# Probing Quantum Hall Pseudospin Ferromagnet by Resistively Detected NMR

G. P. Guo(1) Y. J. Zhao(1) T. Tu(1<EMAIL_ADDRESS>X. J. Hao(1) G. C.
Guo(1) H. W. Jiang(2<EMAIL_ADDRESS>(1) Key Laboratory of Quantum
Information, University of Science and Technology of China, Chinese Academy of
Sciences, Hefei 230026, P. R. China
(2) Department of Physics and Astronomy, University of California at Los
Angeles, 405 Hilgard Avenue, Los Angeles, CA 90095, USA

###### Abstract

Resistively Detected Nuclear Magnetic Resonance (RD-NMR) has been used to
investigate a two-subband electron system in a regime where quantum Hall
pseudo-spin ferromagnetic (QHPF) states are prominently developed. It reveals
that the easy-axis QHPF state around the total filling factor $\nu=4$ can be
detected by the RD-NMR measurement. Approaching one of the Landau level (LL)
crossing points, the RD-NMR signal strength and the nuclear spin relaxation
rate $1/T_{1}$ enhance significantly, a signature of low energy spin
excitations. However, the RD-NMR signal at another identical LL crossing point
is surprisingly missing which presents a puzzle.

###### pacs:

73.43.Nq, 71.30.+h, 72.20.My

The multi-component electron systems have been continuously drawing intensive
research interest because of its novel ground states and excitations DasSarma
. In experimental systems, different Landau levels (LLs) can be tuned to cross
by varying gate voltage, charge density, magnetic field or the magnetic field
tilted angle to the sample. Electron-electron correlations become particularly
prominent when two or more sets of LLs with different layer, subband, valley,
spin, or Landau level indices are brought into degeneracy DasSarma ;
Wescheider1999 ; Shayegan2000 ; Hirayama2001 ; Jiang2005 ; Jiang2006 ;
Tsui2006 ; Shayegan2006 . Recent experiments in single quantum well with two
subbands occupied systems Jiang2005 ; Jiang2006 , showed evidence of the
formation of quantum Hall pseudospin ferromagnets (QHPFs) due to the
interactions of the two subbands (termed as pseudospins) around the LLs
crossing point. The QHPFs taking place at total filling factor $\nu=3,5$ and
$\nu=4$ are easy-plane or easy-axis QHPFs respectively, depending on the
details of the two subbands configurations. In spite of various theoretical
models MacDonald2000 ; DasSarma2003 ; Hao2008  motivated by these findings, a
comprehensive understanding is not yet achieved. Thus far, experimental and
theoretical studies all focused on the pseudospin freedom. However, in this
work we would address the unique spin excitations in the QHPF states.

To address the question whether spin states in two-subband systems in nature,
measurements other than the conventional transport and optical means are
needed. Since the Zeeman energy of nuclear spin is about $3$ orders of
magnitude smaller than that of electron spin, exchange of spin angular
momentum between the electron and nuclear spin is allowed only when the
electron system supports spin excitations with low energy. The nuclear spin
relaxation rate $1/T_{1}$ thus probes the density of states at low energy of
the electron spin system that cannot be accessed by other means. The
resistively detected NMR technique has recently emerged as an effective method
to probe collective spin states in the fractional quantum Hall regime
KlitzingNMR1 ; KlitzingNMR2 , the Skyrmion spin texture close to the filling
factor $1$ PortalNMR ; TsuiNMR , the role of electron spin polarization in the
phase transition of a bilayer system EisensteinNMR ; HirayamaNMR1 , and the
ferromagnetic state accompanied by collective spin excitations of a two-
subband system JiangNMR . Here we use this technique to study spin freedom and
its relation with pseudospin in the vicinity of the QHPF states at filling
factor $\nu=3,4,5$. It reveals that the easy-axis QHPF state at $\nu=4$ is
sensitive to the RD-NMR measurement. As approaching to one LL crossing point
at $\nu=4$ where the easy-axis QHPF phase is well developed, the RD-NMR signal
strength and the nuclear spin relaxation rate $1/T_{1}$ enhance quickly which
may be due to the low energy spin excitations there. Furthermore, the RD-NMR
signal can be suppressed anomaly at another identical LL crossing point of
$\nu=4$.

The sample was grown by molecular-beam epitaxy and consists of a symmetrical
modulation-doped $24$ nm wide single GaAs quantum well bounded on each side by
Si $\delta$-doped layers of AlGaAs with doping level $n_{d}=10^{12}$ cm-2.
Heavy doping creates a very dense 2DEG, resulting in the filling of two
subbands in the well. As determined from the Hall resistance data and
Shubnikov-de Haas oscillations in the longitudinal resistance, the total
density is $n=8.0\times 10^{11}$ cm-2, where the first and the second subband
have a density of $n_{1}=6.1\times 10^{11}$ cm-2 and $n_{2}=1.9\times 10^{11}$
cm-2. The sample has a low-temperature mobility $\mu=4.1\times 10^{5}$ cm2/V
s, which is extremely high for a 2DEG with two filled subbands. A $100$ $\mu$m
wide Hall bar with $270$ $\mu$m between voltage probes was patterned by
standard lithography techniques. A NiCr top gate was evaporated on the top of
the sample, approximately $350$ nm away from the center of the quantum well.
By applying a negative gate voltage on the NiCr top gate, the electron density
can be varied continuously. Several turns of NMR coil were wound around the
sample, which was placed in a Top-Loading Dilution Refrigerator with a base
temperature of $15$ mK. A small radio frequency (rf) magnetic field generated
by the coil with a matching frequency $f=\gamma H_{0}$ will cause NMR for 75As
nuclei, where the gyromagnetic ratio $\gamma=7.29$ MHz/T. The resistance was
measured using quasi-dc lock-in technique with $11.3$ Hz.

Figure 1: (a) The longitudinal resistance $R_{xx}$ in the density ($n$) -
magnetic field ($B_{\bot}$) phase diagram at filling factor $\nu=3,4,5$, which
are measured at the base temperature. (b) Schematic drawing of the crossing
between different indices Landau levels and resulting easy-plane or easy-axis
pseudo-spin states at points B, D and A, C, as correspondingly marked in Fig.
2a.

Figure 2: (a) The NMR signals phase diagram of the sample at $\nu=3,4,5$. The
cross and circle symbols in the map denote the places where the NMR signals
are measured. The ’$\times$’ mean places where there are no NMR signals, while
the ’$\circ$’ show the places where the NMR signals are observed. And the size
of ’$\circ$’ symbols give a schematic illustration of the strength of NMR
signals. The dashed line L1 is the trace along which we measured NMR signal as
shown in Fig. 5. (b) Typical resistively detected NMR spectrum measured around
point C and A, B, D.

In the present work, we refer the first and second subbands, to as symmetric
and antisymmetric states. In the pseudo-spin language, one of them can be
labeled as pseudo-spin up ($\Uparrow$) and the other as pseudo-spin down
($\Downarrow$). When a magnetic field $B_{\bot}$ is applied, the energy
spectrum of the quantum well discretizes into a sequence of Landau levels. We
label the single-particle levels ($i,N,\sigma$), which $i$
($=\Uparrow,\Downarrow$), $N$, and $\sigma$ ($=\uparrow,\downarrow$) are the
pseudo-spin, orbital and spin quantum numbers. In the present work we have
concentrated our study around the filling factor $\nu=3,4,5$, where the
filling factor $\nu$ denotes the number of filled Landau levels. The
longitudinal resistance $R_{xx}$ in the density ($n$) - perpendicular magnetic
field ($B_{\bot}$) plane exhibits a square-like structure around $\nu=3,4,5$,
as shown in Fig. 1a. The most noticeable feature of the square-like structure
is the disappearance of the extended states (i.e., bright lines) on its four
boundaries, marked by A, B, C, D in Fig. 1a. Here point A corresponds to the
degeneracy point of $\left|(\Uparrow,1,\downarrow)\right\rangle$ and
$\left|(\Downarrow,0,\uparrow)\right\rangle$, point B corresponds to that of
$\left|(\Uparrow,1,\uparrow)\right\rangle$ and
$\left|(\Downarrow,0,\uparrow)\right\rangle$, point C corresponds to that of
$\left|(\Uparrow,1,\uparrow)\right\rangle$ and
$\left|(\Downarrow,0,\downarrow)\right\rangle$, point D corresponds to that of
$\left|(\Uparrow,1,\downarrow)\right\rangle$ and
$\left|(\Downarrow,0,\downarrow)\right\rangle$, as illustrated schematically
in the Landau level fan diagram Fig. 1b. The disappearance and result square
structure represents a pseudo-spin ferromagnet, which is due to the opening
pseudo-spin gaps of easy-plane or easy-axis pseudo-spin ferromagnetic states,
respectively at the level crossing points of B, D and A, C, as depicted in
Fig. 1b Jiang2005 ; Jiang2006 ; Hirayama2001 ; MacDonald2000 .

