- Email: [email protected]
Topological entropy of Chebyshev polynomials and related real mappings
JOURNAL
OF MATHEMATICAL
Topological
ANALYSIS
AND
APPLICATIONS
816-822 (1973)
Entropy of Chebyshev Polynomials and Related Real Mappings MICHAEL
Department
43,
of Mathematics,
Wayne
D. WEISS
State
Submitted
University,
Detroit,
Michigan
48202
by Ky Fan
Topological entropy was originally defined in [l]. If 01and ,8 are open covers of a compact space X, let OLv /3 = {A n B 1A E 01and B E,8}, and define N(a) asthe cardinality of a subcover of (II of minimal cardinality. If T is a continuous self mapping of X, lim,,,(l /KZ)log iV(Vyz,r Th) can be shown to exist and is denoted h(a, T). The topological entropy of T, h(T), is defined assup h(or, T), where the supremum is taken over all open covers OL of X. A basic property of topological entropy is that it is an invariant of topologically conjugate mappings. Using intricate methods, Adler and McAndrew proved in [2] that the topological entropy of the vth Chebyshev polynomial is log V, and Bowen [4] generalizedthis result. In Section 1, we use a theorem of Goodwyn to give a short and natural proof of the theorem of Adler and McAndrew. In Section 2, we consider classesof real mappings which share certain properties of Chebyshev polynomials and obtain upper bounds for the entropy of “at mostp-to-l” and Lipschitzian mappings. The notation and terminology of this paper will follow that of [l], with the exception that arbitrary covers of a compact spacewill occasionally be usedin place of open covers. This paper is basedon a portion of the author’s doctoral dissertation [lo], completed and distributed in 1968-1969. [I I] refers to the results contained herein.
1.
ENTROPY
OF
CHEBYSHEV
POLYNOMIALS
LEMMA 1.1. Let (X, d) be a compactmetric space,01a finite cover of X (by arbitrary sets),and T: X -+ X a continuousmap. If diam(v~~~ T-h) + 0 as n -+ co, then h(T) < h(or,T). Copyright All rights
@ 1973 by Academic Press, Inc. of reproduction in any form reserved.
816
TOPOLOGICAL
817
ENTROPY
Proof. Given any open cover /3 of X, choose n so llarge that diarn(V~~~ T-%x) is lessthan the Lebesguenumber of /3. Then /3< Vyi, T-%, so that l@,T)
1.2. h( TV) = log V.
Proof. Let S, = cos-l o TVo cos on [0, ~1. Clearly h(T,) = h(S,). Note that
whenever kc+ < x < (k + 1) n/v,
k E(0, l,..., v - l}.
We shall prove h(S,) = log V. Since S, is merely the identity, we may assumev > 1. Define a cover cxof [0, 7r] by 01= ([h/v, (k + 1) VT/v)j k E(0, l)...) v - 2)) u ([(v - 1) z-/v, Tr]}. We can easily verify by induction on n that, for any n E kJ, VyL, SLOOP consists of precisely v” intervals Ij , where Ij has endpoints jn/vn and (j + 1) T/P, j = 0, l,..., v%- 1. Hence, diarn(vIli Syicl) + 0 as n-+ 00, so that, by U.l), h(S,) < h(a, S,) = li+i(log v”)/n = log V. To obtain the reverse inequality, we observe that S, preservesnormalized Lebesguemeasurem and employ the measure-theoreticentropy h, (cf. [9]). Since Vz, Srior is the partition of [0, ~1 into points, it follows from [9, Proposition 9.41 that h,(S,) = &(a, S,) = lii$log v”)/n = log v. However, by Goodwyn’s bound theorem [5], h(S,) > h,(S,). h(S”) = log v.
Hence,
WEISS
818
2. SOME GENERALIZATIONS DEFINITION
2.1.
A function j is at mostp-to-l if card j-l(y)
< p when-
ever y E rangej. The Chebyshev polynomial TV is at most v-to-l. In fact, its graph risesand falls exactly Y times. Furthermore, TV satisfiesh(T,) < log v. This leads us to the following. THEOREM 2.2. Supposej is a continuousmapping of the real interval [a, b] into itself. If j is at mostp-to-l, thm h( j) < log p.
