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by: Maverick Koelpin


Maverick Koelpin
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This 16 page Class Notes was uploaded by Maverick Koelpin on Tuesday October 13, 2015. The Class Notes belongs to MATH 2030 at Louisiana State University taught by Staff in Fall. Since its upload, it has received 15 views. For similar materials see /class/222643/math-2030-louisiana-state-university in Mathematics (M) at Louisiana State University.

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Date Created: 10/13/15
Metrized Sequence Spaces Neal Stoltzl us3 3Louisiana State University Lectures Math 2030 a Sequence Space 0 Definitions 0 Metric Spaces 0 Shift Map 0 Conjugacy Theorem Definition The sequence space on two symbols 0 1 is the set of all tuntions F N 0 123 a 01 The space is often denoted 0 1N or 2N We will write sequences in the form 30313233 e 9 010101 000 111 101 001 0001 00001 000001000000 Distance Definition The distance between two sequences 5 30313233 and t 10111213 is the series 00 I 7 t ds t Z 0 Note that the diameter largest distance in sequence space is d000011111121412 1 17 MA Metric Space Definition A metric space is a set X together with a function d X x X a R satisfying 0 Positivity dX y 2 O and dX y 0 it and only if X y 9 Symmetry dX7y dy X 0 Triangular inequality dX 2 g dX y dyz Proximity Theorem Lets st ti 6 X be sequences lfs t for i 0 12 n then ds t g 217 Conversely ifds t lt 217 then s t fori g n The shImeap a X a X is defined on sequence by a s si 13233 dropping so and shifting the remaining symbols to the left as SH The fixed points are s O and s 1 the allzero and allone sequences Note that as s 1 implies ts1s39henceso 3132 A periodic point of not necessarily least period n must have 34 s for all i therefore 3 sin si2n si3n si4n w 3 3mm for all positive integers k Only the first 17 symbols so 31 s1 arbitrary and each can be only 0 or 1 Hence there are 2quot points of period 17 two choices for so times two choices for s1 Metric Spaces Definition Afunction F X dX a Y dy is continuous at X0 6 X provided Given a positive real number s gt 0 there is a number 6 gt O with the property if X e X satisfies 0 lt dXX0X lt 6 then dyFXo FX lt e Note that the definition of continuity for functions between two metric spaces eg sequence spaces generalizes the calculus definition where the real numbers R are provided with the distance da b lai bl Continuity of the Shift The shift map is continuous on sequence space 2 Proof To show continuity of a at a sequence x we must demonstrate the following Given a positive real number s gt O we must produce a number 6 gt O with the property if 0 lt dt X lt 6then daxat lt e First we find an integer nwith 217 lt e and define 6 21 Now if t e X and dtx lt 6 by Proximity the first n 2 symbols of t and x agree After the shift the first n 1 symbols of at and ax agree Again by Proximity daxat lt 217 lt e gacy of Shift and the Quadratic Map If c lt 75432 then there is a homeomorphism biective continuous With continuous inverse S A a X With equality of the following compositions S o 00X o o 8X for all X e A In words we say that the shift map on sequence space is conjugate via 8 which we will call the itinerary function to iteration of the quadratic map 00 on the nonwandering set A Definitio of the itinerary function 8 Recall the definition of the interval A1 C 7p7p where p is the repelling fixed point for 00 c lt 72 A1 r e 7ppl00r lt 7p See the Mathematica notebook for illustrations The interval A1 contains zero and is symmetric about the origin Why The complement of A1 in 7p7p consists of two disjoint intervals 0 1 with 0 on the left to be specific Definition The itinerary map 8X 30313233 is defined by the condition sn 0 if and only if 03X e lo and similarly sn 1 if and only if 03X 6 1 The biggest maximal under inclusion domain of S is the subset r E 7ppngr e IO U 1 n 012 Proof of the Co J gacy Theorem Proof We must show the following o The map 8 to sequence space X is onetoone o The map 8 to sequence space X is onto 0 The map 8 to sequence space X is continuous o The inverse of the map Sis continuous For the first property Suppose 8X 8y that is Q X 02y are both in the same interval 0 or 1 for all i In particular for i O X and y are both in the same interval Now the restriction of 00 to either interval is onetoone and the hypothesis on 0 implies the derivativeQ cX gt u gt 1 for some constant In gt 1 Consider the interval X y C ISO the length of 00X Qcy gt u lengthX y ply 7 xi but the fact that the derivative stretches length by the factor p Upon iteration ngX 7 Qgyl gt pnly 7 xi This can hold for all 17 only it ly 7 xi O and X y because limnaoopquot 00 and 00X Qcy C 7pp atinite length interval Proof of Second Property To show Sis onto we need to find a point in IO U 1 whose itinerary under 00 is the given sequence 3 Define the sequence of intervals st1Sn X E E ISMquot 0717 quot 397n 50 Qll51sn39 This gives a sequence of nonempty closed intervals since 00 maps any interval not containing zero bijectively to another such interval The sequence is nested because 50515n sgs1msn1 ogl ISn C 5051sn1 By the basic fact that the intersection of any nested family of nonempty closed intervals of real numbers is nonempty a basic property equivalent to completeness we have shown that Sis onto Proof of Third Property The verification of the continuity of 8 uses the Proximity theorem in a manner reminiscent of the proof of continuity of the shift map 0 Given 8X 031 and e gt 0 choose nwith 217 lt e Considerthe 2quot1 disjoint intervals mm for all possible length n 1 sequences 1011 In Choose 6 gt O smallest than the minimum distance between the endpoints of these intervals If two points in A are in one of these intervals and distance lt 6 then they are in the same such interval So if ix iyl lt 6 then X and y are both in the same SOSWWSH and by the proximity theorem dSX Sy lt 217 lt e The proof of the continuity of 8 1 is similar Metric Space ivalence Shift and Quadratic map The sequence map 8 is a continuous bijection from the nonwandering set A to sequence space X with continuous inverse which conjugates iteration of 00 with the shift map Therefore any property depending only on the metric space structure of one dynamical system is also a property of the other


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