RD-NMR, performed in the proximity of the square structure, reveals prominent
(absent) NMR signal at different regions. In order to get a clear signal and
minimize heat effect, most of experiments were carried out with a rf power of
$0$ dBm. The ac current $I_{ac}$ was $50$ nA, and a large dc current
$I_{dc}=250$ nA were applied to enhance the NMR signal. All the measurements
were carried out at temperature below $120$ mK. The measurement result under
the same condition are shown in Fig. 2a, the cross and circle symbols in the
map denote the places where the NMR signals are measured. The cross ’$\times$’
means the places where there are no NMR signals, while the circle ’$\circ$’
shows the places where the NMR signals are observed. And the size of ’$\circ$’
symbols give a schematic illustration of the strength of NMR signals. From
this map we found that the NMR signals only occur at the upper arm of the
square structure around crossing point C, while we didn’t find any signal at
the lower arm of this square structure around another crossing point A and its
two sides around crossing point B and D.

Now we focused on the region around the LL crossing point C, where pronounced
NMR signals were observed. Typical NMR lines around point C are shown in Fig.
2b. The relative change of $R_{xx}$ is typically about 1% at resonance. Upon
resonance, $R_{xx}$ in all NMR lines shows a sharp decrease followed by a much
slower relaxation process back to its original value, which is characterized
by the nuclear spin relaxation time owing to the interaction with the electron
spin system, $T_{1}$, as will be discussed below. In these experiments, we
have changed the rf amplitude from $-15$ dBm to $2$ dBm. Even very weak, the
NMR signal can be recognized at $-15$ dBm.

We believe the RD-NMR described here is due to the electron and nuclear spin
flip-flop effect JiangNMR . For the two dimensional electron system in GaAs,
the contact hyperfine interaction with the polarized nuclei acts as an
effective magnetic field $B_{N}$ for the electron spin. The effective electron
spin-flip energy is then reduced, $E_{z}=g^{\ast}\mu_{B}BS_{z}+A\left\langle
I_{z}\right\rangle S_{z}=g^{\ast}\mu_{B}(B+B_{N})S_{z}$ as $g^{\ast}<0$. When
the NMR resonance condition is matched, the nuclear spins are depolarized and
the electron Zeeman energy increases consequently. Since $R_{xx}$ is dependent
on the thermally activated energy gap $E_{a}$,
$R_{xx}\propto\exp(-E_{a}/2k_{B}T)$, the NMR is manifested by a drop in
$R_{xx}$, as shown by all the NMR lines in Fig. 2b. This allows the nuclear
spin polarization to be sensitively detected by a change in the transport
coefficient of the electron system $R_{xx}$.

The above observations reveals the spin excitation in the square structure is
of intrinsic interest and is well correlated with the spin excitations of the
easy-axis QHPF states. At point C, when the two competing pseudospin (up and
down) states acquire the same energy and leads to easy-axis anisotropy, they
separate into domains with opposite pseudospin states Hirayama2001 ; Jiang2006
; MacDonald2000 ; MacDonald2001 . On the other hand, the pseudospin up and
down states have opposite spins. As a result, magnetic domains form and the
electronic state within each domain is described as an Ising-like QH
ferromagnet with either one of two possible spin orientations. As the applied
current forces electrons to scatter between adjacent domains with different
spin but almost degenerate energy, the nuclei in the neighborhood can become
polarized and probed by the RD-NMR measurement. However at other crossing
point B and D, the QHPF states are easy-plane, which means that the two
degenerate Landau levels are mixing and no spin magnetization formation. Since
easy-plane QHPF state can not spontaneously separate into magnetic domains,
there is no nuclear polarization and the NMR signals are destroyed.

To support the mechanism of the polarized nuclear spins, current dependence of
the NMR signal was studied. In this measurement, the sample resistance was
measured with a low ac current of $20$ nA, while ramping the dc current in a
wide range to bias the sample. The result indicates that the NMR signal is
enhanced by a factor of $8$ in the low current range from $100$ nA up to $250$
nA. The data thus consist with the picture of current induced dynamic
polarization.

Figure 3: Measuring nuclear spin relaxation time $T_{1}$ around point C by
recording time evolution of $R_{xx}$ irradiated by rf, initially off
resonance, on resonance and finally off resonance. $T_{1}$ is determined by an
exponential fit to the experiment data.

Figure 4: (a) Plot of the resistively detected NMR signal ratio $\Delta
R_{xx}/R_{xx}$ (black square), nuclear spin relaxation rate $1/T_{1}$ (blue
circle) against gate voltage $V_{g}$ along the line L1 (in Fig. 2a). (b) Plot
of electron activation energy gap $E_{a}$ against gate voltage $V_{g}$ along
the same line.

To gain more support of our observation of the nature of the spin in the easy-
axis QHPF states, we studied the coupling between the nuclei and the electrons
by measuring the nuclear spin relaxation time $T_{1}$, at various positions
near the crossing point C. First, rf was tuned into resonance, and $R_{xx}$
shows a sharp decrease due to the nuclear depolarization. Then, the frequency
was switched back to off resonance. Nuclear spins that have once flopped
hardly relax back because of their longer relaxation time $T_{1}$, which is on
the order of minutes, relative to that of the electrons. Hence, $R_{xx}$
slowly relaxes back to its original value, and $T_{1}$ can be derived by
fitting $R_{xx}$ to the relation $R_{xx}=\alpha+\beta\exp(-t/T_{1})$. Fig. 3
shows the data around point C to determine $T_{1}$.

Further insight is gained by investigating the NMR signals along the line L1
(please see Fig. 2a). As depicted in Fig. 4a, our measurement shows a clear
peak of NMR ratio $\Delta R_{xx}/R_{xx}$ at the crossing point C where the
easy-axis pseudo-spin ferromagnetic states is well developed. The obtained
values of nuclear spin relaxation rate $1/T_{1}$ along line L1 are also
plotted in Fig. 4a. $1/T_{1}$ rapidly increases from nearly zero to $8\times
10^{-3}$ (1/s) toward to the crossing point C, as electron becomes the pseudo-
spin ferromagnetic states. For comparison, in Fig. 4b we also show the
electron activation energy gap $E_{a}$ along the line L1. The single particle
energy difference $E_{z}$ acts as effective Zeeman energy, and $E_{a}$ shows a
slope of $5$ times greater than the single particle Zeeman gap $E_{z}$. This
unusual behavior is likely to be caused by the easy-axis ferromagnetism
Jiang2006 ; Hirayama2001 . These quantities all show an obvious change as
approaching to the crossing point and demonstrate that $1/T_{1}$ is a
sensitive indicator of the pseudo-spin ferromagnetic formation. The similarity
between these phenomenon strongly suggest that an intimate link between the
spin and pseudo-spin in the easy-axis pseudo-spin ferromagnetic states.

Interestingly, the data shown in Fig. 4b shows that the slop of activation
energy gap $E_{a}$ to single particle Zeeman gap is as large as $5$, which
implies many spin flips within the magnetic domain walls and support low
energy mode of spin excitations Eisenstein1995 ; MacDonald2001 . As
approaching to the crossing point C, there are low energy spin excitations
which give new channel to relax the nuclear spin through the electron and
nuclear spin flip-flop process. Thus the NMR signal ratio $\Delta
R_{xx}/R_{xx}$ and the nuclear spin relaxation rate $1/T_{1}$ enhanced.

Despite the fact that the bulk of the results can be understood within the
framework of pseudo-spin quantum Hall ferromagnetism, there is still an
apparent puzzle. While we can find very strong NMR signals at the upper arm of
the square structure around point C, there is no detectible signal at the
lower arm of this square structure around point A. Since the two points have
equivalent LLs crossing configurations, one would expect that they are the
same easy-axis QHPF states and should produce similar NMR responses. In
principle, the NMR signal can be suppressed by spin-orbital coupling
HirayamaNMR2  or mobility of domains HirayamaNMR3 . However, in our case,
point A and C have identical strength in spin-orbital coupling and disorder.
Therefore, the anomalous suppression of NMR signal at point A may suggest that
there could be some additional physics which has not yet been recognized in
the theory of pseudo-spin quantum Hall ferromagnetism.

In summary, RD-NMR has been measured in a two-subband electron system around
the LLs crossing points at total filling factor $\nu=3,5$ and $4$ where easy-
plane or easy-axis QHPF states are well developed. It reveals that the easy-
axis quantum Hall pseudospin state of $\nu=4$ is sensitive to the RD-NMR
measurement. As approaching to one LL crossing point at $\nu=4$, the RD-NMR
signal strength and the nuclear spin relaxation rate $1/T_{1}$ enhance quickly
which may be due to the low energy spin excitations. At another identical LL
crossing point of $\nu=4$, the RD-NMR signal is found to be suppressed and
remains as a puzzle to be understood. Of course further study is necessary to
access the detailed mechanism.