Proof. Choose an arbitrary relatively open interval UC [a, b]. We first show that (1) j-l U is a union of p + 1 or fewer disjoint relatively open intervals; and (2) if either j(a) $ U or j(b) $ U, then f-'U disjoint relatively open intervals.
is a union of p or fewer
We can assumeU n rangej is an interval of positive length. Let us call any point t exceptional if t = j(7) for some r E [a, b] which is either a relative maximum or a relative minimum off; otherwise, call t unexceptional.From the fact that j is at most p-to-l, it follows that there are at most countably many exceptional points, so that the set of unexceptional points is densein [a, b]. Hence, we can chooseopen intervals Er C E, C *.. whose endpoints are unexceptional and whose union is the interior of U n range j. For arbitrary k, f -lEk can be representedasa disjoint union .f-lE,
=F
6
fi (xi icn
, Yi),
whereF C (a, b) (possiblyF = m) and xi < yi for eachi E A. Let Ek = (x, y). Since x and y are unexceptional, xi # yj for all i, j E (1. Careful consideration of card j-l{x, y} now reveals that (1) and (2) hold for Ek (with Ek in place of U), from which it follows that they hold for U itself. Using induction, we next prove that, for any n E N, j-*U is a union of or fewer disjoint relatively open intervals. By (l), 1 +p+p2+ “‘i-p” this is true for n = 1. Supposethat j-“U is a union j+lJ = (JE, Vi of or fewer disjoint relatively open intervals U, . Certainly, at l+p+...+pm mostone of theseintervals containsboth j(a) andj(b). ChooseiOED suchthat i E9, i # i,, impliesj(a) $ Ua or j(b) $ U, . By (I), j-lUi, is a union of p + 1 or fewer disjoint relatively open intervals. By (2), for i # i,, , j-lU, is a union of p or fewer disjoint relatively open intervals. Hence, j-crn+l)U is a
TOPOLOGICAL
819
ENTROPY
unionof(p+ l)+p(cardQ1) or f ewer disjoint relatively open intervals. But card 52 < 1 + p + ... + pm; hence,
This completes the induction. For an arbitrary m E IV, let 01, be the set of relatively open intervals in [a, b] with length not exceeding l/m, and let /6&,be a finite subcover of (Y, . If S is a subsetof [a, b], we let X3 be the boundary of S, and if 5 is a cover of [a, 61,we define at = USECAS. Now, for each71E N, n-l
j
card af -+Im < 2(card &) 1 C pk. j=O
k=O
Ifs and t are two consecutive points from the finite set a(Vyli f -j/3,) (viewed as ordered according to increasingmagnitude), then (s, t) must be contained in some element of Vyzi f-jflm . It follows that
n(n + 1) 2
i liYps
if
P = 1,
if
p>l,
whence h(am,f) < h(&,, ,f) < logp. Since the sequence{ana}zZ1is refining, we conclude (cf. [l]) that h(f) < logp. COROLLARY
2.3.
COROLLARY 2.4. h(f) < log degf.
Iff:
[u, b] --f [a, b] is 1-l.
then h(f)
= 0.
If f: [a, b] --t [a, b] is a nonconstantpolynomial, then
PYOO~. It follows from the fundamental theorem of algebra that f is at most (degf)-to-l. We turn now to a property enjoyed by the mapping S, (cf. (1.2)): if x, y E [0, ~1, then / S,(X) - S,(y)1 f v 1x - y 1. The relationship between this Lipschitz property and the fact that h(S) < log v can be generalized as follows. THEOREM 2.5. Let T be a continuousmapping of the compactmetric space (X, d) into itself. Supposethereis a constantK > 0 suchthat
W(x),
T(Y))
d Kd(x, y)
for all x, Y 6 X.
