Pure Point Spectrum on Fractals and Related Questions

Pure point spectrum on fractals and related questions

Alexander Teplyaev part of a joint work with Patricia Alonso-Ruiz, Toni Brzoska, Joe Chen, Michael Hinz, Stanislav Molchanov, Luke Rogers, Rodrigo Trevino et al.

Fractal Geometry and Stochastics 6 30 Sep - 5 Oct 2018, Bad Herrenalb, Germany

October 1, 2018 ∗ FGS6

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 1 / 28 outline: 1. Algebraic applications: spectrum of the Laplacian on the Basilica Julia set (with Rogers, Brzoska, George, Jarvis et al. (research in progress)). 2. Introduction and motivation: Mathematical Physics and spectral analysis on fractals:

I Weak Uncertainty Principle (Okoudjou, Saloﬀ-Coste, Strichartz, T., 2008)

I Laplacians on fractals with spectral gaps gaps have nicer Fourier series (Strichartz, 2005)

I Bohr asymptotics on inﬁnite Sierpinski gasket (with Chen, Molchanov, 2015).

I Singularly continuous spectrum of a self-similar Laplacian on the half-line (with Chen, 2016). 3. Dynamical systems: canonical diﬀusions on the pattern spaces of aperiodic Delone sets (with Alonso-Ruiz, Hinz, Trevi˜no,2018). This is a part of the broader program to develop probabilistic, spectral and vector analysis on singular spaces by carefully building approximations by graphs or manifolds.

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 2 / 28 Asymptotic aspects of Schreier graphs and Hanoi Towers groups Rostislav Grigorchuk 1, Zoran Suniˇ k´ Department of Mathematics, Texas A&M University, MS-3368, College Station, TX, 77843-3368, USA

Received 23 January, 2006; accepted after revision +++++

Presented by Etienne´ Ghys

Abstract

We present relations between growth, growth of diameters and the rate of vanishing of the spectral gap in Schreier graphs of automaton groups. In particular, we introduce a series of examples, called Hanoi Towers groups since they model the well known Hanoi Towers Problem, that illustrate some of the possible types of behavior. To cite this article: R. Grigorchuk, Z. Suniˇ k,´ C. R. Acad. Sci. Paris, Ser. I 344 (2006). b

111 a c b

c 011 211 a

;

; 021 201

b a c b

´ ¼ ¾ µ

¼ ¿

´ ¼ ¿ µ ¼ ¾ c b a 221 001

121 101

a c

¼ ½

½ ¾

220 002 i

d c b b a

´ ½ ¾ µ

´ ¼ ½ µ

;

a c

b 120 020 c a 202 102 b

100 010 212 122

½ ¿ ¾

¿ a c b a c b a c

´ ´ ¾ ½ ¿ ¿ µ µ b a c b a c b

c 000 200 210 110 112 012 022 222 a

;

Figure 1. The automaton generating H(4) and the Schreier graph of H(3) at level 3 / L’automate engendrant H(4) et le graphe de Schreier de H(3) au niveau 3 Asymptotic aspects of Schreier graphs and Hanoi Towers groups Rostislav Grigorchuk 1, Zoran Suniˇ k´ Department of Mathematics, Texas A&M University, MS-3368, College Station, TX, 77843-3368, USA

Received 23 January, 2006; accepted after revision +++++

Presented by Etienne´ Ghys

Abstract

111 a c b

c 011 211 a

;

; 021 201

b a c b

´ ¼ ¾ µ

¼ ¿

´ ¼ ¿ µ ¼ ¾ c b a 221 001

121 101

a c

¼ ½

½ ¾

220 002 i

d c b b a

´ ½ ¾ µ

´ ¼ ½ µ

;

a c

b 120 020 c a 202 102 b

100 010 212 122

½ ¿ ¾

¿ a c b a c b a c

´ ´ ¾ ½ ¿ ¿ µ µ b a c b a c b

c 000 200 210 110 112 012 022 222 a

;

Figure 1. The automaton generating H(4) and the Schreier graph of H(3) at level 3 / L’automate engendrant H(4) et le graphe de Schreier de H(3) au niveau 3 Spectral analysis on fractals

A part of an inﬁnite Sierpi´nskigasket.

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 3 / 28 6 6 5

0 -

Figure: An illustration to the computation of the spectrum on the inﬁnite Sierpi´nskigasket. The curved lines show the graph of the function R(·).

Theorem (Rammal, Toulouse 1983, Béllissard 1988, Fukushima, Shima 1991, T. 1998, Quint 2009) On the infinite Sierpińskigasket the spectrum of the Laplacian consists of a dense −1 set of eigenvalues R (Σ0) of infinite multiplicity and a singularly −1 continuous component of spectral multiplicity one supported on R (JR ).

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 4 / 28 Part 1: Spectral Analysis of the Basilica Graphs (with Luke Rogers, Toni Brzoska, Courtney George, Samantha Jarvis et al.)

The question of existence of groups with intermediate growth, i.e. subexponential but not polynomial, was asked by Milnor in 1968 and answered in the positive by Grigorchuk in 1984. There are still open questions in this area, and a complete picture of which orders of growth are possible, and which are not, is missing. The Basilica group is a group generated by a ﬁnite automation acting on the binary tree in a self-similar fashion, introduced by R. Grigorchuk and A. Zuk in 2002, does not belong to the closure of the set of groups of subexponential growth under the operations of group extension and direct limit. In 2005 L. Bartholdi and B. Virag further showed it to be amenable, making the Basilica group the 1st example of an amenable but not subexponentially amenable group (also “M¨unchhausentrick” and amenability of self-similar groups by V.A. Kaimanovich).

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 5 / 28 The Basilica fractal is the Julia set of the polynomial z 2 − 1. In 2005, V. Nekrashevych described the group as the iterated monodromy group, and there exists a natural way to associate it to the Basilica fractal (Nekrashevych+T., 2008). In Schreier graphs of the Basilica group (2010), Nagnibeda et al. classified up to isomorphism all possible limits of finite Schreier graphs of the Basilica group. In Laplacians on the Basilica Julia set (2010), L. Rogers+T. constructed Dirichlet forms and the corresponding Laplacians on the Basilica fractal in two different ways: by imposing a self-similar harmonic structure and a graph-directed self-simliar structure on the fractal. In 2012-2015, Dong, Flock, Molitor, Ott, Spicer, Totari and Strichartz provided numerical techniques to approximate eigenvalues and eigenfunctions on families of Laplacians on the Julia sets of z 2 + c.

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 6 / 28 Spectral Analysis of the Basilica Graphs

Basilica Julia Set and the Schreier graph Γ4

pictures taken from paper by Nagnibeda et. al. Antoni Brzoska Spectral Properties of the Hata Tree March 28, 2017 39 / 71 Spectral Analysis of the Basilica Graphs

Replacement Rule and the Graphs Gn

Antoni Brzoska Spectral Properties of the Hata Tree March 28, 2017 41 / 71 Spectral Analysis of the Basilica Graphs Distribution of Eigenvalues, Level 13

Antoni Brzoska Spectral Properties of the Hata Tree March 28, 2017 44 / 71 Spectral Analysis of the Basilica Graphs

One can deﬁne a Dirichlet to Neumann map for the two boundary points of the graphs Gn. One can construct a dynamical system to determine these maps (which are two by two matrices). The dynamical system allows us to prove the following. Theorem In the Hausdorﬀ metric, lim sup σ(L(n)) has a gap that contains the n interval (2.5, 2.8). →∞

Conjecture In the Hausdorﬀ metric, lim sup σ(L(n)) has inﬁnitely many gaps. n →∞ Proving the conjecture would be interesting. One would be able to apply the results discovered by R. Strichartz in Laplacians on Fractals with Spectral Gaps have nicer Fourier Series (2005).