This work at USTC was funded by National Basic Research Programme of China
(Grants No. 2006CB921900 and No. 2009CB929600), the Innovation funds from
Chinese Academy of Sciences, and National Natural Science Foundation of China
(Grants No. 10604052 and No. 10874163 and No.10804104). The work at UCLA was
supported by the NSF under Grant No. DMR-0804794.

## References

  * (1) Chap 2 and 5 in Perspectives on Quantum Hall Effects, S. Das Sarma and A. Pinczuk eds., (Wiley, New York, 1997).
  * (2) V. Piazza, V. Pellegrini, F. Beltram, W. Wegscheider, T. Jungwirth, and A. H. MacDonald, Nature 402, 638 (1999).
  * (3) E. P. De Portere, E. Tutuc, S. J. Papadakis, M. Shayegan, Science 290, 1546 (2000).
  * (4) K. Muraki, T. Saku, and Y. Hirayama, Phys. Rev. Lett. 87, 196801 (2001).
  * (5) X. C. Zhang, D. R. Faulhaber and H. W. Jiang, Phys. Rev. Lett. 95, 216801 (2005).
  * (6) X. C. Zhang, I. Martin and H. W. Jiang, Phys. Rev. B 74, 073301 (2006).
  * (7) K. Lai, W. Pan, D.C. Tsui, S. Lyon, M. Muhlberger and F. Schaffler, Phys. Rev. Lett. 96, 076805 (2006).
  * (8) K. Vakili, T. Gokmen, O. Gunawan, Y. P. Shkolnikov, E. P. De Poortere and M. Shayegan, Phys. Rev. Lett. 97. 116803 (2006).
  * (9) T. Jungwirth and A. H. MacDonald, Phys. Rev. B 63, 035305 (2000).
  * (10) D. W. Wang, E. Demler, and S. Das Sarma, Phys. Rev. B 68, 165303 (2003).
  * (11) X. J. Hao et al., arXiv:0807.0297.
  * (12) J. H. Smet et al., Nature (London) 415, 281 (2002).
  * (13) O. Stern et al., Phys. Rev. B 70, 075318 (2004).
  * (14) W. Desrat et al., Phys. Rev. Lett. 88, 256807 (2002).
  * (15) G. Gervais et al., Phys. Rev. Lett. 94, 196803 (2005).
  * (16) I. B. Spielman et al., Phys. Rev. Lett. 94, 076803 (2005).
  * (17) N. Kumada et al., Phys. Rev. Lett. 94, 096802 (2005).
  * (18) X. C. Zhang, G. D. Scott and H. W. Jiang, Phys. Rev. Lett. 98, 246802 (2007).
  * (19) T. Jungwirth and A. H. MacDonald, Phys. Rev. Lett. 87, 216801 (2001).
  * (20) A. Schmeller, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 75, 4290 (1995).
  * (21) K. Hashimoto et al., Phys. Rev. Lett. 94, 146601 (2005).
  * (22) Y. Hirayama et al., Physica E. 20, 133 (2003).


# Robust heteroclinic tangencies

Pablo G. Barrientos  Instituto de Matemática e Estatística, UFF
Rua Prof. Marcos Waldemar de Freitas Reis, s/n, Niterói, Brazil
<EMAIL_ADDRESS>and  Sebastián A. Pérez  Centro de Matematica da
Universidade do Porto
Rua do Campo Alegre 687, 4169-007 Porto, Portugal
<EMAIL_ADDRESS>
###### Abstract.

We construct diffeomorphisms in dimension $d\geq 2$ exhibiting $C^{1}$-robust
heteroclinic tangencies.

###### Key words and phrases:

folding manifolds, robust equidimensional tangencies, robust heterodimensional
tangencies.

###### 2000 Mathematics Subject Classification:

Primary 34D30, 37D10, 37D30, 37G25.

##  1\. Introduction

An important problem in the modern theory of Dynamical Systems is to describe
diffeomorphisms whose qualitative behavior exhibits robustness under (small)
perturbations and how abundant these sets of dynamics can be. Motivated by
this issue, Smale introduced in [15] the hyperbolic diffeomorphisms as
examples of structural stable dynamics (open sets of dynamics which are all of
them conjugated). However, the transverse intersection between the invariant
manifolds of basic sets was soon observed as a necessary condition [20, 13,
17]. The main goal of this article is to study the persistence of the non-
transverse intersection between those manifolds. Namely, we focus in
tangencial heteroclinic orbits.

A diffeomorphism $f$ of a manifold $\mathcal{M}$ has a _heteroclinic tangency_
if there are different transitive hyperbolic sets $\Lambda$ and $\Gamma$,
points $P\in\Lambda$, $Q\in\Gamma$ and $Y\in W^{u}(P)\cap W^{s}(Q)$ such that

$c_{T}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\dim\mathcal{M}-\dim[T_{Y}W^{u}(P)+T_{Y}W^{s}(Q)]>0\quad\text{and}\quad
d_{T}\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\dim
T_{Y}W^{u}(P)\cap T_{Y}W^{s}(Q)>0.$

The number $c_{T}$ is called _codimension of the tangency_ and measures how
far the tangencial intersection is from a transverse intersection. On the
other hand, $d_{T}$ indicates the number of linearly independent common
tangencial directions. Observe that

$c_{T}=d_{T}-k_{T}\qquad\text{with}\quad
k_{T}=\operatorname{ind}(\Lambda)-\operatorname{ind}(\Gamma)$

where $\operatorname{ind}(\Sigma)$ denotes the stable index of a (transitive)
hyperbolic set $\Sigma$. The integer $k_{T}$ is called _signed co-index_.
Notice that when $k_{T}>0$ this number coincides with the classical co-index
between $\Lambda$ and $\Gamma$. Moreover, $k_{T}>0$ if and only if

$\dim T_{Y}W^{u}(P)+\dim T_{Y}W^{s}(Q)>\dim M.$

If $k_{T}=0$, the heteroclinic tangency is called _equidimensional_ and
otherwise _heterodimensional_. Figure 1 illustrates the different types of
heteroclinic tangencies in dimension three.

\begin{overpic}[scale={0.27}]{tangencias.pdf}
\scriptsize\put(-8.0,62.0){\large$W^{s}(Q)$}
\put(54.0,68.0){\large\circle*{3.0}} \put(57.0,56.0){\large$Y$}
\put(75.0,130.0){\large$W^{u}(P)$} \put(-2.0,15.0){\normalsize(a)
heterodimensional} \put(145.0,15.0){\normalsize(b) equidimensional}
\put(282.0,15.0){\normalsize(c) heterodimensional}
\put(135.0,63.0){\large$W^{s}(Q)$} \put(193.0,67.0){\large\circle*{3.0}}
\put(199.0,56.0){\large$Y$} \put(215.0,130.0){\large$W^{u}(P)$}
\put(285.0,63.0){\large$W^{s}(Q)$} \put(341.0,67.0){\large\circle*{3.0}}
\put(348.0,56.0){\large$Y$} \put(360.0,130.0){\large$W^{u}(P)$} \end{overpic}
Figure 1. Heteroclinic tangencies in dimension 3: (a) $c_{T}=2$, $d_{T}=1$ and
$k_{T}=-1$; (b) $c_{T}=1$, $d_{T}=1$ and $k_{T}=0$; (c) $c_{T}=1$, $d_{T}=2$
and $k_{T}=1$.

Heterodimensional tangencies with signed co-index $k_{T}>0$ was introduced in
[9] where interesting dynamics consequences were obtained. Indeed, the authors
showed that the $C^{1}$-unfolding of a three dimensional heterodimensional
tangency (with $k_{T}=1$) leads to $C^{1}$-robustly non-dominated dynamics and
in some cases to very intermingled dynamics related to universal dynamics, for
details see [9, 6]. In the $C^{r}$-topologies with $r>1$, the bifurcation of
such tangencies leads, for instance, to the existence of blender dynamics [8,
10]. Kiriki and Soma in [12] obtain the first examples of $C^{2}$-robust
heterodimensional tangencies with $c_{T}=1$ and $k_{T}=d-2$ in any manifold of
dimension $d\geq 3$. Recently in [3] new examples of $C^{2}$-robust
heterodimensional tangencies with $0<c_{T}\leq\lfloor(d-3)/2\rfloor$ and
$1\leq k_{T}\leq d-2-2c_{T}$ were also constructed in any manifold of
dimension $d\geq 5$. In the same work, $C^{2}$-robust heteroclinic tangencies
with $k\leq 0$ were also obtained. In [12] it was proposed the problem of
constructing $C^{1}$-robust heterodimensional tangencies with $k>0$ in any
dimension greater than $2$. Motivated by this issue, the main result of this
work shows that, in particular, these tangencies can be built persistently
under $C^{1}$-perturbations.