WEISS
820 Then (1) IfK
,< 1, h(T) = 0;
(2) If K > 1 and (X, d) is a subspace of iRs(with respectto the induced metric), then h(T) < s log K. The proof of (2.5) will require somepreliminaries. DEFINITION
2.6. For any E> 0, let
N,(X) = min{card 5 ) f is a cover of X (by arbitrary sets),diam 5 < 2~) and define HE(X) = log NE(X). H,(X) 1s . called the c-entropy of X (cf. [7, 81). THEOREM 2.7 (cf. [7, p. 3001). Jf (X, d) is a s&pace of FP, then there existsa constantC suchthat for all su$6cientlysmallpositiveE, N,(X) < C( 1I<>“. LEMMA
2.8. Suppose(X, d) is a subspace of W. Then,for any K > 1 and
P > 0, lim sup -fr H~a,6Kfl-I,(X) < Slog K. n-am Proof. Assume K > 1, By (2.7), there exists a (positive) constant C such that for all sufficiently large n, N (p,6p-l)(X) < c (7)“. The result follows. Proof of (2.5). For anyp > 0, let CQ,={A IACX,Aopen,diamA
[,={SISCX,diamS
821
TOPOLOGICAL ENTROPY
To prove (1) of (2.5), assumeK < 1, and observe that (p/3) < (p/3Kj) for eachj = 0, l,..., 71- 1. Hence,
so that
Therefore,
Since p was arbitrary, it follows that h( 7’) = 0. To prove (2), assumethat (X, d) is a subspaceof R?and K > 1. Note that (p/3Kn-l) < (p/3Kj) for each j = 0, I ,..., n - 1. Hence, n-1 rv/3Kj,
<
&,3K”-‘,
’
so that and
IV (:"' T-$,) j=O
<
Ntp,3Kn-1j(X).
Hence, by (2.8), h(yl PT) < lim sup $ HCP,GK,+zr(X)< s log K. n+m Since p was arbitrary, we conclude that h(T) < s log K. We remark that results analogousto (2.5), but with somewhat different hypotheses, appeared in [3, 61 subsequent to completion of the results reported in [lo]. COROLLARY 2.9. If
f: [a, b] -+ [a, b] is diSferentiubZe,then h(f)
< max{O, log SUP If’
I).
Proof. Apply the mean value theorem to (2.5). ACKNOWLEDGMENT The author wishes to thank Professor Yuji Ito of Brown University for supervising the research which led to this work.
822
WEISS REFERENCES
1. R. ADLER, A. KONHEIM, AND M. MCANDREW, Math. sm. 114 (1965), 309-319. 2. R. ADLER AND M. MCANDREW, The entropy Amer. Math. Sot. 121 (1966), 236-241. 3. R. BOWEN, Entropy for group endomorphisms Amer. Math. Sot. 153 (1971), 401-414. 4. R. BOWEN, Topological entropy and Axiom A, Symposia in Pure Mathematics, (S.-S. Chern pp. 23-41, Amer. Math. Sot., Providence, RI, 5. L. GOODWYN, Topological entropy bounds
Topological of Chebyshev
entropy,
Trans. Amer.
polynomials,
and homogeneous
spaces,
Trans. Trans.
in “Global Analysis,” Proceedings and S. Smale, Eds.), Vol. XIV, 1970. measure-theoretic entropy, Proc.
Amer. Math. Sot. 23 (1969), 679-688. 6. S. ITO, An estimate from above for the entropy and the topological entropy of a Cl-diffeomorphism, Proc. Japan Acad. 46 (1970), 226-230. 7. A. KOLMOGOROV AND V. TIHOMIROV, r-entropy and e-capacity of sets in functional spaces, Amer. Math. Sot. Transl. 17 (1961), 277-364. 8. G. LORENTZ, Metric entropy, widths, and superpositions of functions, Amer.
Math. Monthly 69 (1962), 469-485. 9. V. ROHLIN, Lectures on the entropy theory Russian Math. Surveys 22 (1967), l-52. 10. M. WEISS, Restricted ergodic theory, Ph.D. 11. M. WEISS, Restricted
topological Dissertation, topological
Math. Sot. 17 (1970), 973.
entropy Brown entropy
of measure-preserving
transformations,
and its application to problems in University, June, 1970. and real-mappings, Notices Amer.
