Explicit forms for and some functional analysis behind a family of multidimensional continued fractions triangle partition maps and their associated transfer operators

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Signed (student author) Signatures redacted Signed (faculty advisor) Signed (2d advisor, if a p p l i c a b l e ) - - - - - - - - - - - - - - - - - - t;: l< i>l•'c i-l Fo riv'S .j-c ,Ana ly S (.s 'B d1ih li'( ct Thesis title ~c 111lly ot M~:il±;rl(~k _,,.,. : P"rt,·tu;.r M,.psDate JZ /I ~,./ T~"[...]... original TRIP map in [4] (the “Banach space approach” and the “Hilbert space approach”) to show that the transfer operators corresponding to many additional TRIP maps are also nuclear of trace class zero or possess spectral gaps In particular, we have used the Banach space approach to show that several additional TRIP maps have spectral gaps We have also used the Hilbert space approach to show that all transfer. .. generated by each of the TRIP maps to help visualize the action of the maps on The partition diagrams for all 216 maps are presented in Appendix F while the TRIP diagrams are presented in Appendix G Several similar diagrams had already been constructed in [5] and [2], but here we present diagrams for all 216 TRIP maps 5.1 Sample Partition and TRIP Diagram Calculation Let us explain how to arrive at... original input pair to its k th iterate 25 Chapter 5 Partition Diagrams and TRIP Diagrams Define a partition diagram as a visual representation of { ∞ k }k=0 induced on by a TRIP map Given a TRIP sequence (a1 , a2 , ) induced by some TRIP map, define a TRIP a a diagram as a visual representation of triangles of the form BF1 1 F0 F1 2 F0 It is helpful to have both the partition and TRIP diagrams... Schweiger’s Multidimensional Continued Fractions [16] A particular family of multidimensional continued fraction algorithms – TRIP maps – has been used to construct maps such that a number being a cubic irrational (real and algebraic of degree 3) corresponds to a certain kind of periodicity under those maps [3] This thesis will explore the functional analysis behind this family of multidimensional continued fractions. .. functional analysis behind these TRIP maps, discussing their explicit form and ergodic properties – as well as the form, spectrum, and nuclearity of the associated transfer operators 1.8 Polynomial- and Non-Polynomial-Growth TRIP Maps We see that the transfer operator corresponding to Te,e,e has a particularly nice form, in that the denominator of the factor 1 (1+kx+y)3 is (non-trivially) polynomial... Te,e,e had already been calculated in [5], but here we present the explicit forms associated with all 216 TRIP maps 6.1 Sample Transfer Operator Calculation While the calculation of the explicit form of any non-polynomial-growth transfer operators is incredibly involved, the calculation for certain polynomial-growth transfer operators is manageable We present one such polynomial-growth example below, calculating... found in Chapter 17 19 Chapter 2 Explicit Form of TRIP Maps 3 The explicit form of Tσ,τ0 ,τ1 (x, y) has been calculated for all (σ, τ0 , τ1 ) ∈ S3 These explicit forms are presented in Appendix A Several explicit forms had already been calculated in [5] and [2], but here we present all 216 explicit forms 2.1 Sample TRIP Map Calculation We will go through a sample calculation of the from of Te,23,e... transfer operators for which we have found corresponding eigenfunctions of eigenvalue 1 are nuclear of trace class zero The Banach space approach involves finding an appropriate Banach space V on which LT acts, showing that LT is a linear map from V to V, and showing that the largest eigenvalue of LT is 1 and has multiplicity 1 To get at the Hilbert space approach, we consider a transfer operator related... detail; for n = 6, we obtain the TRIP diagram shown below: 28 Chapter 6 Explicit Form of all Transfer Operators LTσ,τ ,τ 0 1 We have calculated the explicit form of 1 LTσ,τ0 ,τ1 f (x, y) = (a, b):Tσ,τ0 ,τ1 (a, b)=(x,y) |Jac(Tσ,τ0 ,τ1 (a, b))| f (a, b) 3 for all (σ, τ0 , τ1 ) ∈ S3 The forms of |Jac(Tσ,τ0 ,τ1 (a, b))| and LTσ,τ0 ,τ1 f (x, y) are presented in Appendices D and E The explicit forms associated. .. Sequences and TRIP Tree Sequences The application of F0 and F1 to (the R3 representation of) any triangle in (the R3 representation of) can subdivide partitions it into two triangles Hence, instead of using A0 and A1 , we using F0 and F1 Using these F0 and F1 , we can define 216 triangle partition 3 maps (TRIP maps, for short), each for one of the 216 permutation triplets in S3 Let us make this definition . ' . ·, ' (. re .;Mar9h 1010 Signatures redacted Explicit Forms for And Some Functional Analysis Behind A Family of Multidimensional Continued Fractions – Triangle Partition Maps. Polynomial-Growth in Combo TRIP Maps 42 10 Origin of Partition Geometry 47 11 Functional Analysis Behind Transfer Operators: Banach Space Approach 53 11.1 Transfer Operators as Linear Maps on Appropriate. spectral gaps. In particular, we have used the Banach space approach to show that several additional TRIP maps have spectral gaps. We have also used the Hilbert space approach to show that all transfer