######  Theorem A.

Every manifold of dimension $d\geq 2$ admits a diffeomorphism $f$ having a
$C^{1}$-robust heteroclinic tangency of codimension $c_{T}=1$ and signed co-
index $0\leq k\leq d-2$.

By constraints of the dimension, in surfaces, we only get equidimensional
tangencies. In higher dimensions, we construct both type of heteroclinic
tangencies: equidimensional and heterodimensional with all possible signed co-
index between 0 and $d-1$.

Theorem A will be proved in Section 2, by providing a local construction close
to the classical examples given by Abraham and Smale [1], Simon [14] and
Asaoka in [2]. In Section 3 we will give a different proof of Theorem A using
ideas of the recent work [4] studying the differential cocyle in the tangent
space. These new ideas allow us to generalize the construction for large
codimension ($c_{T}\geq 2$) in some particular cases. Namely, we get the
following result.

######  Theorem B.

Given integers $c_{T}\geq 1$ and $s>c_{T}$, there are diffeomorphisms $f$ of
the $d$-dimensional torus $\mathbb{T}^{d}$ with $d=c_{T}\cdot(s+1)$ having a
$C^{1}$-robust heterodimensional tangency of codimension $c_{T}$ and signed
co-index $k_{T}=s-c_{T}>0$.

The proof of the above theorem will be carried on in Section 4. Finally, in
Section 5 we conclude the work with a section of open questions and future
directions.

##  2\. Geometric construction of $C^{1}$-robust heteroclinic tangencies

Let $\Lambda$ be a Plykin attractor in a disc with three holes [16]. Let $Q$
be a saddle in the complement of this disc as in Figure 2. To do possible the
construction we need to assume that $Q$ belongs to a Plykin repellor $\Gamma$.
This figure illustrates the two-dimensional version of the Asaoka’s argument
[2] (see also [14]) providing a $C^{1}$-robust equidimensional tangency in any
surface between the stable manifold of $\Lambda$ and the unstable manifold of
$Q$.

\begin{overpic}[scale={0.22}]{Plykin.pdf}
\scriptsize\put(65.0,158.0){\huge$\Lambda$} \put(201.0,110.0){\circle*{4.0}}
\put(200.0,88.0){\large$P$} \put(232.0,110.0){\large\circle*{3.0}}
\put(228.0,90.0){\large$y$} \put(315.0,108.0){\circle*{4.0}}
\put(324.0,88.0){\large$Q$} \end{overpic} Figure 2.  Equidimensional tangency
in dimension $d=2$

Using this idea, we built a diffeomorphism $f$ on any manifold $\mathcal{M}$
of dimension $d\geq 2$ having a hyperbolic attractor $\Lambda$ whose
attracting region is a connected set and foliated by $(d-1)$-dimensional
stable submanifolds. After that, we consider a fixed point $P$ in $\Lambda$
and another fixed point $Q$ of $f$ of stable index $d-1-k$ where $0\leq k\leq
d-2$ creating a heteroclinic tangency between $W^{s}(P)$ and $W^{u}(Q)$, so
that $W^{u}(Q)$ and $W^{s}(\Lambda)$ meet transversely. The $C^{1}$\-
persistence of this last intersection provides a $C^{1}$-robust heteroclinic
tangency associated with $\Lambda$ and $Q$.

###  2.1. Construction

We now give the details of our construction. Since our argument is local, we
can put $\mathcal{M}=\mathbb{R}^{d}$ with $d\geq 2$. First, we take a two-
dimensional diffeomorphism $h$ with a Plykin attractor $\Sigma$ constructed in
local coordinates inside a disk $D\subset\mathbb{R}^{2}$ with three holes. We
consider a $C^{r}$-diffeomorphism $f:\mathcal{M}\to\mathcal{M}$ with $r\geq 1$
such that for a small $\varepsilon>0$, the restriction of $f$ to the set
$D_{\varepsilon}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}[-\varepsilon,\varepsilon]^{d-2}\times D$ is given by

$f=g\times h\quad\text{where}\quad g(t)=\lambda t\ \ \text{for}\ \
t\in[-\varepsilon,\varepsilon]^{d-2}\ \ \text{and}\ \ 0<\lambda<1.$ (1)

Thus, the set

$\Lambda\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\\{0^{d-2}\\}\times\Sigma=\bigcap_{n\geqslant
1}f^{n}(D_{\varepsilon})$ (2)

is a hyperbolic attractor of $f$ and $D_{\varepsilon}$ is a trapping region of
$f$, i.e. $f(D_{\varepsilon})\subset\mathrm{interior}(D_{\varepsilon})$. The
structural stability of $\Lambda$ provides the existence of a
$C^{1}$-neighborhood $\mathcal{V}$ of $f$ such that for each
$g\in\mathcal{V}$, the continuation $\Lambda_{g}$ of $\Lambda$ has by trapping
region the set $D_{\varepsilon}$. We remark that the local stable manifolds
$W^{s}_{loc}(x)=W^{s}(x)\cap D_{\varepsilon}$ for $x\in\Lambda$ provide a
foliation of the set $D_{\varepsilon}$ by leaves (plaques) of dimension $d-1$.
It is not hard to verify that this property also holds for any diffeomorphism
$g$ in $\mathcal{V}$. We will denote by $W^{s}_{loc}(x,g)$ the stable local
manifold at $x$ for $g$.

Now we build the robust heteroclinc tangency of elliptic type. Recall that a
heteroclinic tangency $y\in W^{u}(Q)\cap W^{s}(P)$, is of elliptic type if
there is a neighborhood $U$ of $y$ contained in either, $W^{u}(Q)$ or
$W^{s}(P)$, say $W^{u}(Q)$, such that any point in $U-\\{y\\}$ belongs to the
same side of the tangent space $T_{y}W^{u}(Q)$. We consider a fixed point
$P\in\Lambda$ and a small open ball $B$ centered at $P$ such that
$\overline{B}$ is contained in $D_{\varepsilon}$. We observe that for every
$g\in\mathcal{V}$, $B$ is foliated by

$\mathcal{F}_{g}(x)\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}W^{s}_{loc}(x,g)\cap B,\quad x\in\Lambda_{g}.$

Consider a hyperbolic fixed point $Q\not\in D_{\varepsilon}$ of $f$ with
stable index $d-1-k$. By means of a homotopic deformation, we force to the
$(k+1)$-dimensional unstable manifold $W^{u}(Q)$ intersects non-transversely
the stable manifold $W^{s}(P)$ in a heteroclinic tangency of elliptic type,
namely $y$. Taking a suitable iterated if necessary, we can assume that $y$ is
in $B$.

\begin{overpic}[scale={0.7}]{S-2.pdf}
\scriptsize\put(108.0,100.0){\large$P_{g}$}
\put(60.0,122.0){\large$\mathcal{F}_{g}(\bar{t})$}
\put(163.0,150.0){\large$U_{g}$} \put(162.0,108.0){\normalsize$y_{g}$}
\put(133.0,175.0){\large$\gamma_{g}(t)$} \put(165.0,117.0){\circle*{2.0}}
\put(120.0,10.0){\large(a)} \put(330.0,175.0){\large$\gamma_{g}(t)$}
\put(267.0,125.0){\large$\mathcal{F}_{g}(\bar{t})$}
\put(360.0,135.0){\large$U_{g}$} \put(306.0,100.0){\large$P_{g}$}
\put(360.0,101.0){\normalsize$y_{g}$} \put(362.0,108.0){\circle*{2.0}}
\put(315.0,10.0){\large(b)} \end{overpic} Figure 3. (a) $C^{1}$-robust
equidimensional tangency. (b) $C^{1}$-robust heterodimensional tangency.

Thus, this last diffeomorphism, that again we call $f$, has a heteroclinic
tangency of codimension $c_{T}=1$ and signed co-index $k_{T}=k$ with $0\leq
k\leq d-2$, associated with the saddles $P\in\Lambda$ and $Q$.

