Conjugacy of one-dimensional maps

[12/67%]
Show
LaTeX
Problem X.1.9 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

Maps f:Δ1→Δ1f: \Delta_{1} \rightarrow \Delta_{1} and g:Δ2→Δ2g: \Delta_{2} \rightarrow \Delta_{2} (Δ1,Δ2\Delta_{1}, \Delta_{2} being intervals on the line) are called topologically conjugate if there exists a continuous bijection (homeomorphism) h:Δ2→Δ1h: \Delta_{2} \rightarrow \Delta_{1} such that f=h∘g∘h−1f=h \circ g \circ h^{-1} (i.e., h∘g=f∘hh \circ g=f \circ h). In this case we write f∼gf \sim g and call hh the conjugating map. It is assumed that the maps are continuous, defined on intervals, and map them into themselves.

Prove that the relation of topological conjugacy is an equivalence relation, i.e.

?
(a)

f∼ff \sim f;

(b)

f∼gf \sim g if and only if g∼fg \sim f;

(c)

if f1∼f2,f2∼f3f_{1} \sim f_{2}, f_{2} \sim f_{3}, then f1∼f3f_{1} \sim f_{3}.

Problem X.1.10 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

Maps f:Δ1→Δ1f: \Delta_{1} \rightarrow \Delta_{1} and g:Δ2→Δ2g: \Delta_{2} \rightarrow \Delta_{2} (Δ1,Δ2\Delta_{1}, \Delta_{2} being intervals on the line) are called topologically conjugate if there exists a continuous bijection (homeomorphism) h:Δ2→Δ1h: \Delta_{2} \rightarrow \Delta_{1} such that f=h∘g∘h−1f=h \circ g \circ h^{-1} (i.e., h∘g=f∘hh \circ g=f \circ h). In this case we write f∼gf \sim g and call hh the conjugating map. It is assumed that the maps are continuous, defined on intervals, and map them into themselves.

Prove that topologically conjugate maps simultaneously possess or fail to possess the following properties:

?
(a)

injectivity;

(b)

monotonicity of a definite type;

(c)

one-sided boundedness (without specifying its type);

(d)

boundedness.

Problem X.1.11 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

Maps f:Δ1→Δ1f: \Delta_{1} \rightarrow \Delta_{1} and g:Δ2→Δ2g: \Delta_{2} \rightarrow \Delta_{2} (Δ1,Δ2\Delta_{1}, \Delta_{2} being intervals on the line) are called topologically conjugate if there exists a continuous bijection (homeomorphism) h:Δ2→Δ1h: \Delta_{2} \rightarrow \Delta_{1} such that f=h∘g∘h−1f=h \circ g \circ h^{-1} (i.e., h∘g=f∘hh \circ g=f \circ h). In this case we write f∼gf \sim g and call hh the conjugating map. It is assumed that the maps are continuous, defined on intervals, and map them into themselves.

Prove that topologically conjugate maps have

?
(a)

the same number of intervals of monotonicity;

(b)

the same number of fixed points.

Problem X.1.12 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

Maps f:Δ1→Δ1f: \Delta_{1} \rightarrow \Delta_{1} and g:Δ2→Δ2g: \Delta_{2} \rightarrow \Delta_{2} (Δ1,Δ2\Delta_{1}, \Delta_{2} being intervals on the line) are called topologically conjugate if there exists a continuous bijection (homeomorphism) h:Δ2→Δ1h: \Delta_{2} \rightarrow \Delta_{1} such that f=h∘g∘h−1f=h \circ g \circ h^{-1} (i.e., h∘g=f∘hh \circ g=f \circ h). In this case we write f∼gf \sim g and call hh the conjugating map. It is assumed that the maps are continuous, defined on intervals, and map them into themselves.

Prove that if f∼gf \sim g, then the intervals on which the inequality f(x)>xf(x)>x holds are carried by the conjugating map hh into intervals on which (depending on the character of the monotonicity of hh) the same or the opposite strict inequality holds for the map gg. Prove that if the maps ff and gg are defined on R\mathbb {R} and one of them is odd, then the conjugating bijection hh can be chosen to be increasing.

?
Problem X.1.13 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

Maps f:Δ1→Δ1f: \Delta_{1} \rightarrow \Delta_{1} and g:Δ2→Δ2g: \Delta_{2} \rightarrow \Delta_{2} (Δ1,Δ2\Delta_{1}, \Delta_{2} being intervals on the line) are called topologically conjugate if there exists a continuous bijection (homeomorphism) h:Δ2→Δ1h: \Delta_{2} \rightarrow \Delta_{1} such that f=h∘g∘h−1f=h \circ g \circ h^{-1} (i.e., h∘g=f∘hh \circ g=f \circ h). In this case we write f∼gf \sim g and call hh the conjugating map.