Antoni Brzoska Spectral Properties of the Hata Tree March 28, 2017 45 / 71 Spectral Analysis of the Basilica Graphs

Inﬁnite Blow-ups of Gn

Deﬁnition

Let kn n N be a strictly increasing subsequence of the natural numbers. { } ∈ For each n, embed Gkn in some isomorphic subgraph of Gkn+1 . The corresponding infinite blow-up is G := n 0Gk . ∞ ∪ ≥ n Assumption The infinite blow-up G satisfies: ∞ For n 1, the long path of Gkn 1 is embedded in a loop γn of Gkn . ≥ − Apart from lkn 1 and rkn 1 , no vertex of the long path can be the − − 3, 6, 9 or 12 o’clock vertex of γn.

The only vertices of Gkn that connect to vertices outside the graph

are the boundary vertices of Gkn .

Antoni Brzoska Spectral Properties of the Hata Tree March 28, 2017 46 / 71 Spectral Analysis of the Basilica Graphs

Antoni Brzoska Spectral Properties of the Hata Tree March 28, 2017 47 / 71 Spectral Analysis of the Basilica Graphs

Theorem

(kn) (jn) (1) σ(L `2 ) = σ(L0 ). | a,kn,γn ( ) ( ) (2) The spectrum of L ∞ is pure point. The set of eigenvalues of L ∞ is

(jn) 1 σ(L ) = c− 0 , 0 jn { } n 0 n 0 [≥ [≥ (n) where the polynomials cn are the characteristic polynomials of L0 , as deﬁned in the previous proposition. ( ) (3) Moreover, the set of eigenfunctions of L ∞ with ﬁnite support is complete in `2.

Antoni Brzoska Spectral Properties of the Hata Tree March 28, 2017 48 / 71 The geometry behind the pure point spectrum on fractals (use local symmetries:-) Malozemov-T. 1995 Barlow-Kigami 1997 Kigami 1998 T. 1998 Sabot 2000 Kajino 2013-2014 (heat kernel oscillations) Hauser 2018

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 7 / 28 Malozemov-T. 1995

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 8 / 28 0 2 Lemma 6.2: ( ,i, ,i) is a regular Dirichlet form on L (K, µη ˜ ˜ ) with dis- R Ki Vi E DR 0 0 | \ crete non negative spectrum and for u, v ,1 ,6 we have ∈ DR ⊕ · · · ⊕ DR 6 (u, v) = ,i(u ˜ , v ˜ ) ER ER |Ki |Ki i=1 By σ we denote the 120◦ rotation and by τ the reﬂection at the bisecting line trough X p1 as you can see in Figure 6. 0 Proof : From [18, Theorem 10.3] [11, Theorem 4.4.3] we know that ( ,i, ,i) is a ER DR regular Dirichlet form on L2(K, µ ). The spectrum is discrete since 0 η K˜ i V˜i ,i | \ 0 DR ⊂ (see [8, Theo. 4, Chap. 10]). The ,i are orthogonal to each other with DR DR 2 respect to as well as the inner product of L (K, µη). Therefore we have the desired equality.ER

0 Let ϕ be any eigenfunction of ( ,1, ,1) with measure µη and η (0, 1] with eigenvalue λ. We can use this ϕ toER constructDR a Dirichlet-Neumann eigenfunction∈ on SSG. Byϕ ˜ we denote the reﬂection of ϕ along the bisecting line trough p1. I.e. ϕ˜ = ϕ τ. We glue these functions together in the following fashion which you can see in Figure◦ 8.

ϕ1 := ϕ ϕ := ϕ˜ 2 − ϕ := ϕ σ2 3 ◦ ϕ := ϕ˜ σ2 4 − ◦ ϕ := ϕ σ 5 ◦ ϕ := ϕ˜ σ Figure 6: Symmetries of the SSG. 6 − ◦

We divide K into six parts in the following manner illustrated in Figure 7.

Figure 7: Dividing the SSG into six parts. Figure 8: Gluing ϕ.

˜ ˜ ˜ ˜ 6 The individual parts are denoted by Ki and the intersectionsWe denoteVi the:= K functioni (Ki+1 on K as Φ := i=1 ϕi. Thanks to symmetry ϕi is ˜ ˜ ∩ ˜ 0 ∪ Ki 1) where i isElias taken Hauser modulo 6. 2018 That (based means V onani is Utaeigenfunction the intersection Freiberg of (andof K,ii, Patriciawith,i) with Alonso-Ruiz measure µη K˜ 2012-2018)and eigenvalue λ. And the − ˜ ER DR P | i the bisecting lines indicated in Figure 7. On Vi weDirichlet introduce conditions Dirichlet and boundary Lemma 6.2 ensure that Φ 0 . We now want to show, conditions. There are countably inﬁnitely many of those points and all but one lie ∈ DR that Φ itself is a Dirichlet-Neumann eigenfunction i.e. (Φ, v) = λ(Φ, v)µη for all in the middle of one-dimensional lines. ER

0 ˜ ,i := u : u V˜ 0, supp(u) Ki DR { ∈ DR | i ≡ ⊂ } 17

Denote the parts of the quadratic forms by ,i := 0 0 . R R D ,i×D ,i Sasha Teplyaev (UConn) E PureE | pointR spectrumR on fractals and related questions October 1, 2018 ∗ FGS6 9 / 28

16 w w w TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT HH ¨ H ¨ TT H ¨¨ H ¨ TT H ¨ TT HH ¨ H ¨ TT H ¨¨ H¨ TT ¨ H TT ¨ HH ¨ H TT ¨¨ H ¨ H TT ¨ H TT ¨ HH ¨ H TT ¨¨ H ¨ H TT ¨ H TT ¨ HH ¨ H TT ¨¨ H ¨ HTT ¨ HTT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT 10 TT TT TT TT TT TT TT TT TT TT TT TT TT w w w TT w w w TT TT TT TT TT TT TT TT TT T. 1998, 2007: http://www.math.uconn.edu/TT ~teplyaev/gregynog/TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT TT

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 10 / 28 Part 2: Motivation coming from Mathematical Physics

6/14/2014 François Englert - Wikipedia, the free encyclopedia François Englert From Wikipedia, the free encyclopedia

François Baron Englert (French: [ɑ̃ɡlɛʁ]; born 6 November 1932) is a Belgian theoretical physicist and 2013 Nobel prize François Englert laureate (shared with Peter Higgs). He is Professor emeritus at the Université libre de Bruxelles (ULB) where he is member of the Service de Physique Théorique. He is also a Sackler Professor by Special Appointment in the School of Physics and Astronomy at Tel Aviv University and a member of the Institute for Quantum Studies at Chapman University in California. He was awarded the 2010 J. J. Sakurai Prize for Theoretical Particle Physics (with Gerry Guralnik, C. R. Hagen, Tom Kibble, Peter Higgs, and Robert Brout), the Wolf Prize in Physics in 2004 (with Brout and Higgs) and the High Energy and Particle Prize of the European Physical Society (with Brout and Higgs) in 1997 for the mechanism which unifies short and long range interactions by generating massive gauge vector bosons. He has made contributions in statistical physics, quantum field theory, cosmology, string theory and supergravity.[4] He is the recipient of the 2013 Prince of Asturias Award in technical and scientific research, François Englert in Israel, 2007 together with Peter Higgs and the CERN. Born 6 November 1932 [1] Englert was awarded the 2013 Nobel Prize in Physics, Etterbeek, Brussels, Belgium together with Peter Higgs for the discovery of the Higgs Nationality Belgian mechanism.[5] Fields Theoretical physics Institutions Université Libre de Bruxelles Tel Aviv University[2][3] Contents Alma mater Université Libre de Bruxelles Notable awards Francqui Prize (1982) Sasha1 Ea Teplyaevrly life (UConn) Pure point spectrum on fractalsWo andlf Pr relatedize in Ph questionsysics (2004) October 1, 2018 ∗ FGS6 11 / 28 Sakurai Prize (2010) 2 Academic career Nobel Prize in Physics (2013) 3 The Brout–Englert–Higgs–Guralnik–Hagen– Kibble mechanism[7] 4 Major awards 5 References 6 External links