On the other hand, by definition, there exists a neighborhood $U$ of $y$
contained in $W^{u}(Q)$ such that $U-\\{y\\}$ is contained in
$W^{s}(\Lambda)\pitchfork W^{u}(Q)$. See Figure 3. We will see that the
$C^{1}$-persistence of these last transverse intersections provides a
$C^{1}$-robust heteroclinic tangency associated with $\Lambda$ and $Q$.
Besides, for each $g\in\mathcal{V}$, we consider a small curve
$\gamma_{g}:t\in(-r,r)\mapsto\gamma_{g}(t)\in\Lambda_{g}\subset
D_{\varepsilon}$ parameterizing a small local unstable manifold of the
continuation $P_{g}=\gamma_{g}(0)$ of $P$ such that

$B=\bigcup_{t\in(-r,r)}\mathcal{F}_{g}(t)\quad\text{where}\quad\mathcal{F}_{g}(t)=\mathcal{F}_{g}(\gamma_{g}(t)).$

Since $y$ is a heteroclinic tangency of elliptic type between $W^{u}(Q)$ and
$W^{s}(P)$ we can assume that (see Figure 3)

$\begin{split}&U\cap\mathcal{F}_{f}(0)=\\{y\\},\\\
&U\cap\mathcal{F}_{f}(t)\subset W^{u}(Q)\pitchfork
W^{s}(\gamma_{f}(t))\quad\mbox{for all $t\in(0,r)$},\\\
&U\cap\mathcal{F}_{f}(t)=\emptyset\quad\mbox{for all
$t\in(-r,0)$}.\end{split}$

Our conditions imply that for each $g\in\mathcal{V}$, the set

$I_{g}=\\{t\in(-r,r):U_{g}\pitchfork\mathcal{F}_{g}(t)\neq\emptyset\\}$

is inferiorly bounded where $U_{g}$ is a continuation in $W^{s}_{loc}(Q_{g})$
of the neighborhood $U$. Thus, if $\bar{t}$ is the infimum of $I_{g}$ then
$U_{g}$ and $\mathcal{F}_{g}(\bar{t})$ meet in a heteroclinic tangency $y_{g}$
of codimension $c_{T}=1$ and signed co-index $k_{T}=k$ with $0\leq k\leq d-2$.
This completes the proof of Theorem A.

\begin{overpic}[scale={0.7}]{S-4.pdf}\scriptsize\put(282.0,125.0){\Large$P$}
\put(172.0,146.0){\Large$Q$} \end{overpic} Figure 4. Heterodimensional
tangency constructed from a deformation of diffeomorphism $f$ of
$\mathbb{T}^{3}$ locally defined as the product $g\times h$ where $h$ has a
DA-attractor in $\mathbb{T}^{2}$ and $g$ is a contraction.

##  3\. Differential construction of $C^{1}$-robust heteroclinic tangencies

In this section we will prove again Theorem A but now using a different
argument. This different approach allows us to generalize the result to get
robust heterodimensional tangencies of large codimension in the next section.
In order to explain the idea behind of this new approach we will consider
again the situation described in Figure 2.

\begin{overpic}[scale={0.74}]{Codigo-2.pdf}
\scriptsize\put(37.0,90.0){{\Large$\mathcal{S}\subset W^{u}(Q)$}}
\put(200.0,30.0){{\Large$W^{s}(P)$}}
\put(25.0,195.0){{\Large$\mathcal{C}^{s}_{\alpha}$}}
\put(185.0,215.0){\Large{$E$}} \put(182.0,177.0){\Large{$E^{s}$}}
\put(201.0,75.0){\Large{$T_{x}\mathcal{S}$}} \put(105.0,36.0){\circle*{3.0}}
\put(103.0,45.0){\Large$y$} \put(142.0,47.0){\circle*{3.0}}
\put(138.0,55.0){\Large$x$} \put(330.0,163.0){\normalsize{$(P,E^{s})$}}
\put(363.0,185.0){\Large{$W_{loc}^{s}(\Lambda^{s})$}}
\put(280.0,70.0){\Large$\mathcal{S}^{s}$}
\put(362.0,115.0){\normalsize{$y^{s}=(y,E^{s})$}} \end{overpic} Figure 5.
Figure on the left shows the folding manifold $\mathcal{S}$ on $\mathcal{M}$
and how cover the cone $\mathcal{C}^{s}_{\alpha}$ on $\mathbb{R}^{d}$. The
other shows the transverse intersection between $\mathcal{S}^{s}$ and the
stable manifold of $\Lambda^{s}$ on $\mathbb{R}^{d}\times G(s,d)$

By considering, if necessary, local coordinates around $P$, define the
projective cocycle $f^{s}(x,E)=(f(x),Df(x)E)$ where $x\in\mathbb{R}^{2}$ and
$E$ belongs to the space $G(1,2)$ of one-dimensional vector space in
$\mathbb{R}^{2}$. Recall that the $\Lambda$ is a hyperbolic attractor of $f$
with splitting $E^{s}\oplus E^{u}$. Hence, the set $\Lambda^{s}=\Lambda\ltimes
E^{s}=\\{(x,E):x\in\Lambda,\,E=E^{s}(x)\\}$ is a hyperbolic set of $f^{s}$
where the direction corresponding to the variable in $G(1,2)$ is uniformly
expanding. Thus, $W^{s}(\Lambda^{s})=W^{s}(\Lambda)\ltimes E^{s}=\\{(x,E):x\in
W^{s}(\Lambda),E=E^{s}(x)\\}$ is a two-dimensional manifold in the three-
dimensional space $\mathbb{R}^{2}\times G(1,2)$ as it is showed in Figure 5.
On the other hand the unstable manifold of $Q$ contains a folding manifold
that we denote by $\mathcal{S}$. That is, a small piece of the unstable
manifold contained the point $y$ in its interior. Namely, this manifold folds
with respect to the stable cone-field of $f$ at the point $P\in\Lambda$ as it
is represented Figure 5. That is, by considering linear transport to the
origin of $\mathbb{R}^{2}$, the union of tangent spaces $T_{x}S$ where
$x\in\mathcal{S}$ cover the cone
$\mathcal{C}^{s}_{\alpha}=\\{(x,y):|y|\leq\alpha|x|\\}$ for some small
$\alpha>0$. This property allows us to see the set $\mathcal{S}^{s}=S\ltimes
TS=\\{(x,E):x\in S,E=T_{x}\mathcal{S}\\}$ as a graph of a function
$E\in\mathcal{C}^{s}_{\alpha}\mapsto x=x(E)\in\mathcal{S}$. In other words, as
a one-dimensional manifold in $\mathbb{R}^{2}\times G(1,2)$ which is the image
of a graph over $G(1,2)$ and thus transversally intersecting
$W^{s}(\Lambda^{s})$ at the point $(y,E^{s})$ where
$E^{s}=E^{s}(P)=\mathbb{R}\times\\{0\\}$. Since this intersection is
transversal, it persists for any small perturbation. In particular, for any
small perturbation $g$ of $f$, we get a intersection point between
$\mathcal{S}^{s}$ and $W^{s}(\Lambda_{g}^{s})$ where $\Lambda_{g}^{s}$ is the
continuation of $\Lambda^{s}$ for cocycle $g^{s}$ induced by $g$ in
$\mathbb{R}^{2}\times G(1,2)$. Notice that this intersection point between
$\mathcal{S}^{s}$ and $W^{s}(\Lambda^{s}_{g})$ provides the tangency point and
direction between $\mathcal{S}$ and a stable manifold $W^{s}(z)$ for some
$z\in\Lambda_{g}$. Therefore, we get a robust tangency.

###  3.1. Construction

Now we will give the formal details. Recall the $C^{r}$-diffeomorphism
$f:\mathcal{M}\to\mathcal{M}$ in (1) and the attractor
$\Lambda=\\{0^{d-2}\\}\times\Sigma$ in (2). This set has a well defined
hyperbolic structure $T_{\Lambda}\mathcal{M}=E^{s}\oplus E^{u}$ where the
stable bundle $E^{s}$ of $\Lambda$ is $(d-1)$-dimensional. Observe that
$E^{s}$ can be uniquely extended to a continuous $Df$-invariant fiber bundle,
which we also denote by $E^{s}$, over each leaf $W^{s}_{loc}(x)$,
$x\in\Lambda$, and so to the whole set $D_{\varepsilon}$. Moreover, from the
hyperbolicity of $\Lambda$, we have that $E^{s}$ varies continuously with
respect to the point $x\in D_{\varepsilon}$ and the diffeomorphism $g$ in a
small $C^{1}$ neighborhood $\mathcal{V}$ of $f$. Thus, for each
$g\in\mathcal{V}$ the set $D_{\varepsilon}$ is foliated by $(d-1)$-dimensional
(local) stable manifolds of $\Lambda_{g}$ which are tangent to the bundle
$E^{s}_{g}$ continuation of $E^{s}$.