Determine whether the maps ff and gg defined on R\mathbb {R} are topologically conjugate in the following cases:

?
(a)

f(x)=x2−af(x)=x^{2}-a, g(x)=1−ax2(a≠0);g(x)=1-a x^{2} \quad (a \neq 0) ;

(b)

f(x)=cos⁡xf(x)=\cos x, g(x)=sin⁡xg(x)=\sin x;

(c)

f(x)=cos⁡xf(x)=\cos x, g(x)=−cos⁡xg(x)=-\cos x;

(d)

f(x)=sin⁡xf(x)=\sin x, g(x)=−sin⁡xg(x)=-\sin x;

(e)

f(x)=x+sin⁡xf(x)=x+\sin x, g(x)=x+cos⁡xg(x)=x+\cos x.

Problem X.1.14 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

Maps f:Δ1→Δ1f: \Delta_{1} \rightarrow \Delta_{1} and g:Δ2→Δ2g: \Delta_{2} \rightarrow \Delta_{2} (Δ1,Δ2\Delta_{1}, \Delta_{2} being intervals on the line) are called topologically conjugate if there exists a continuous bijection (homeomorphism) h:Δ2→Δ1h: \Delta_{2} \rightarrow \Delta_{1} such that f=h∘g∘h−1f=h \circ g \circ h^{-1} (i.e., h∘g=f∘hh \circ g=f \circ h). In this case we write f∼gf \sim g and call hh the conjugating map.

Let α>0,α≠1,fα(x)=xα,gα(x)=x−α(x>0)\alpha >0, \alpha \neq 1, f_{\alpha }(x)=x^{\alpha }, g_{\alpha }(x)=x^{-\alpha }(x>0). For which α\alpha and β\beta are the functions topologically conjugate

?
(a)

fαf_{\alpha } and fβf_{\beta };

(b)

gαg_{\alpha } and gβg_{\beta };

(c)

fαf_{\alpha } and gβg_{\beta } ?

Problem X.1.15 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

Maps f:Δ1→Δ1f: \Delta_{1} \rightarrow \Delta_{1} and g:Δ2→Δ2g: \Delta_{2} \rightarrow \Delta_{2} (Δ1,Δ2\Delta_{1}, \Delta_{2} being intervals on the line) are called topologically conjugate if there exists a continuous bijection (homeomorphism) h:Δ2→Δ1h: \Delta_{2} \rightarrow \Delta_{1} such that f=h∘g∘h−1f=h \circ g \circ h^{-1} (i.e., h∘g=f∘hh \circ g=f \circ h). In this case we write f∼gf \sim g and call hh the conjugating map.

?
(a)

Let f(x)=x2,g(x)=x2+ax+b(x∈R)f(x)=x^{2}, g(x)=x^{2}+a x+b(x \in \mathbb {R}). Describe the set of pairs (a,b)(a, b) for which f∼gf \sim g.

(b)

For which a∈Ra \in \mathbb {R} can one find b∈Rb \in \mathbb {R} such that the maps f(x)=1−ax2f(x)=1-a x^{2} and g(x)=bx(1−x)(x∈R)g(x)=b x(1-x)(x \in \mathbb {R}) are topologically conjugate? Describe the set BB of all numbers bb that arise in this way.

(c)

Prove that the maps gg and g~:[0,1]→[0,1],g(x)=2g~(x)==3x(1−x)\widetilde{g}:\left[0,1\right] \rightarrow [0,1], g(x)=2 \widetilde{g}(x)= =3 x(1-x), are not topologically conjugate.

(d)

For which a∈Ra \in \mathbb {R} and which interval Δ=[p,q]⊂R\Delta =\left[p, q\right] \subset \mathbb {R} is the map f:Δ→Δ,f(x)=1−ax2f: \Delta \rightarrow \Delta , f(x)=1-a x^{2}, topologically conjugate to the map g:[0,1]→[0,1]g:\left[0,1\right] \rightarrow [0,1] of the form g(x)=bx(1−x)g(x)=b x(1-x)?

Problem X.1.16 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

Maps f:Δ1→Δ1f: \Delta_{1} \rightarrow \Delta_{1} and g:Δ2→Δ2g: \Delta_{2} \rightarrow \Delta_{2} (Δ1,Δ2\Delta_{1}, \Delta_{2} being intervals on the line) are called topologically conjugate if there exists a continuous bijection (homeomorphism) h:Δ2→Δ1h: \Delta_{2} \rightarrow \Delta_{1} such that f=h∘g∘h−1f=h \circ g \circ h^{-1} (i.e., h∘g=f∘hh \circ g=f \circ h). In this case we write f∼gf \sim g and call hh the conjugating map.