Early life

http://en.wikipedia.org/wiki/Fran%C3%A7ois_Englert 1/5 Nuclear Physics B280 [FS 18] (1987) 147-180 North-Holland, Amsterdam

METRIC SPACE-TIME AS FIXED POINT OF THE RENORMALIZATION GROUP EQUATIONS ON FRACTAL STRUCTURES

F. ENGLERT, J.-M. FRI~REx and M. ROOMAN2 Physique Thkorique, C.P. 225, Universitb Libre de Bruxelles, 1050 Brussels, Belgium

Ph. SPINDEL Facultb des Sciences, Universitb de l'Etat it Mons, 7000 Mons, Belgium

Received 19 February 1986

We take a model of foamy space-time structure described by self-similar fractals. We study the propagation of a scalar field on such a background and we show that for almost any initial conditions the renormalization group equations lead to an effective highly symmetric metric at large scale.

1. Introduction

Quantum gravity presents a potential difficulty which persists in any unification program which incorporates gravity in the framework of a local field theory in dimensions d > 4. In all such theories a local O(d-1,1) space-time symmetry is quite generally assumed at the outset as a "kinematical" symmetry of the classical action. Such an extrapolation from relatively large distances, where the symmetry 0(3,1) is tested to a genuine local property is questionable. Indeed, the unbounded- ness of the Einstein curvature term in the analytically continued euclidean action signals violent fluctuations near the Planck scale. Hence a "foamy" fractal space-time structure is expected [1], from which the average metric below this scale should emerge in a dynamical way. There is no obvious reason why a smooth effective metric should at all be generated, and even if it were, why it should bear any relation to the "bare" symmetrical local metric imposed on the "fundamental"

1 Chercheur qualifi~ du FNRS. 2 Chercheur IISN.

0619-6823/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) 150 F. Englert et al. / Metric space-time

Fig. 1. The first two iterations of a 2-dimensional 3-fractal.

tive integers vi (i = 1 ..... d) such that their sum Y~./d=lP i is less or equal to n. All these points are contained in the hypertetrahedron bounded by the coordinate hyperplanes and the E~a=lVi= n hyperplane. We distinguish interior points and points belonging to a k-face (k < d), that is points characterized by a set of coordinates vj which contains d-k subsets s such that ~, ~svi = 0 (mod n). Every point belongs to the boundary of at least one sub-hypertetrahedron and two points are called neighbours if they belong to the same sub-hypertetrahedron. One goes from a point to one of its neighbours by one of the elementary translations t i and lij defined as: _+ ti: v~--+ v~:, where v~ = v k if k :~ i,

v" = v i + 1 ;

l q : v}--+ vj , where v'k = v k if i 4= k --t=j ;

v" = v i + 1,

v~ = vj- 1. (2.1)

In general, an interior point admits d(d + 1) neighbours reached by the 2d translations ___ti and the d(d-1) lq translations. If a point belongs to a k-face of the hypertetrahedron, some of these operations reach a point outside the initial hypertetrahedron. Actually, points belonging to a k-face have only d(k + 1) neighbours. 11

i~ °

\ \ fix N MINKOWSKIAN N EUCLIDEAN

METRICS METRICS 03 ~ '~

iI'x EUCLIDEAN I \ METRICS I "x

/ c~=- P.. ~- / /3+1 /

/ / / / / / / / / / /

Fig. 5. The plane of 2-parameterhomogeneous metrics on the Sierpinski gasket. The hyperbole a = /3/(,8 + 1) separates the domain of euclidean metrics from minkowskianmetrics and corresponds- except at the origin - to 1-dimensionalmetrics. ML, M 2, Ma denote unstable minkowskian fixed geometries while E corresponds to the stable euclidean fixed point. The unstable fixed points 01, 0 2 and 0 3 associated to 0-dimensional geometries are located at the origin and at infinity on the (a,/~) coordinates axis. The six straight lines are subsets invariant with respect to the recursion relation but repulsivein the region where they are dashed. The first points of two sequencesof iterations are drawn. Note that for one of them the 10th point (a = -56.4,/3 = -52.5) is outside the frame of the figure. F. Englert et al. / Metric space-time 177

I,I

1,1 ,II

0 II,II

Fig. 10. A metrical representation of the two first iterations of a 2-dimensional 2-fractal corresponding to the euclidean fixed point. Vertices are labelled according to fig. 4. angles of the cell without its base, that is 57r, minus the sum of the angles not belonging to the cell and touching the 3 exterior vertices of the cell, that is 6~r - ~r = 5~r. We find thus that the curvature of a cell is zero, which is consistent with the assumption that the space surrounding the cell is flat. Though the exact value of the curvature at each vertex of a cell is subject to some arbitrariness, because of the arbitrariness showed in the previous section of the normalization of the ?~i9's at successive levels, one easily verifies that, for the homogeneous metrics considered here, all the non-zero cancelling curvatures are located at the cell boundaries. The vertices belonging to the p and (p + 1) levels ot fractalization have negative curvature, the others have positive curvature. Consider now a metric n-fractal, n >> 1, cutoff after the first iteration (or equivalently a (p- 1) triangle in a fractal cutoff at the pth level). The result is a triangular lattice. Because the integrated curvature of any cell is zero, the inside of the lattice is correctly described on the average by a locally flat metric. From

170 R. MEYERS, R.S. STRICHARTZ AND A. TEPLYAEV

Ö Ö Þ

Ö Ö Ö

a a

¿ ½

Õ Õ

½ ¿ Figure 6.4. Geometric interpretation of Proposition 6.1.

7. Effective resistance metric, Green’s function and capacity of points We first recall from [Ki4] some facts about limits of resistance networks. Although we state all the results of this section for the Sierpiński gasket, they can be applied to general pcf fractals with only minor changes. Let (f, f) be defined by (1.2) for any function f on V , where n is a E ∗ E compatible sequence of Dirichlet forms on Γn.

Proposition 7.1. Every point of V = n 0 Vn has positive capacity. ∗ ≥ Proof. Let x V . Then x Vn for someSn. The capacity of x with respect to is the same∈ ∗ as that with∈ respect to because of the{ compatibility} of E En the sequence of networks. The latter capacity is positive because Vn is a finite set. The effective resistance is defined for any x, y V by ∈ ∗ 1 (7.1) R(x, y) = min (u, u): u(x) = 1, u(y) = 0 − . u{E } Here the minimum is taken over all functions on V . Note that x, y Vn ∗ ∈ for large enough n and that (7.1) does not change if is replaced by n, because of the compatibility condition (see [Ki4], PropositionE 2.1.11).E By Theorem 2.1.14 in [Ki4], R(x, y) is a metric on V . In what follows we will write R-continuity, R-closure etc. for continuity, closure∗ etc. with respect to the effective resistance metric R. It is known that if (u, u) < then u is R-continuous ([Ki4], Theorem 2.2.6(1)). The main ingredientE ∞ in the proof of this fact is the inequality (7.2) u(x) u(y) 2 R(x, y) (u, u). | − | ≤ E Let Ω be the R-completion of V . We can conclude from (7.2) that if u is a function on V such that (u, u∗) < then u has a unique continuation ∗ E ∞ PHYSICAL REVIEW LETTERS week ending PRL 95, 171301 (2005) 21 OCTOBER 2005

The Spectral Dimension of the Universe is Scale Dependent

J. Ambjørn,1,3,* J. Jurkiewicz,2,† and R. Loll3,‡ 1The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark 2Mark Kac Complex Systems Research Centre, Marian Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, PL 30-059 Krakow, Poland 3Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, NL-3584 CE Utrecht, The Netherlands (Received 13 May 2005; published 20 October 2005) We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, deﬁned through state sums of causal triangulated geometries. While four dimensional on large scales, the quantum universe appears two dimensional at short distances. We conclude that quantum gravity may be ‘‘self-renormalizing’’ at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction.