Fix $s\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}d-1$. On the set
$D_{\varepsilon}$ we have defined a stable cone-field
$\mathcal{C}^{s}_{\alpha}$ of dimension $s$ and size $\alpha>0$ satisfying

$E^{s}(x)\in\mathcal{C}^{s}_{\alpha}(x)\subset
T_{x}\mathbb{R}^{d}\quad\text{and}\quad
Df^{-1}(x)\mathcal{C}^{s}_{\alpha}(f(x))\subset\mathcal{C}^{s}_{\alpha}(x)\quad\mbox{for
all $x\in D_{\varepsilon}$}.$

In what follows, for notational simplicity, we omit the subscript $\alpha$ in
the notation $\mathcal{C}^{s}_{\alpha}$. The $s$-dimensional cone-field
$\mathcal{C}^{s}$ can be seen as an open set of the Grassmannian manifold
$G_{s}(\mathbb{R}^{d})=\mathbb{R}^{d}\times G(s,d)$ where $G(s,d)$ is set of
the $s$-planes in $\mathbb{R}^{d}$. Observe that in the case $d=2$, this
Grassmannian manifold is the projective space. Now consider the differential
cocycle induced by $f$ on $G_{s}(\mathbb{R}^{d})$ given by

$f^{s}:G_{s}(\mathbb{R}^{d})\to G_{s}(\mathbb{R}^{d}),\quad
f^{s}(x,E)=(f(x),Df(x)E).$

Observe that $f^{s}$ is a $C^{r-1}$-diffeomorphism of $G_{s}(\mathbb{R}^{d})$
with $r\geq 2$. Since $E^{s}$ is a repelling point of $Df$,

$\Lambda^{s}=\Lambda\ltimes E^{s}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\\{(x,E):x\in\Lambda,\ E=E^{s}(x)\\}$

is hyperbolic set of $f^{s}$ with stable index equals to $\dim E^{s}=s$.
Namely, the splitting of $\Lambda^{s}$ is of the form $E^{s}\oplus E^{u}\oplus
E^{uu}$ where $E^{s}\oplus E^{u}$ corresponds with the splitting of $\Lambda$
for $f$ and $E^{uu}$ with the directions over $G(s,d)$. On the other hand, the
local stable manifold $W^{s}_{loc}(\Lambda^{s})$ of $\Lambda^{s}$ contains the
set

$D^{s}_{\varepsilon}=D_{\varepsilon}\ltimes
E^{s}\stackrel{{\scriptstyle\scriptscriptstyle\rm def}}{{=}}\\{(x,E):x\in
D_{\varepsilon},\ E=E^{s}(x)\\}.$

This is a manifold of codimension the dimension of $G(s,d)$.

We now construct the heteroclinic tangency. First, we give a more formal
notion of heterodimensional tangency between any two manifolds.

######  Definition 3.1.

Let $\mathcal{L}$ and $\mathcal{N}$ be two submanifolds of $\mathcal{M}$. We
say that $\mathcal{L}$ and $\mathcal{N}$ has a heteroclinic tangency at
$x\in\mathcal{L}\cap\mathcal{N}$ if $c_{T}=d_{T}-k_{T}>0$ where

$d_{T}=\dim T_{x}\mathcal{L}\cap T_{x}\mathcal{N}\quad\text{and}\quad
k_{T}=\dim\mathcal{L}+\dim\mathcal{N}-\dim\mathcal{M}.$

The numbers $d_{T}=d_{T}(x,\mathcal{L},\mathcal{N})$,
$c_{T}=c_{T}(x,\mathcal{L},\mathcal{N})$ and
$k_{T}=k_{T}(x,\mathcal{L},\mathcal{N})$ are called, respectively, dimension,
codimension and signed co-index of the tangency between $\mathcal{L}$ and
$\mathcal{N}$ at $x$. The tangency is said to be _heterodimensional_ if
$k_{T}\not=0$ and _equidimensional_ if $k_{T}=0$.

For simplicity and clarity of the exposition we restrict the construction to
the case of signed co-index $k_{T}=s-1=d-2$. By means of a similar argument
one can also get the other possible co-index in Theorem A. We will consider
two types of tangencies: elliptical (see Section 2) and of saddle type. We
recall that a tangency $y\in W^{u}(Q)\cap W^{s}(P)$ is of saddle type if every
neighborhood $U$ of $y$ contained in either, $W^{u}(Q)$ or $W^{s}(P)$, say
$W^{u}(Q)$, intersects each connected component of $\mathbb{R}^{d}\setminus
T_{y}W^{u}(Q)$.

######  Example 3.2.

Consider a diffeomorphism having two periodic saddles $P$ and $Q$ such that
$0^{d}\in W^{s}_{loc}(P)\subset\mathbb{R}^{s}\times\\{0\\}$, with $0^{d}\neq
P$ and $\dim W^{s}(Q)=s$. Assume that, $\mathcal{S}([-1,1]^{s})\subset
W^{u}(Q)$, where

$\mathcal{S}:(t_{1},\dots,t_{s})\mapsto(t_{1},\dots,t_{s},t_{1}^{2}+\dots+t_{s}^{2})$
(3)

or

$\mathcal{S}:(t_{1},\dots,t_{s})\mapsto(t_{1},\dots,t_{s},t_{1}t_{2}+\dots+t_{s-1}t_{s}).$
(4)

Then $0^{d}$ is a heteroclinic tangency of elliptic type between $W^{s}(P)$
and $W^{s}(Q)$ choosing $\mathcal{S}$ as in (3) and the saddle type if
$\mathcal{S}$ is as in (4).

As it is usual, we identified the embedding $\mathcal{S}$ (as those described
above) with its image. Now, using the $s$-dimensional manifold $\mathcal{S}$
in (3) and (4) we create a tangency between the leaves the foliation of
$D_{\varepsilon}$ by stable manifold (of dimension $s$) of $\Lambda$. Fix a
fixed point $P\in\Lambda$ and consider $y\in W^{s}_{loc}(P)$ with $y\neq P$.
Modifying slightly the construction of the attractor if necessary, we can
consider coordinates $(t_{1},\dots,t_{d})$ in neighborhood of $P$ such that

  * •

$P$ is identified with $(1^{d-1},0)$ and $y$ with $0^{d}$,

  * •

the local unstable manifold $W^{u}_{loc}(P)$ is $t_{1}=\dots=t_{d-1}=1$,

  * •

for each $z=(1^{d-1},t)\in W^{u}_{loc}(P)$, the local stable manifold
$W^{s}_{loc}(z)$ is $t_{d}=t$; and

  * •

the bundle $E^{s}$ is trivial on this neighborhood.

Hence, in this local coordinates we can assume that
$E^{s}=\mathbb{R}^{s}\times\\{0\\}$ and

$\mathcal{C}^{s}=\\{(u,v)\in\mathbb{R}^{s}\oplus\mathbb{R}:\|v\|<\alpha\,\|u\|\\}\cup\\{0^{d}\\}$

where $\alpha>0$ is a small constant.

At this coordinates, the folding manifolds $\mathcal{S}$ in (3) and (4)
intersect $W^{s}(P)$ at $y$ in a heteroclinic tangency of codimension
$c_{T}=1$ and signed co-index $k_{T}=s-1=d-2$. The next result state that this
tangency persist under perturbations.

######  Proposition 3.3.

The folding manifold $\mathcal{S}$ has a heteroclinic tangency with the stable
foliation of $\Lambda$ which persists under small $C^{1}$-perturbations of
$f$.

The proof of this proposition makes use of the following result:

######  Lemma 3.4.

The set $\mathcal{S}^{s}=S\ltimes
TS\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\\{(x,E):x\in\mathcal{S},\ E=T_{x}\mathcal{S}\\}$ is a manifold of
dimension $\dim G(s,d)$ embedded as a disc in $\mathbb{R}^{d}\times G(s,d)$.
Namely it is a graph of a function of the form $E\in\mathcal{C}^{s}\mapsto
x=x(E)\in\mathcal{S}$.

Let us postpone for a while the proof of lema, to conclude the proof of the
proposition.

###### Proof of Proposition 3.3.

Since $\mathcal{S}$ tangentially meets $W^{s}_{loc}(P)$ at $y$, we get that
$\mathcal{S}^{s}$ topologically transversally intersect
$W^{s}_{loc}(\Lambda^{s})$ at $y^{s}=(y,E^{s})$. The (topological)
transversality follows from Lemma 3.4 since $\mathcal{S}^{s}$ is a disc of
$\dim G(s,d)$ and $W^{s}_{loc}(\Lambda^{s})$ is a manifold of codimension
$\dim G(s,d)$. See Figure 5.

Now consider a diffeomorphism $g$ $C^{1}$-close to $f$. Observe that the
cocycle $g^{s}$ is a homeomorphism of ${G}_{s}(\mathbb{R}^{d})$ only
$C^{0}$-close to $f^{s}$. However, $\Lambda^{s}_{g}=\Lambda_{g}\ltimes
E^{s}_{g}$ is still a topological hyperbolic set for $g^{s}$ where
$\Lambda_{g}$ and $E^{s}_{g}$ are the continuation of $\Lambda$ and $E^{s}$
for $g$. Thus, the set $W^{s}(\Lambda^{s}_{g})$ contains a manifold
$C^{0}$-close to $W^{s}_{loc}(\Lambda^{s})$ of codimension $\dim G(s,d)$. Thus
we still have a transversal intersection between $\mathcal{S}^{s}$ and
$W^{s}_{loc}(\Lambda^{s}_{g})$. Observe that if $(x,E)\in\mathcal{S}^{s}\cap
W^{s}_{loc}(\Lambda^{s}_{g})$ then $x\in\mathcal{S}\cap
W^{s}_{loc}(\Lambda_{g})$, $E=E^{s}_{g}(x)$ and $E=T_{x}\mathcal{S}$. Thus
$E^{s}_{g}(x)=T_{x}\mathcal{S}$. This provides a tangency between
$\mathcal{S}$ and the stable foliation of $\Lambda_{g}$ concluding the proof
of the proposition. ∎

######  Remark 3.5.

Proposition 3.3 also holds for any small enough $C^{1}$-perturbation of
$\mathcal{S}$. To see this, if the perturbation is $C^{1}$-close then we have
a change of variable $C^{1}$-close to the identity sending the perturbed
manifold to the folding manifold $\mathcal{S}$. Hence we get a new
diffeomorphism $g$ which is $C^{1}$-close to $f$. Thus, applying Proposition
3.3 we get a tangency.