Prove the topological conjugacy of the continuous maps f,g:[−1,1]→[−1,1]f, g:\left[-1,1\right] \rightarrow [-1,1], given as follows:

?
(a)

f(x)=1−2∣x∣,g(x)=1−2x2(x∈[−1,1])f(x)=1-2\left|x\right|, g(x)=1-2 x^{2} \quad (x \in [-1,1]);

(b)

f=fn(n∈N)f=f_{n}(n \in \mathbb {N}) is the ``sawtooth'' map taking values (−1)k+n(-1)^{k+n} at the points −1+2kn,0⩽k⩽n-1+\frac{2 k}{n}, 0 \leqslant k \leqslant n, and linear on the intervals between them (Fig. 14), and g=Tng=T_{n} is the nn-th Chebyshev polynomial, defined by the equality Tn(x)=cos⁡(narccos⁡x)T_{n}(x)=\cos (n \arccos x) (see Problem I.2.20 and Fig. 15);

Fig. 14Fig. 14

Fig. 15Fig. 15

(c)

prove that the functions from part a) are conjugate as maps from R\mathbb {R} to R\mathbb {R}.

Problem X.1.17 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

Maps f:Δ1→Δ1f: \Delta_{1} \rightarrow \Delta_{1} and g:Δ2→Δ2g: \Delta_{2} \rightarrow \Delta_{2} (Δ1,Δ2\Delta_{1}, \Delta_{2} being intervals on the line) are called topologically conjugate if there exists a continuous bijection (homeomorphism) h:Δ2→Δ1h: \Delta_{2} \rightarrow \Delta_{1} such that f=h∘g∘h−1f=h \circ g \circ h^{-1} (i.e., h∘g=f∘hh \circ g=f \circ h). In this case we write f∼gf \sim g and call hh the conjugating map.

Let f:[a1,b1]→[a1,b1],g:[a2,b2]→[a2,b2]f:\left[a_{1}, b_{1}\right] \rightarrow \left[a_{1}, b_{1}\right], g:\left[a_{2}, b_{2}\right] \rightarrow \left[a_{2}, b_{2}\right] be homeomorphisms of intervals having no fixed interior points. Prove that f∼gf \sim g. Verify that the statement remains valid if the closed intervals are replaced by open intervals.

?
Problem X.1.18 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

Maps f:Δ1→Δ1f: \Delta_{1} \rightarrow \Delta_{1} and g:Δ2→Δ2g: \Delta_{2} \rightarrow \Delta_{2} (Δ1,Δ2\Delta_{1}, \Delta_{2} being intervals on the line) are called topologically conjugate if there exists a continuous bijection (homeomorphism) h:Δ2→Δ1h: \Delta_{2} \rightarrow \Delta_{1} such that f=h∘g∘h−1f=h \circ g \circ h^{-1} (i.e., h∘g=f∘hh \circ g=f \circ h). In this case we write f∼gf \sim g and call hh the conjugating map.

Classify, up to topological conjugacy, the homeomorphisms of an interval onto itself having a finite set of fixed points, among which are the endpoints of the interval.

?
Problem X.2.18 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.2 "Measure-Preserving Transformations"

Prove that if a transformation T:[a,b]→[a,b]T:\left[a, b\right] \rightarrow [a, b] preserves some finite absolutely continuous measure, then TT is topologically conjugate to some transformation of the same interval preserving Lebesgue measure.

?
Problem 1.50 from Steeb W., Tanski I., Hardy Y. "Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications" [STH12]More problems in Chapter 1 "Groups"

Given two manifolds MM and NN, a bijective map ϕ\phi from MM to NN is called a diffeomorphism if both ϕ:M→N\phi : M \rightarrow N and its inverse ϕ−1\phi^{-1} are differentiable. Let f:R→Rf : \mathbb {R} \rightarrow \mathbb {R} be given by the analytic function

f(x)=4x(1−x) f(x) = 4x(1-x)

and the analytic function g:R→Rg : \mathbb {R} \rightarrow \mathbb {R} be given by

g(x)=1−2x2. g(x) = 1-2x^{2} .
?
(i)

Can one find a diffeomorphism ϕ:R→R\phi : \mathbb {R} \rightarrow \mathbb {R} such that

g=ϕ∘f∘ϕ−1 ? g = \phi \circ f \circ \phi ^{-1} \ ?
(ii)

Consider the diffeomorphism ψ:R→R\psi : \mathbb {R} \rightarrow \mathbb {R}

ψ(x)=sinh⁡(x). \psi (x) = \sinh (x) .

Calculate ψ∘g∘ψ−1\psi \circ g \circ \psi^{-1}.

References

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.
[STH12] Steeb W., Tanski I., Hardy Y. Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications. World Scientific Publishing, 2012.