DOI: 10.1103/PhysRevLett.95.171301 PACS numbers: 04.60.Gw, 04.60.Nc, 98.80.Qc

Quantum gravity as an ultraviolet regulator?—A shared tral dimension, a diffeomorphism-invariant quantity ob- hope of researchers in otherwise disparate approaches to tained from studying diffusion on the quantum ensemble quantum gravity is that the microstructure of space and of geometries. On large scales and within measuring ac- time may provide a physical regulator for the ultraviolet curacy, it is equal to four, in agreement with earlier mea- inﬁnities encountered in perturbative quantum ﬁeld theory. surements of the large-scale dimensionality based on the PHYSICAL REVIEW LETTERS week ending PRL 95, 171301 (2005) 21 OCTOBER 2005

d logP b lattice theory. Because of the perturbative nonrenormaliz- 2 a (10) d log c ability of gravity, this is expected to be quite subtle. CDT provides a concrete framework for addressing this issue agrees best with the data. In Fig. 1, the curve and we will return to it elsewhere. However, since from 119 (1) can be assigned the length dimension two, and since we D 4:02 (11) S 54 expect the short-distance behavior of the theory to be governed by the continuum gravitational coupling GN,it has been superimposed on the data, where the three con- is tempting to write the continuum version of (10) as stants were determined from the entire data range 2 40; 400. Although both b and c individually are slightly 1 1 altered when one varies the range of , their ratio b=c as PV 2 ; (16) 1 const: GN= well as the constant a remain fairly stable. Integrating relation (10), we have where const. is a constant of order one. Using the same 1 naı¨ve dimensional transmutation, one finds that our ‘‘uni- P ; a=2 b=2c (12) verse’’ of 181.000 discrete building blocks has a spacetime 1 c= 4 volume of the order of 20lPl in terms of the Planck implying a behavior length lPl, and that the diffusion with 400 steps corresponds to a linear diffusion depth of 20l , and is there- a=2 for large ; Pl P ab=c=2 (13) fore of the same magnitude. The relation (16) describes for small : a universe whose spectral dimension is four on scales Our interpretation of Eqs. (12) and (13) is that the quantum large compared to the Planck scale. Below this scale, geometry generated by CDT does not have a self-similar the quantum-gravitational excitations of geometry lead structure at all distances, but instead has a scale-dependent to a nonperturbative dynamical dimensional reduction spectral dimension which increases continuously from a to two, a dimensionality where gravity is known to be b=c to a with increasing distance. renormalizable. Taking into account the variation of a in Eq. (10) when using various cuts min;max for the range of , as well as different weightings of the errors, we obtain the asymptotic value *Electronic address: [email protected] DS 14:02 0:1; (14) Email address: [email protected] †Electronic address: [email protected] which means that the spectral dimension extracted from ‡Electronic address: [email protected] the large- behavior (which probes the long-distance [1] J. Ambjørn, J. Jurkiewicz, and R. Loll, Phys. Rev. Lett. 93, structure of spacetime) is compatible with four. On the 131301 (2004). other hand, the ‘‘short-distance spectral dimension,’’ ob- [2] J. Ambjørn, J. Jurkiewicz, and R. Loll, Phys. Lett. B 607, tained by extrapolating Eq. (12) to ! 0 is given by 205 (2005). [3] J. Ambjørn, J. Jurkiewicz, and R. Loll, Phys. Rev. D 72, DS 01:80 0:25; (15) 064014 (2005). and thus is compatible with the integer value two. [4] J. Ambjørn, D. Boulatov, J. L. Nielsen, J. Rolf, and Y. Watabiki, J. High Energy Phys. 02 (1998) 010. Discussion.— The continuous change of spectral dimen- [5] J. Ambjørn, J. Jurkiewicz, and Y. Watabiki, Nucl. Phys. sion described in this Letter constitutes to our knowledge B454, 313 (1995). the first dynamical derivation of a scale-dependent dimen- [6] J. Ambjørn, K. N. Anagnostopoulos, T. Ichihara, L. Jen- sion in full quantum gravity. (In the so-called exact renor- sen, and Y. Watabiki, J. High Energy Phys. 11 (1998) 022. malization group approach to Euclidean quantum gravity, a [7] J. Ambjørn and K. N. Anagnostopoulos, Nucl. Phys. B497, similar reduction has been observed recently in an 445 (1997). Einstein-Hilbert truncation [12].) It is natural to conjecture [8] D. ben-Avraham and S. Havlin, Diffusion and Reactions in it will provide an effective short-distance cutoff by which Fractals and Disordered Systems (Cambridge University the nonperturbative formulation of quantum gravity em- Press, Cambridge, England, 2000). ployed here, causal dynamical triangulations, evades the [9] T. Jonsson and J. F. Wheater, Nucl. Phys. B515, 549 ultraviolet infinities of perturbative quantum gravity. (1998). [10] J. Ambjørn, J. Jurkiewicz, and R. Loll, Phys. Rev. Lett. 85, Contrary to current folklore (see [13] for a review), this 924 (2000). is done without appealing to short-scale discreteness or [11] J. Ambjørn, J. Jurkiewicz, and R. Loll, Nucl. Phys. B610, abandoning geometric concepts altogether. 347 (2001). Translating our lattice results to a continuum notation [12] O. Lauscher and M. Reuter, Phys. Rev. D 65, 025013 requires a ‘‘dimensional transmutation’’ to dimensionful (2002). quantities, in accordance with the renormalization of the [13] L. J. Garay, Int. J. Mod. Phys. A 10, 145 (1995).

171301-4 JHEP12(2011)012 w d Springer October 26, 2011 December 2, 2011 Models of Grav- : November 18, 2011 : ure of space-time : , and the UV-fixed d is work we carry out a and walk dimension 10.1007/JHEP12(2011)012 Received s = 2+ Published d Accepted eralized dimensions are ap- : a classical regime where w ural explanation for the ap- doi: al Triangulations (EDT), and is shown to be in very good ctral dimension reported from , d ) sion d y safe Quantum Einstein Grav- (2+ Published for SISSA by d/ = 2 , s d = 4. On the length scales covered by three-dimensional w , d 2 d/ = s d 1110.5224 Models of Quantum Gravity, Renormalization Group, Lattice The emergence of fractal features in the microscopic struct = 2, a semi-classical regime where [email protected] w d, d = SISSA 2011 [email protected] Institute of Physics, UniversityStaudingerweg of 7, Mainz, D-55099 Mainz, Germany E-mail: s c ArXiv ePrint: Asymptotic Safety. Keywords: ity, Nonperturbative Effects Monte Carlo simulations, theagreement with resulting the spectral data. dimension parent This puzzle comparison between also the provides short aCausal distance Dynamical nat behavior Triangulations (CDT), of Euclidean the Dynamic spe associated with the effectiveity space-times (QEG). of We discover asymptoticall threeproximately scaling constant regimes for whered these an gen extended range of length scales