OF MATHEMATICAL
Topological
ANALYSIS
AND
APPLICATIONS
816-822 (1973)
Entropy of Chebyshev Polynomials and Related Real Mappings MICHAEL
Department
43,
of Mathematics,
Wayne
D. WEISS
State
Submitted
University,
Detroit,
Michigan
48202
by Ky Fan
Topological entropy was originally defined in [l]. If 01and ,8 are open covers of a compact space X, let OLv /3 = {A n B 1A E 01and B E,8}, and define N(a) asthe cardinality of a subcover of (II of minimal cardinality. If T is a continuous self mapping of X, lim,,,(l /KZ)log iV(Vyz,r Th) can be shown to exist and is denoted h(a, T). The topological entropy of T, h(T), is defined assup h(or, T), where the supremum is taken over all open covers OL of X. A basic property of topological entropy is that it is an invariant of topologically conjugate mappings. Using intricate methods, Adler and McAndrew proved in [2] that the topological entropy of the vth Chebyshev polynomial is log V, and Bowen [4] generalizedthis result. In Section 1, we use a theorem of Goodwyn to give a short and natural proof of the theorem of Adler and McAndrew. In Section 2, we consider classesof real mappings which share certain properties of Chebyshev polynomials and obtain upper bounds for the entropy of “at mostp-to-l” and Lipschitzian mappings. The notation and terminology of this paper will follow that of [l], with the exception that arbitrary covers of a compact spacewill occasionally be usedin place of open covers. This paper is basedon a portion of the author’s doctoral dissertation [lo], completed and distributed in 1968-1969. [I I] refers to the results contained herein.
1.
ENTROPY
OF
CHEBYSHEV
POLYNOMIALS
LEMMA 1.1. Let (X, d) be a compactmetric space,01a finite cover of X (by arbitrary sets),and T: X -+ X a continuousmap. If diam(v~~~ T-h) + 0 as n -+ co, then h(T) < h(or,T). Copyright All rights
@ 1973 by Academic Press, Inc. of reproduction in any form reserved.
816
TOPOLOGICAL
817
ENTROPY
Proof. Given any open cover /3 of X, choose n so llarge that diarn(V~~~ T-%x) is lessthan the Lebesguenumber of /3. Then /3< Vyi, T-%, so that l@,T)
1.2. h( TV) = log V.
Proof. Let S, = cos-l o TVo cos on [0, ~1. Clearly h(T,) = h(S,). Note that
whenever kc+ < x < (k + 1) n/v,
k E(0, l,..., v - l}.
We shall prove h(S,) = log V. Since S, is merely the identity, we may assumev > 1. Define a cover cxof [0, 7r] by 01= ([h/v, (k + 1) VT/v)j k E(0, l)...) v - 2)) u ([(v - 1) z-/v, Tr]}. We can easily verify by induction on n that, for any n E kJ, VyL, SLOOP consists of precisely v” intervals Ij , where Ij has endpoints jn/vn and (j + 1) T/P, j = 0, l,..., v%- 1. Hence, diarn(vIli Syicl) + 0 as n-+ 00, so that, by U.l), h(S,) < h(a, S,) = li+i(log v”)/n = log V. To obtain the reverse inequality, we observe that S, preservesnormalized Lebesguemeasurem and employ the measure-theoreticentropy h, (cf. [9]). Since Vz, Srior is the partition of [0, ~1 into points, it follows from [9, Proposition 9.41 that h,(S,) = &(a, S,) = lii$log v”)/n = log v. However, by Goodwyn’s bound theorem [5], h(S,) > h,(S,). h(S”) = log v.
Hence,
WEISS
818
2. SOME GENERALIZATIONS DEFINITION
2.1.
A function j is at mostp-to-l if card j-l(y)
< p when-
ever y E rangej. The Chebyshev polynomial TV is at most v-to-l. In fact, its graph risesand falls exactly Y times. Furthermore, TV satisfiesh(T,) < log v. This leads us to the following. THEOREM 2.2. Supposej is a continuousmapping of the real interval [a, b] into itself. If j is at mostp-to-l, thm h( j) < log p.
