbits can be transformed to each other to overlap in the spatial sense, the periods of such systems cannot be analogously matched, thus failing to satisfy the topological conjugacy criterion while satisfying the topological equivalence criterion.

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Smooth and orbital equivalence More equivalence criteria can be studied if the flows, $\backslash phi$ and $\backslash psi$, arise from differential equations.

Two dynamical systems defined by the differential equations, $\backslash dot\{x\}\; =\; f(x)$ and $\backslash dot\{y\}\; =\; g(y)$, are said to be smoothly equivalent if there is a diffeomorphism, $h:\; X\; \backslash to\; Y$, such that

$f(x)\; =\; M^\{-1\}(x)\; g(h(x))\; \backslash quad\backslash text\{where\}\backslash quad\; M(x)\; =\; \backslash frac\{\backslash mathrm\{d\}h(x)\}\{\backslash mathrm\{d\}x\}.$

In that case, the dynamical systems can be transformed into each other by the coordinate transformation, $y\; =\; h(x)$.

Two dynamical systems on the same state space, defined by $\backslash dot\{x\}\; =\; f(x)$ and $\backslash dot\{x\}\; =\; g(x)$, are said to be orbitally equivalent if there is a positive function, $\backslash mu\; :\; X\; \backslash to\; \backslash mathbb\{R\}$, such that $g(x)\; =\; \backslash mu(x)\; f(x)$. Orbitally equivalent system differ only in the time parametrization.

Systems that are smoothly equivalent or orbitally equivalent are also topologically equivalent. However, the reverse is not true. For example, consider linear systems in two dimensions of the form $\backslash dot\{x\}\; =\; Ax$. If the matrix, $A$, has two positive real eigenvalues, the system has an unstable node; if the matrix has two complex eigenvalues with positive real part, the system has an unstable focus (or spiral). Nodes and foci are topologically equivalent but not orbitally equivalent or smoothly equivalent,

because their eigenvalues are different (notice that the Jacobians of two locally smoothly equivalent systems must be similar, so their eigenvalues, as well as algebraic and geometric multiplicities, must be equal).

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Generalizations of dynamic topological conjugacy There are two reported extensions of the concept of dynamic topological conjugacy:

1. Analogous systems defined as isomorphic dynamical systems
2. Adjoint dynamical systems defined via adjoint functors and natural equivalences in categorical dynamics.

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See also

• Commutative diagram

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