Theorem A follows from the above proposition and remark by considering that
the folding manifold is contained in the unstable manifold of a hyperbolic
fixed point of $f$ of unstable index $s=d-1$. Observe that the codimension of
the tangency is given by the formula $c_{T}=d_{T}-k_{T}$ where $d_{T}$ is the
number of tangent directions and $k_{T}$ is the co-indice between the
hyperbolic set involved. In this case, $c_{T}=s-(s-1)=1$ and $k_{T}=s-1=d-2$.

To complete our construction we give the proof of Lemma 3.4.

###### Proof of Lemma 3.4.

To prove that $\mathcal{S}^{s}$ is an embedded disc in $\mathbb{R}^{d}\times
G(s,d)$ we need to show that $\mathcal{S}^{s}$ is a graph of an injective
function of the form

$\mathcal{S}^{s}:E\in\mathcal{C}^{s}\mapsto x=x(E)\in G(s,d).$

To do this, we must associate to $E$ an unique point $x\in\mathcal{S}$ such
that $E=T_{x}\mathcal{S}$. In other words, we need to show that

$\mathcal{C}^{s}\subset\bigcup_{x\in\mathcal{S}}T_{x}\mathcal{S}.$

As above, we are standing that $\mathcal{C}^{s}$ is a small open set in
$G(s,d)$ centered at $E^{s}=\mathbb{R}^{s}\times\\{0\\}$ and the tangent space
$T_{x}\mathcal{S}$ as a vector space of $\mathbb{R}^{d}$. Analytically, we
need to solve the following problem: given $E\in\mathcal{C}^{s}$ we look for
$t=(t_{1},\dots,t_{s})$ such that $E=T_{x}\mathcal{S}$ where
$x=\mathcal{S}(t)$.

In order to do the calculation, we choose the elliptic form of the folding
manifold given in (3). For folding manifold of saddle type in (4) the argument
is similar. Hence,

$T_{x}\mathcal{S}:(t^{\prime}_{1},\dots,t^{\prime}_{s})\in\mathbb{R}^{s}\mapsto(t^{\prime}_{1},\dots,t^{\prime}_{s},2t_{1}t^{\prime}_{1}+\dots+2t_{s}t^{\prime}_{s})\in\mathbb{R}^{d},\quad\text{where
$x=\mathcal{S}(t_{1},\dots,t_{s})$.}$

We write $E=\operatorname{span}\langle v_{1},\dots,v_{s}\rangle$ where
$v_{i}=(a_{1i},\dots,a_{di})$ for $i=1,\dots,s$. Hence $E=T_{x}\mathcal{S}$
if, and only if, $v_{i}\in T_{x}\mathcal{S}$ for all $i=1,\dots,s$.
Equivalently, if

$t^{\prime}_{ji}=a_{ji}\quad\text{for $j=1,\dots,s$ \ \ and}\ \ \
2t_{1}t^{\prime}_{1i}+\dots+2t_{s}t^{\prime}_{si}=a_{di}\quad\text{for all
$i=1,\dots,s$}.$

Hence,

$2\cdot\begin{pmatrix}a_{11}&\dots&a_{1s}\\\ &\ddots&\\\
a_{s1}&\dots&a_{ss}\end{pmatrix}\cdot\begin{pmatrix}t_{1}\\\ \vdots\\\
t_{s}\end{pmatrix}=\begin{pmatrix}a_{d1}\\\ \vdots\\\ a_{ds}\end{pmatrix}$ (5)

That is, we have a square linear system $At=b$ where $A=A(E)$ and $b=b(E)$
depends on the vector space $E$. To find $t$ we need to show that $A$ is an
invertible matrix. To do this, we will take as the vector space $E$ the center
$E^{s}=\mathbb{R}^{s}\times\\{0\\}=\operatorname{span}\langle e_{1},\dots
e_{s}\rangle$ of $\mathcal{C}^{s}$ where $e_{i}$ denotes the vector with a $1$
in the $i$-th coordinate and 0’s elsewhere. We get in this case that
$A(E^{s})=2\cdot I_{s}$ where $I_{s}$ is the identity square matrix of order
$s$. Thus $\det A(E^{s})\not=0$. Then by the continuity for all
$E\in\mathcal{C}^{s}$ close to $E^{s}$ we uniquely solve (5) and thus we find
$t=(t_{1},\dots,t_{s})$ such that $E=T_{x}\mathcal{S}$ where
$x=\mathcal{S}(t)$. This completes the proof of the lemma. ∎

##  4\. $C^{1}$-robust heterodimensional tangencies of large codimension

Fix $c_{T}\geq 1$ and $s>c_{T}$. Set $d=c_{T}\cdot(s+1)$. A hyperbolic set
$\Lambda$ of a diffeomorphism of a manifold $M$ is said to be a _codimension
one expanding attractor_ if for every $x\in\Lambda$, holds that
$W^{u}(x)\subset\Lambda$ and $\dim W^{u}(x)=\dim M-1$. Let us take a
codimension one expanding hyperbolic attractor $\Lambda$ of a diffeomorphism
$h$ on a manifold of dimension $n=d-s+1$. In order to avoid the problem of
classifying the manifold that support these kind of attractors, we set
$\Sigma$ as the _Derived from Anosov_ (by short $DA$-attractor) in the
$n$-torus $\mathbb{T}^{n}$, see [15]. After that, we will consider a $C^{r}$
diffeomorphism $f$ of $\mathbb{T}^{d}$ locally defined on
$D_{\varepsilon}=[-\varepsilon,\varepsilon]^{s-1}\times\mathbb{T}^{n}$ for a
fixed small $\varepsilon>0$ and $r\geq 2$ as

$f=g\times h\quad\text{where}\quad g(t)=\lambda t\ \ \text{for}\ \
t\in[-\varepsilon,\varepsilon]^{s-1}\ \ \text{and}\ \ 0<\lambda<1.$

Notice that the set $\Lambda=\\{0^{s-1}\\}\times\Sigma$ is a hyperbolic
attractor of $f$ whose basin of attraction contains $D_{\varepsilon}$.
Moreover, $E^{s}=\mathbb{R}^{s-1}\times\tilde{E}^{s}$ is the stable bundle of
$\Lambda$ where $\tilde{E}^{s}$ is the one-dimensional stable bundle of
$\Sigma$ for $h$. Thus, $s=\dim E^{s}$. Analogously as in previous sections,
this bundle can be uniquely extended to a $Df$-invariant bundle over
$D_{\varepsilon}$ which we also denote by $E^{s}$. Consequently the set
$D_{\varepsilon}$ is foliated by $s$-dimensional stable manifolds of $\Lambda$
which are tangent to $E^{s}$. This allows us to consider a stable cone-field
$\mathcal{C}^{s}$ of dimension $s$ defined in whole $D_{\varepsilon}$. As in
Section 3, we defined the differential cocycle $f^{s}$ induced by $f$ on
$G_{s}(\mathbb{R}^{d})$. Similarly, we have that the set
$\Lambda^{s}_{f}=\Lambda\ltimes E^{s}$ is also a hyperbolic set of $f^{s}$
with stable index equals to $s$ and whose local stable manifold
$W^{s}_{loc}(\Lambda^{s}_{f})$ contains the set
$D^{s}_{\varepsilon}=D_{\varepsilon}\ltimes E^{s}$. Thus, this manifold has by
codimension the dimension of $G(s,d)$.