Proof. Choose an arbitrary relatively open interval UC [a, b]. We first show that (1) j-l U is a union of p + 1 or fewer disjoint relatively open intervals; and (2) if either j(a) $ U or j(b) $ U, then f-'U disjoint relatively open intervals.
is a union of p or fewer
We can assumeU n rangej is an interval of positive length. Let us call any point t exceptional if t = j(7) for some r E [a, b] which is either a relative maximum or a relative minimum off; otherwise, call t unexceptional.From the fact that j is at most p-to-l, it follows that there are at most countably many exceptional points, so that the set of unexceptional points is densein [a, b]. Hence, we can chooseopen intervals Er C E, C *.. whose endpoints are unexceptional and whose union is the interior of U n range j. For arbitrary k, f -lEk can be representedasa disjoint union .f-lE,
=F
6
fi (xi icn
, Yi),
whereF C (a, b) (possiblyF = m) and xi < yi for eachi E A. Let Ek = (x, y). Since x and y are unexceptional, xi # yj for all i, j E (1. Careful consideration of card j-l{x, y} now reveals that (1) and (2) hold for Ek (with Ek in place of U), from which it follows that they hold for U itself. Using induction, we next prove that, for any n E N, j-*U is a union of or fewer disjoint relatively open intervals. By (l), 1 +p+p2+ “‘i-p” this is true for n = 1. Supposethat j-“U is a union j+lJ = (JE, Vi of or fewer disjoint relatively open intervals U, . Certainly, at l+p+...+pm mostone of theseintervals containsboth j(a) andj(b). ChooseiOED suchthat i E9, i # i,, impliesj(a) $ Ua or j(b) $ U, . By (I), j-lUi, is a union of p + 1 or fewer disjoint relatively open intervals. By (2), for i # i,, , j-lU, is a union of p or fewer disjoint relatively open intervals. Hence, j-crn+l)U is a
TOPOLOGICAL
819
ENTROPY
unionof(p+ l)+p(cardQ1) or f ewer disjoint relatively open intervals. But card 52 < 1 + p + ... + pm; hence,
This completes the induction. For an arbitrary m E IV, let 01, be the set of relatively open intervals in [a, b] with length not exceeding l/m, and let /6&,be a finite subcover of (Y, . If S is a subsetof [a, b], we let X3 be the boundary of S, and if 5 is a cover of [a, 61,we define at = USECAS. Now, for each71E N, n-l
j
card af -+Im < 2(card &) 1 C pk. j=O
k=O
Ifs and t are two consecutive points from the finite set a(Vyli f -j/3,) (viewed as ordered according to increasingmagnitude), then (s, t) must be contained in some element of Vyzi f-jflm . It follows that
n(n + 1) 2
i liYps
if
P = 1,
if
p>l,
whence h(am,f) < h(&,, ,f) < logp. Since the sequence{ana}zZ1is refining, we conclude (cf. [l]) that h(f) < logp. COROLLARY
2.3.
COROLLARY 2.4. h(f) < log degf.
Iff:
[u, b] --f [a, b] is 1-l.
then h(f)
= 0.
If f: [a, b] --t [a, b] is a nonconstantpolynomial, then
PYOO~. It follows from the fundamental theorem of algebra that f is at most (degf)-to-l. We turn now to a property enjoyed by the mapping S, (cf. (1.2)): if x, y E [0, ~1, then / S,(X) - S,(y)1 f v 1x - y 1. The relationship between this Lipschitz property and the fact that h(S) < log v can be generalized as follows. THEOREM 2.5. Let T be a continuousmapping of the compactmetric space (X, d) into itself. Supposethereis a constantK > 0 suchthat
W(x),
T(Y))
d Kd(x, y)
for all x, Y 6 X.