Abstract: Martin Reuter and Frank Saueressig point regime where Fractal space-times under thea microscope: renormalization group view on Monte Carlo data is a common themedetailed in renormalization many group approaches study to of quantum the gravity. spectral In dimen th arXiv:1110.5224v1 [hep-th]arXiv:1110.5224v1 24 Oct 2011 [hep-th] 24 Oct 2011 smttcSafety. Dyn Asymptotic pr Euclidean behavior (CDT), also Triangulations distance comparison Dynamical short Causal This the from between puzzle data. apparent the the with for agreement good resulti very the in simulations, Carlo Monte three-dimensional by ayapoce oqatmgaiy nti okw ar out dimension carry spectral the we of work study this In gravity. quantum to struct approaches microscopic many the in features fractal of emergence The lsia eiewhere regime classical f constant a approximately are dimensions generalized d these We (QEG). Gravity Einstein Quantum safe asymptotically of + 2 smttcSafety. Dyn Asymptotic pr Euclidean behavior (CDT), also Triangulations distance comparison Dynamical short Causal This the from between puzzle data. apparent the the with for agreement good resulti very the in simulations, Carlo Monte three-dimensional by out dimension carry spectral the we of work study this In gravity. quantum to struct approaches microscopic many the in features fractal of emergence The lsia eiewhere regime classical f constant a approximately are dimensions generalized d these We (QEG). Gravity Einstein Quantum safe asymptotically of + 2 eomlzto ru iwo ot al data Carlo Monte on view Group Renormalization A eomlzto ru iwo ot al data Carlo Monte on view Group Renormalization A d d n h Vfie on eiewhere regime point UV-fixed the and , n h Vfie on eiewhere regime point UV-fixed the and , rca pc-ie ne h microscope: the under space-times Fractal rca pc-ie ne h microscope: the under space-times Fractal d d s s tuigre ,D509Miz Germany Mainz, D-55099 7, Staudingerweg = tuigre ,D509Miz Germany Mainz, D-55099 7, Staudingerweg [email protected] = [email protected] nttt fPyis nvriyo Mainz of University Physics, of Institute atnRue n rn Saueressig Frank and Reuter Martin nttt fPyis nvriyo Mainz of University Physics, of Institute atnRue n rn Saueressig Frank and Reuter Martin d [email protected] ,d d, d [email protected] ,d d, s s n akdimension walk and n akdimension walk and w w ,asm-lsia eiewhere regime semi-classical a 2, = ,asm-lsia eiewhere regime semi-classical a 2, = Abstract Abstract d d s s = = d/ d/ d d 2 w 2 w d , d , soitdwt h ffciespace-times effective the with associated r fsaetm sacmo hm in theme common a is space-time of ure soitdwt h ffciespace-times effective the with associated w ra xeddrneo eghscales: length of range extended an or r fsaetm sacmo hm in theme common a is space-time of ure gseta ieso ssont be to shown is dimension spectral ng w ra xeddrneo eghscales: length of range extended an or gseta ieso ssont be to shown is dimension spectral ng .O h eghsae covered scales length the On 4. = .O h eghsae covered scales length the On 4. = fteseta ieso reported dimension spectral the of soe he cln eie where regimes scaling three iscover mclTinuain ET,and (EDT), Triangulations amical fteseta ieso reported dimension spectral the of soe he cln eie where regimes scaling three iscover mclTinuain ET,and (EDT), Triangulations amical ealdrnraiaingroup renormalization detailed a ealdrnraiaingroup renormalization detailed a vdsantrlexplanation natural a ovides vdsantrlexplanation natural a ovides d d s s 2 = 2 = d/ d/ 2+ (2 MZ-TH/11-35 2+ (2 MZ-TH/11-35 d ) d d , ) d , w w = = Weak Uncertainty Principle (Kasso Okoudjou, Laurent Saloff-Coste, T., 2008) 1 The R Heisenberg Uncertainty Principle is equivalent, if kf kL2 = 1, to

2 2 2 0 2 1 |x − y| |f (x)| |f (y)| dx dy · |f (x)| dx > 8 ZR ZR ZR On a metric measure space (K, d, µ) with an energy form E a weak uncertainty principle

Varγ (u) E(u, u) > C (1) holds for u ∈ L2(K) Dom(E)

T γ 2 2 Varγ (u) = d(x, y) |u(x)| |u(y)| dµ(x) dµ(y). (2) ZZK×K provided either that d is the eﬀective resistance metric, or some of the suitable Poincare inequalities are satisﬁed.

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 12 / 28 Laplacians on fractals with spectral gaps gaps have nicer Fourier series (Robert Strichartz, 2005)

If the Laplacian has an inﬁnite sequence of exponentially large spectral gaps and the heat kernel satisﬁes sub-Gaussian estimates, then the partial sums of Fourier series (spectral expansions of the Laplacian) converge uniformly along certain special subsequences.

U.Andrews, J.P.Chen, G.Bonik, R.W.Martin, T., Wave equation on one-dimensional fractals with spectral decimation. J. Fourier Anal. Appl. 23 (2017) http://teplyaev.math.uconn.edu/fractalwave/

An introduction given in 2007: http://www.math.uconn.edu/~teplyaev/gregynog/AT.pdf

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 13 / 28 PHYSICAL REVIEW B VOLUME 31, NUMBER 3 1 FEBRUARY 1985

Energy spectrum for a fracta1 lattice in a magnetic field

r Jayanth R. Banavar Schlumberger Doll-Research, Old Quarry Road, Ridgefield, Connecticut 06877-4108

Leo Kadanoff Department of Physics, Uniuersity of Chicago, Chicago, Illinois 60637

A. M. M. Pruisken Schlumberger Doll -Research, Old Quarry Road, Ridgefield, Connecticut 06877-4I08 (Received 10 September 1984)

To simulate a kind of magnetic field in a fractal environment we study the tight-binding Schrodinger equation on a Sierpinski gasket. The magnetic field is represented by the introduction of a phase onto each hopping matrix element. The energy levels can then be determined by either direct diagonalization or recursive methods. The introduction of a phase breaks all the degeneracies which exist in and dominate the zero-field solution. The spectrum in the field may be viewed as considerably broader than the spectrum with no field. A novel feature of the recursion relations is that it leads to a power-law behavior of the escape rate. Green's-function arguments suggest that a majority of the eigenstates are truly extended despite the finite order of ramification of the fractal lattice.

I. INTRODUCTION by the path. The very simplest model is chosen by taking all bonds Hamiltonians defined on fractal surfaces have been the to have exactly the same phase. We make this choice by ' subject of many papers lately. In particular, Domany allowing all bonds in the direction of the arrows in Fig. 1 ' et al. have given a detailed analysis of the spectrum of to have a matrix element f= foe'~, and all bonds opposite eigehstates of the Schrodinger equation defined on a two- to the arrows to have f=foe '~, with fo real and posi- dimensional (2D) Sierpinski gasket. This calculation is tive. Although this choice gives a natural bond pattern, based upon exact decimination techniques which provide the magnetic flux pattern is far less natural. All the ele- a powerful semianalytical method of finding detailed mentary upward-pointing triangles, like those labeled 3 in properties of the Hamiltonian. the figure, have the very same flux, N+ —3P. However, Problems of fractal lattices are reminiscent of (but very using the same convention for the sign of the flux, the much simpler than) those which are conventionally stud- smallest downward-pointing triangles labeled 8 have flux ied for electron localization. For this situation the locali- @~——3tb, while larger downward-pointing triangles have zation. can be rather fully understood. One learns, for ex- larger negative flux, for example, Nc ——6P. Hence, the ample, that, although there are a few extended states, magnetic field pattern studied is quite nontrivial. most states are very highly localized, on finitely ramified fractals. Furthermore, almost all the states have a very high degree of degeneracy. Of course, the observed localization phenomena on these systems are fundamentally different from Anderson localization. For one thing, these lattices give a finite order of ramification. For another, if one chooses to describe the self-similar structure by configurational disorder (e.g., by cutting some bonds) then the resulting "disorder" is highly correlated. Nonetheless, these systems form interesting test cases which are worth studying. In this paper we study magnetic field effects on electronic motion through a 2D Sierpinski gasket (Fig. 1). Following Alexander, we describe the motion by giving an electronic wave function at each mode of the lattice, The tight-binding Hamiltonian fixes the hopping matrix element between neighboring sites to have a magnitude which is the same for all nearest-neighbor sites C II ~ I f ~ and zero otherwise. A magnetic field is defined by giving FIG. 1. Fragment of the Sierpinski gasket. The phase of the the value of the phase on each bond so that the sum of hopping matrix is equal to P in the direction of the arrow and phases along a closed path is the magnetic flux enclosed —P otherwise.

31 1388 1985 The American Physical Society PHYSICAL REVIEW B VOLUME 31, NUMBER 3 1 FEBRUARY 1985

Energy spectrum for a fracta1 lattice in a magnetic field

r Jayanth R. Banavar Schlumberger Doll-Research, Old Quarry Road, Ridgefield, Connecticut 06877-4108

Leo Kadanoff Department of Physics, Uniuersity of Chicago, Chicago, Illinois 60637

A. M. M. Pruisken Schlumberger Doll -Research, Old Quarry Road, Ridgefield, Connecticut 06877-4I08 (Received 10 September 1984)

31 1388 1985 The American Physical Society

Recent refs related to Dirac operators on fractals D. Guido, T. Isola, Spectral triples for nested fractals. J. Noncommut. Geom. 11 (2017)

M. Hinz, L. Rogers, Magnetic ﬁelds on resistance spaces. J. Fractal Geom. 3 (2016)

D. Kelleher, M. Hinz, A. Teplyaev, Metrics and spectral triples for Dirichlet and resistance forms. J. Noncommut. Geom. 9 (2015) arXiv:1309.5937

M. Hinz, A. Teplyaev, Dirac and magnetic Schrodinger operators on fractals. J. Funct. Anal. 265 (2013), arXiv:1207.3077

M. Ionescu, L. G. Rogers, A. Teplyaev, Derivations and Dirichlet forms on fractals. Journal of Functional Analysis, 263 (2012) arXiv:1106.1450 note especially Theorem 5.24

V. Nekrashevych and A. Teplyaev, Groups and analysis on fractals. Analysis on Graphs and its Applications, Proc. Symposia Pure Math., AMS, 77 (2008) note especially Theorem 3.3

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 14 / 28 Half-line example - - - - - - - - - - - t1 qpt pq t qp t qp t pq t pq t qp t pq t qp t Figure: Transition probabilities in the pq random walk. Here p ∈ (0, 1) and q = 1 − p.

f (0) − f (1), if x = 0 −m(x) (∆pf )(x) =  f (x) − qf (x − 1) − pf (x + 1), if 3 x ≡ 1 (mod 3)    f (x) − pf (x − 1) − qf (x + 1), if 3−m(x)x ≡ 2 (mod 3)   Theorem (J.P.Chen, T., 2016)

p 6 1 `2 If = 2 , the Laplacian ∆p on (Z+) has purely singularly continuous spectrum. z(z2−3z+(2+pq)) The spectrum is the Julia set of the polynomial R(z) = pq , which is a topological Cantor set of Lebesgue measure zero.

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 15 / 28 Bohr asymptotics

For 1D Sch¨odingeroperator

Hψ = −ψ00 + V (x)ψ, x ≥ 0 (3)

if V (x) → +∞ as x → +∞ then (H. Weyl), the spectrum of H in L2([0, ∞), dx) is discrete and, under some technical conditions,

1 ∞ N(λ, V ) := #{λi (H) ≤ λ} ∼ (λ − V (x))+ dx. (4) π 0 Z p This is known as the Bohr’s formula. It can be generalized for Rn.

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 16 / 28 Theorem (Fractal Bohr’s formula (Joe Chen, Stanislav Molchanov, T., J. Phys. A: Math. Theor. (2015)))

On inﬁnite Sierpinski-type fractafolds, under mild assumptions,

N(V , λ) lim = 1, (5) λ→∞ g(V , λ) where

ds /2 1 g(V , λ) := (λ − V (x)) G log(λ − V (x))+ µ∞(dx), (6) + 2 Z K∞ where G is the Kigami-Lapidus periodic function, obtained via a renewal theorem.

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 17 / 28 Part 3: Canonical diﬀusions on the pattern spaces of aperiodic Delone sets (Patricia Alonso-Ruiz, Michael Hinz, T., Rodrigo Trevi˜no) A subset Λ ⊂ Rd is a Delone set if it is uniformly discrete:

∃ε > 0 : |~x − ~y| > ε ∀~x, ~y ∈ Λ and relatively dense:

d ∃R > 0 : Λ ∩ BR (~x) =6 ∅ ∀~x ∈ R . A Delone set has ﬁnite local complexity if ∀R > 0∃ ﬁnitely many clusters d P ,..., P such that for any ~x ∈ there is an i such that the set BR (~x) ∩ Λ 1 nR R is translation-equivalent to Pi . A Delone set Λ is aperiodic if Λ − ~t = Λ implies ~t = ~0. It is repetitive if for d any cluster P ⊂ Λ there exists RP > 0 such that for any ~x ∈ R the cluster

BRP (~x) ∩ Λ contains a cluster which is translation-equivalent to P. These sets have applications in crystallography (≈ 1920), coding theory, approximation algorithms, and the theory of quasicrystals.

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 18 / 28 Electron diﬀraction picture of a Zn-Mg-Ho quasicrystal

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 19 / 28 Penrose tiling

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 20 / 28 pattern space of a Delone set

d Let Λ0 ⊂ R be a Delone set. The pattern space (hull) of Λ0 is the closure of the set of translates of Λ0 with respect to the metric %, i.e.

~ d ΩΛ0 = ϕ~t (Λ0): t ∈ R . Deﬁnition d ~ Let Λ0 ⊂ R be a Delone set and denote by ϕ~t (Λ0) = Λ0 − t its translation by d the vector ~t ∈ R . For any two translates Λ1 and Λ2 of Λ0 deﬁne %(Λ1, Λ2) = ~ ~ ~ −1/2 inf{ε > 0 : ∃ ~s, ~t ∈ Bε(0) : B 1 (0) ∩ ϕ~s (Λ1) = B 1 (0) ∩ ϕ~(Λ2)} ∧ 2 ε ε t

Assumption

The action of Rd on Ω is uniquely ergodic: Ω is a compact metric space with the unique Rd -invariant probability measure µ.

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 21 / 28 Topological solenoids (similar topological features as the pattern space Ω):

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 22 / 28 Theorem ~ ~ d (i) If W = (Wt )t≥0 is the standard Gaussian Brownian motion on R , then for any Λ ∈ Ω the process X Λ := ϕ (Λ) = Λ − W~ is a conservative Feller t W~ t t diﬀusion on (Ω,%). Λ (ii) The semigroup Pt f (Λ) = E[f (Xt )] is

2 self-adjoint on Lµ, Feller but not strong Feller.

Its associated Dirichlet form is regular, strongly local, irreducible, recurrent, and has Kusuoka-Hino dimension d.