WEISS
820 Then (1) IfK
,< 1, h(T) = 0;
(2) If K > 1 and (X, d) is a subspace of iRs(with respectto the induced metric), then h(T) < s log K. The proof of (2.5) will require somepreliminaries. DEFINITION
2.6. For any E> 0, let
N,(X) = min{card 5 ) f is a cover of X (by arbitrary sets),diam 5 < 2~) and define HE(X) = log NE(X). H,(X) 1s . called the c-entropy of X (cf. [7, 81). THEOREM 2.7 (cf. [7, p. 3001). Jf (X, d) is a s&pace of FP, then there existsa constantC suchthat for all su$6cientlysmallpositiveE, N,(X) < C( 1I<>“. LEMMA
2.8. Suppose(X, d) is a subspace of W. Then,for any K > 1 and
P > 0, lim sup -fr H~a,6Kfl-I,(X) < Slog K. n-am Proof. Assume K > 1, By (2.7), there exists a (positive) constant C such that for all sufficiently large n, N (p,6p-l)(X) < c (7)“. The result follows. Proof of (2.5). For anyp > 0, let CQ,={A IACX,Aopen,diamA
[,={SISCX,diamS
821
TOPOLOGICAL ENTROPY
To prove (1) of (2.5), assumeK < 1, and observe that (p/3) < (p/3Kj) for eachj = 0, l,..., 71- 1. Hence,
so that
Therefore,
Since p was arbitrary, it follows that h( 7’) = 0. To prove (2), assumethat (X, d) is a subspaceof R?and K > 1. Note that (p/3Kn-l) < (p/3Kj) for each j = 0, I ,..., n - 1. Hence, n-1 rv/3Kj,
<
&,3K”-‘,
’
so that and
IV (:"' T-$,) j=O
<
Ntp,3Kn-1j(X).
Hence, by (2.8), h(yl PT) < lim sup $ HCP,GK,+zr(X)< s log K. n+m Since p was arbitrary, we conclude that h(T) < s log K. We remark that results analogousto (2.5), but with somewhat different hypotheses, appeared in [3, 61 subsequent to completion of the results reported in [lo]. COROLLARY 2.9. If
f: [a, b] -+ [a, b] is diSferentiubZe,then h(f)
< max{O, log SUP If’
I).
Proof. Apply the mean value theorem to (2.5). ACKNOWLEDGMENT The author wishes to thank Professor Yuji Ito of Brown University for supervising the research which led to this work.
822
WEISS REFERENCES
1. R. ADLER, A. KONHEIM, AND M. MCANDREW, Math. sm. 114 (1965), 309-319. 2. R. ADLER AND M. MCANDREW, The entropy Amer. Math. Sot. 121 (1966), 236-241. 3. R. BOWEN, Entropy for group endomorphisms Amer. Math. Sot. 153 (1971), 401-414. 4. R. BOWEN, Topological entropy and Axiom A, Symposia in Pure Mathematics, (S.-S. Chern pp. 23-41, Amer. Math. Sot., Providence, RI, 5. L. GOODWYN, Topological entropy bounds
Topological of Chebyshev
entropy,
Trans. Amer.
polynomials,
and homogeneous
spaces,
Trans. Trans.
in “Global Analysis,” Proceedings and S. Smale, Eds.), Vol. XIV, 1970. measure-theoretic entropy, Proc.
Amer. Math. Sot. 23 (1969), 679-688. 6. S. ITO, An estimate from above for the entropy and the topological entropy of a Cl-diffeomorphism, Proc. Japan Acad. 46 (1970), 226-230. 7. A. KOLMOGOROV AND V. TIHOMIROV, r-entropy and e-capacity of sets in functional spaces, Amer. Math. Sot. Transl. 17 (1961), 277-364. 8. G. LORENTZ, Metric entropy, widths, and superpositions of functions, Amer.
Math. Monthly 69 (1962), 469-485. 9. V. ROHLIN, Lectures on the entropy theory Russian Math. Surveys 22 (1967), l-52. 10. M. WEISS, Restricted ergodic theory, Ph.D. 11. M. WEISS, Restricted
topological Dissertation, topological
Math. Sot. 17 (1970), 973.
entropy Brown entropy
of measure-preserving
transformations,
and its application to problems in University, June, 1970. and real-mappings, Notices Amer.