(iii) The semigroup (Pt )t>0 does not admit heat kernels with respect to µ. It does have Gaussian heat kernel with respect to the d not-σ-ﬁnite (no Radon-Nykodim theorem) pushforward measure λΩ

−1 p d (t, h (Λ )) if Λ ∈ orb(Λ ), R Λ1 2 2 1 pΩ(t, Λ1, Λ2) = (7)  0 otherwise. 

(iv) There are no semi-bounded or L1 harmonic functions (”Liouville-type”). Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 23 / 28 no classical inequalities

Useful versions of the Poincare, Nash, Sobolev, Harnack inequalities DO NOT HOLD, except in orbit-wise sense.

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 24 / 28 spectral properties Theorem 2 The unitary Koopman operators U~t on L (Ω, µ) deﬁned by U~t f = f ◦ ϕ~t commute with the heat semigroup

U~t Pt = Pt U~t

hence commute with the Laplacian ∆, and all spectral operators, such as the unitary Schr¨odingersemigroup.

... hence we may have continuous spectrum (no eigenvalues) under some assumptions even though µ is a probability measure on the compact set Ω.

Under special conditions Pt is connected to the evolution of a Phason: “Phason is a quasiparticle existing in quasicrystals due to their speciﬁc, quasiperiodic lattice structure. Similar to phonon, phason is associated with atomic motion. However, whereas phonons are related to translation of atoms, phasons are associated with atomic rearrangements. As a result of these rearrangements, waves, describing the position of atoms in crystal, change phase, thus the term “phason” (from the wikipedia)”. Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 25 / 28 Phason evolution Corollary 2 The unitary Koopman operators U~t on L (Ω, µ) deﬁned by U~t f = f ◦ ϕ~t commute with the heat semigroup

U~t Pt = Pt U~t

hence commute with the Laplacian ∆, and all spectral operators, including the unitary Schr¨odingersemigroup ei∆t

i∆t i∆t U~t e = e U~t

Recent physics work on phason (“accounts for the freedom to choose the origin”): Topological Properties of Quasiperiodic Tilings (Yaroslav Don, Dor Gitelman, Eli Levy and Eric Akkermans Technion Department of Physics) https://phsites.technion.ac.il/eric/talks/ J. Bellissard, A. Bovier, and J.-M. Chez, Rev. Math. Phys. 04, 1 (1992).

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 26 / 28 TopologicalYaroslav Properties Don, Eli Levy of and Quasiperiodic Eric Akkermans Tilings Department of Physics, Technion – Israel Institute of Technology, Haifa, Israel

AbstractTopological properties of finite quasiperiodic tilings are examined. We study two spe- StructuralAnother way to define Phase a tiling – isPhason by using a as characteristic a Gauge function. Field We consider the Spectral properties Phase: are also Scattering accessible from Matrixthe continuous Approach wave equation, CutAn alternative and Project method to build Scheme quasiperiodic tilings is the Cut & Project scheme. The cific physical quantities: (a) the structure factor related to the Fourier transform of the following choice [5, 6]: 00 2 2 procedure is as follows [10]. ( ) 0 ( ) ( ) 0 ( ) R structure; (b) spectral properties (using scattering matrix formalism) of the correspond- −ψ x − k v x ψ x = k ψ x ; Cut. Zn ing quasiperiodic Hamiltonian. We show that both quantities involve a phase, whose χ (n; φ) = sign [cos (2πn s + φ) − cos (π s)] : with scattering boundary conditions. 1. Start with an n-dimensional space R = n. windings describe topological numbers. We link these two phases, thus establishing N N N N with s → s = c /d the slope of the C&P scheme, and n = 0 : : : d − 1. 2. Insert “atoms” on the integer lattice Z = . a “Bloch theorem” for specific types of quasiperiodic tilings. ⊥ 3. Divide R into the physical space E and the internal space E such that E ⊕ ⊥ ⊥ The phase φ, called a phason, accounts for the gauge freedom to choose the origin. It E = R and E ∩ E = ∅. ` N −→ −→o r (k) t(k) −→{ −→{ is taken discretely as φ → φ = 2π`/d . ←− ←− ←− ←− 4. To resolve ambiguity for E, choose an initial location c ∈ R such that E passes o t(k) r (k) { { ⊥ The scattering−→ S-matrix is←− defined by = ≡ S iγ,1 with through c. There is no such requirement for E . Substitution Rules – 1D Tilings 0 0 0 i # i # e i0γ2 Define a binary substitution rule by Let s (n) = χ (n; 0). Let T [s (n)] = s (n + 1) be the translation operator. Define −→ −→ I −→r = R e and ←−r = R e . It is unitary and can be diagonalized to S 7→ 0 e α β α β 2iδ(k) Project. n I   1 2 σ(a) = aγ bδ a 7→ aγ bδ 0 n ⇐⇒ s so that det S = e with the total phase shift δ (k) = (γ (k) + γ (k)) /2 independent 1. Inspect the hypercube = [−0:5; 0:5) . σ(b) = a b b 7→ a b :  0  ` ⊥ 0  T [s ]  0 0 of φ. We are interested in the chiral phase, 2. The window is its projection on the internal space W = π ( n). Σ = N =⇒ Σ (n; `) = T [s (n)] : α β  d ···−1  3. The strip is the product with the physical space S = W ⊗ E. γ δ 0 −→ ←− Associate occurrence matrix: M = Consider only primitive matrices: T [s ] α (k; φ) = # (k; φ) − # (k; φ) : 4. Choose only the points inside the strip S∩Z, and project them onto the physical 1 • Largest eigenvalue λ > 1 (Perron-Frobenius) 1 space, Y = π (S ∩ Z). Consider now the row permuted Σ c Py∈Y • Left and right first eigenvectors are strictly positive m(`) −1 Using the Krein-Schwinger formula [7] allows to relate the change of density of states 5. The atomic density is given by ρ (x) ≡ ρ (x) = δ (x − y) with x ∈ E. 1 0 N Σ (n; `) = T [s (n)] ; m (`) = ` cN (mod d ): to the scattering data, Note the implicit dependency of Y on c. Distribution of letters underlies distribution of atoms: 0 1 d 0 1 2 3 4 5 6 7 x x x x x x x x ` N N ` 1 % (k) − % (k) = Im ln det S (k) : a b b a b a b Lemma. For φ = 2π`/d with n; ` = 0 : : : d − 1 one has χ (n; φ ) = Σ (n; `). 2π dk • • • • • • • • ::: ` For 1D systems, define the phason This defines a discrete phason φ for the structure. So that the integrated density of states is 1 N N ⊥ Define atomic density The structure of Σ (d × d ) is that of a torus: 0 φ = 2π b/W b ∈ E ; X k N (k) − N (k) = δ (k) /π: ρ (x) = k δ (x − x ) ; where W is the window above. k k+1 k a;b The total phase shift δ (k) is independent of the phason φ unlike the chiral phase with distances for a and b given by δ = x − x = d : The slope s is given by k α (k; φ), whose winding for values of k inside the gaps is given by [8], Let d¯ be the mean distance and u the deviations from the mean. Define 2π p;q 1/s = 1 + cot β: k k a b g Z x = d¯ k + δ u ; δ ≡ d − d : =⇒ α 1 ∂α k = k ; φ k W = 0 dφ = 2q: P −iξx 2 2π ∂φ Let g (ξ) = k e be the diffraction pattern, and S (ξ) = |g (ξ)| the structure 0 factor. Using ξ = 2π/d¯, the Bragg peaks are located atZ [1] −N Useful Tools m;N 1 0 In periodic structures, topological numbers are described as Chern numbers. This does ξ = m λ ξ ; m; N ∈ : not happen in quasiperiodic tilings, since there exists no notion of a Brillouin zone. 1 But alternative tools exist to describe topological properties of quasiperiodic tilings. We consider the following families: The discrete Fourier transform of Σ about n reads N We now enumerate some of them. 2 d −1 Pisot. The second eigenvalue |λ |<1. X −ξn m(`)ξ 2 1 0 1 T Non-Pisot. The second eigenvalue |λ | ≥ 1. G (ξ; `) ≡ n=0 ω Σ (n; `) = ω ς (ξ) : • Tiling space T (dependent on λ or s) and its hull Ω . k 1 1 Fluctuations u are unbounded [2]; there are no Bragg peaks [3]. 2 T n 0 • Čech cohomology Hˇ (Ω ), simplicial cohomology H (Γ ) • The structure factor S (ξ; φ) = |ς (ξ)| is φ-independent. N and Bratteli graphs [11, 12]. Examine the following examples: 1 1 1 √ Here we used the Fibonacci sequence (s = 1/τ) with d = 233 sites. 1 0 0 T Fibonacci. a 7→ ab , b 7→−1 a. It is Pisot, M = , λ = ( 5 + 1)/2 ≡ τ the golden • The phase of G (ξ; `) reads 2 p;q 0 • K -theory, K (Ω ) group and the abstract gap labeling theorem [4, 13]. ratio and λ = −τ . Bragg peaks are located at ξ = (p + q/τ) ξ . m(`)ξ Q ` N In C&P language, s = 1/τ and Z Θ (ξ; `) ≡ arg ω = φ ξ/c (mod 2π): • The Bloch theorem described before can be given an interpolation for 1D C&P RelationIn 1D C&P structures, between The locationsboth Phases: of Bragg peaks a “Bloch for a diffraction Theorem” spectrum corre- tilings (for an irrational slope s∈ / ) by means of the “commutative diagram”: p;q 0 p;q N ! / ξ /ξ = p + qs p; q ∈ : Corollary. For any diffraction peak (discrete Bragg peak) ξ = qc one has the spond to the spectral density of states, Z O ∼ O p;q = 1 1 Z (discrete) winding number at ξ , 1 1 p;q N p;q Θ (φ) α (φ) Thue-Morse. a 7→ ab , b 7→ ba. Here it is Pisot, M−N= , ξ /d = p + qs = N p; q ∈ : 1 2 m;N 0 λ = 2 and λ = 0. Bragg peaks: ξ = m 2 ξ ; m; N ∈ . p;q 2π Z / Z Z Θ ξ = N ` q; d Drawing the integrated density of states N (red line) on top the structural phase 2 1 ψ p;q p;q ∼ C&P 0 C&P ∼ Θ (ξ; φ), shows numerically the relation between ξ and N . = Hˇ (Ω ) K (Ω ) = ⊕ µ(s) ∗ hence 2π C τ p;q / p;q Z Z Z Z Z Spectral Properties of Tilings ξ 1 ∂Θ ξ = ξ ; φ ∼ Consider a 1D tight-binding equation, W = 0 dφ = q: = 2π ∂φ k+1 k−1 k k k ⊕ s ⊕ s − (ψ + ψ ) + V ψ = 2Eψ : 1 0 The topological features are contained in Hˇ or K groups. The gaps in the integrated density of states are given by the gap labeling theorem [4], Z −N Z Z Z Conclusions m;N 1 1 • We have defined two types of phases—a structural and spectral one—whose N = m λ (mod 1); m; N ∈ : windings unveil topological features of quasiperiodic tilings. c p;q p;q Both ξ and N are isomorphic to ⊕ s . The second , corresponding to q, can 1 Here, c is the gcd of λ and its corre- be derived independently from the windings of both the structural phase Θ (φ) and the • We found a relation between these two phases, which can be interpreted as a sponding eigenvectors in both M and chiral phase α (φ). Since these phases account for windings, they are isomorphic Bloch-like theorem. 2 the collared M . N Here we used the Fibonacci sequence (s = 1/τ) with d = 89 sites. The winding ∼ • We have considered here a subset of tilings, which are known as Sturmian numbers are indicated by the red numbers above. Θ (φ) = α (φ): (C&P) words. Our results can be extended to a broader families of tilings in one In C&P sequences, Z dimension, and to tiles in higher dimensions (D > 1). Remark. The analysis above for the winding numbers is done for rational approxima- The winding gives a topological interpretation to these phases. This result can be p;q N N N N N = p + qs (mod 1) p; q ∈ : tions s = c /d . It holds by construction for the irrational case s → s. viewed as a Bloch theorem for quasiperiodic tilings [9]. • All these features have been observed experimentally [5, 6].

[1] J. M. Luck, C. Godrèche, A. Janner, and T. Janssen, J. Phys. A 26, 1951 (1993). [6] A. Dareau, E. Levy, M. B. Aguilera, R. Bouganne, E. Akkermans, F. Gerbier, and J. Beugnon, Phys. Rev. Lett. 119, [11] L. A. Sadun, Vol. 46, University Lecture Series (American Mathematical Society, 2008). Contact Information References[2] J. M. Dumont, in Springer proceedings in physics, edited by J.-M. Luck, P. Moussa, and M. Waldschmidt, (Springer 215304 (2017). [12] J. E. Anderson, and I. F. Putnam, Ergod. Th. Dynam. Sys. 18, 509 (1998). Email: [email protected] Berlin Heidelberg, 1990), pp. 185–194. [7] E. Akkermans, G. V. Dunne, and E. Levy, in Optics ofhttps://arxiv.org/abs/1509.04028 aperiodic structures: fundamentals and device applications, [13] J. Kellendonk, and I. F. Putnam, in Directions in mathematical quasicrystals, Vol. 13, edited by M. Baake, and R. V. [3] C. Godrèche, and J. M. Luck, Phys. Rev. B 45, 176 (1992). edited by L. Dal Negro, (Pan Stanford Publishing, 2014), pp. 407–449. Moody, CRM Monograph Series (American Mathematical Society, 2000), pp. 186–215. [4] J. Bellissard, A. Bovier, and J.-M. Chez, Rev. Math. Phys. 04, 1 (1992). [8] E. Levy, A. Barak, A. Fisher, and E. Akkermans, (2015) . [14] E. Bombieri, and J. E. Taylor, J. Phys. Colloq. 47, C3–19–C3 (1986). This work was supported by the Israel Science Foundation Grant No. 924/09. [5] F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. Le Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, [9] Y. Don, D. Gitelman, E. Levy, and E. Akkermans, (in preperation), 2018. We thank C. Schochet for useful discussions. Phys. Rev. B 95, 161114 (2017). [10] M. Duneau, and A. Katz, Phys. Rev. Lett. 54, 2688 (1985).

Helmholtz, Hodge and de Rham

Theorem Assume d = 1. Then the space L2(Ω, µ, R1) admits the orthogonal decomposition 2 1 L (Ω, µ, R ) = Im ∇ ⊕ R(dx). (8)

In other words, the L2-cohomology is 1-dimensional, which is surprising because the de Rham cohomology is not one dimensional. M. Hinz, M. R¨ockner,T., Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on fractals, Stoch. Proc. Appl. (2013). M. Hinz, T., Local Dirichlet forms, Hodge theory, and the Navier-Stokes equation on topologically one-dimensional fractals, Trans. Amer. Math. Soc. (2015,2017). Lorenzo Sadun. Topology of tiling spaces, volume 46 of University Lecture Series. American Mathematical Society, Providence, RI, 2008. Johannes Kellendonk, Daniel Lenz, and Jean Savinien. Mathematics of aperiodic order, volume 309. Springer, 2015.

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 27 / 28 end of the talk :-)

Thank you!

Sasha Teplyaev (UConn) Pure point spectrum on fractals and related questions October 1, 2018 ∗ FGS6 28 / 28