Restricting us to a small ball $B\subset D_{\varepsilon}$, we can assume that
the stable cone is give by

$\mathcal{C}^{s}=\\{(u,v)\in\mathbb{R}^{s}\oplus\mathbb{R}^{d-s}:\|v\|<\alpha\|u\|\\}$
(6)

where $\alpha>0$ is small enough and
$E^{s}=\mathbb{R}^{s}\times\\{0^{d-s}\\}$. We will consider a folding manifold
$\mathcal{S}$ in $B$ folded with respect to $\mathcal{B}$ which we introduce
formally as follows:

######  Definition 4.1.

A manifold $\mathcal{S}$ of dimension $k\geq s$ is called _folding manifold_
in an open ball $B$ folded with respect to the cone $\mathcal{C}^{s}$ if
$\mathcal{S}\subset B$ and

$\overline{\mathcal{C}^{s}}\subset\bigcup_{x\in\mathcal{S}}T_{x}\mathcal{S}.$

We are understanding that $\overline{\mathcal{C}^{s}}$ is closure of the open
set $\mathcal{C}^{s}$ in $G(s,d)$ centered at $E^{s}$ which we see as a cone
in $\mathbb{R}^{d}$ and the tangent space $T_{x}\mathcal{S}$ as a
$k$-dimensional vector space of $\mathbb{R}^{d}$. Taking $\alpha$ tends to
zero we observe that the above definition is in fact an infinitesimal property
of $\mathcal{S}$. Thus without restriction we can assume that the tangent
space of $\mathcal{S}$ covers injectively the closure of $\mathcal{C}^{s}$.
This means that for every $E\in\mathcal{C}^{s}$ we have a unique
$x\in\mathcal{S}$ such that $E\leq T_{x}\mathcal{S}$. Moreover, $x=x(E)$
varies continuously with respect to $E$.

######  Example 4.2.

Take $k=d-c_{T}=c_{T}\cdot s$. Let us consider a $k$-dimensional manifold
$\mathcal{S}$ defined by

$\mathcal{S}:(t_{1},\dots,t_{k})\mapsto(t_{1},\dots,t_{k},\,t_{1}^{2},\,t_{2}^{2},\dots,t^{2}_{c_{T}-1},\,t^{2}_{c_{T}}+\dots+t_{k}^{2}).$
(7)

Hence, we have that

$T_{x}\mathcal{S}:(t^{\prime}_{1},\dots,t^{\prime}_{s})\mapsto(t^{\prime}_{1},\dots,t^{\prime}_{k},\,2t_{1}t^{\prime}_{1},\dots,2t_{c_{T}-1}t^{\prime}_{c_{T}-1},\,2t_{c_{T}}t^{\prime}_{c_{T}}+\dots+2t_{k}t^{\prime}_{k}),$

where $x=\mathcal{S}(t_{1},\dots,t_{k})$. We write
$E=\operatorname{span}\langle v_{1},\dots,v_{s}\rangle\in\mathcal{C}^{s}$
where $v_{i}=(a_{1i},\dots,a_{di})$ for $i=1,\dots,s$. Hence $E\leq
T_{x}\mathcal{S}$ if, and only if, $v_{i}\in T_{x}\mathcal{S}$ for all
$i=1,\dots,s$. Equivalently, if

$t^{\prime}_{ji}=a_{ji}\quad\text{for $j=1,\dots,k$ \ \ and}\ \ \
2t_{\ell}t^{\prime}_{\ell i}=a_{k+\ell\,i}\quad\text{for
$\ell=1,\dots,c_{T}-1$ \ \ and}$
$2t_{c_{T}}t^{\prime}_{c_{T}}+\dots+2t_{k}t^{\prime}_{ki}=a_{di}\quad\text{for
all $i=1,\dots,s$}.$

This defines a linear system of $c_{T}\cdot s$ equations and $k$ variable.
Since $k=c_{T}\cdot s$ we can write the system in the form $At=b$ where
$A=A(E)$ is a square matrix of ordem $k$ and $b=b(E)$ is a vector in
$\mathbb{R}^{k}$ depending on the vector space $E$. To find
$t=(t_{1},\dots,t_{k})$ we need to show that $A$ is an invertible matrix. To
do this, we will take as the vector space $E$ the center
$E^{s}=\mathbb{R}^{s}\times\\{0\\}=\operatorname{span}\langle e_{1},\dots
e_{s}\rangle$ of $\mathcal{C}^{s}$ where $e_{i}$ denotes the vector with a $1$
in the $i$-th coordinate and 0’s elsewhere. We get in this case that $\det
A(E^{s})=2$. Then, by continuity, for all $E\in\mathcal{C}^{s}$ close to
$E^{s}$ we uniquely solve the equation $At=b$ and thus we find
$t=(t_{1},\dots,t_{s})$ such that $E\leq T_{x}\mathcal{S}$ where
$x=\mathcal{S}(t)$. Therefore $\mathcal{S}$ is folding manifold with respect
to $\mathcal{C}^{s}$.

As a consequence of the definition of folding manifold we get the following
lemma:

######  Lemma 4.3.

Let $\mathcal{S}$ be a folding manifold folded with respect to
$\mathcal{C}^{s}$. Then the set

$\mathcal{S}^{s}\stackrel{{\scriptstyle\scriptscriptstyle\rm
def}}{{=}}\\{(x,E):x\in\mathcal{S},\ E\leq T_{x}\mathcal{S}\ \text{with}\ \dim
E=s\\}$

contains a manifold of dimension $\dim G(s,d)$ embedded as a disc in
$\mathbb{R}^{d}\times G(s,d)$.

###### Proof.

From the definition of folding manifold, we have an injective continuous
function $E\in\mathcal{C}^{s}\mapsto x\in\mathcal{S}$ such that $E\leq
T_{x}\mathcal{S}$. This defines a subset of $\mathcal{S}^{s}$ which is an
embedding given by $E\in\mathcal{C}^{s}\mapsto(x,E)\in\mathcal{S}\times
G(s,d)$ proving the lemma. ∎

The following result is the analogous to Proposition 3.3.

######  Proposition 4.4.

Let $\mathcal{S}$ be a folding manifold in $B$ of dimension
$k=d-c_{T}=c_{T}\cdot s$ folded with respect to $\mathcal{C}^{s}$. Then
$\mathcal{S}$ has a heterodimensional tangency of codimension $c_{T}$ and
signed co-index $k_{T}=s-c_{T}>0$ with the stable foliation of $\Lambda$ which
persists under small $C^{1}$-perturbations of $f$.

###### Proof.

By assumption if $x\in B$ then
$E^{s}(x)=\mathbb{R}^{s}\times\\{0^{d-s}\\}\in\mathcal{C}^{s}$. Thus, we have
that $\mathcal{S}$ has a heterodimensional tangency of codimension $c_{T}$
with $W^{s}(z)$ for some $z\in\Lambda$. Indeed, by definition of the folding
manifold $\mathcal{S}$ and the stable bundle $E^{s}$ we find
$x\in\mathcal{S}\subset B$ and $z\in\Lambda$ such that $E^{s}(x)\leq
T_{x}\mathcal{S}$ and $E^{s}(x)=T_{x}W^{s}(z)$. Furthermore, the signed co-
index of the tangency is $k_{T}=s+k-d=k-1=s-c_{T}>0$ and the codimension is
$d_{T}-k_{T}=s-(s-c_{T})=c_{T}$. On the other hand, the point $(x,E^{s}(x))$
belongs to $\mathcal{S}^{c}\cap W^{s}(\Lambda^{s})$. Moreover, from Lemma 4.3,
we have that $\mathcal{S}^{s}$ contains a disc of dimension $\dim G(s,d)$.
Additionally, $W^{s}(\Lambda^{s})$ has codimension $\dim G(s,d)$. Hence
$\mathcal{S}^{s}$ transversally intersect (in a topological sense)
$W^{s}(\Lambda^{s})$.

Arguing as in Proposition 3.3, we still have a transversal intersection
between $\mathcal{S}^{s}$ and $W^{s}(\Lambda^{s}_{g})$ for any $C^{1}$-close
diffeomorphism $g$ to $f$. Thus there is $(x,E)\in\mathcal{S}^{s}\cap
W^{s}(\Lambda^{s}_{g})$. Then $x\in\mathcal{S}\cap W^{s}(\Lambda_{g})$, $E\leq
T_{x}\mathcal{S}$ and $E=T_{x}W^{s}(z)$ for some $z\in\Lambda_{g}$. Similar as
above, this implies that $\mathcal{S}$ and $W^{s}(z)$ has a heterodimensional
tangency of codimension $c_{T}$ and signed co-index $k_{T}=s-c_{T}$ concluding
the proof of the proposition. ∎

###### Proof of Theorem B.

It suffices to consider that the folding manifold in Proposition 4.4 is
contained in the unstable manifold of a hyperbolic fixed point of $f$ of
unstable index $k$. ∎

##  5\. Discussion and open questions

The goal of this paper was to construct heteroclinic tangencies which are
robust under $C^{1}$ perturbations. This question was proposed in [12, pag.
3281] where the authors showed the existence of $C^{2}$-robust
heterodimensional tangencies. To approach this problem we have constructed
$C^{1}$-robust tangencies where one of the hyperbolic sets involved is an
attractor. This limitation prevents that our construction could be carried on
a heterodimensional cycle. A diffeomorphism has a _heterodimensional cycle_
associated with two transitive hyperbolic sets if these sets have different
indices (dimension of the stable bundle) and their invariant manifolds meet
cyclically. This cycle is called non-transverse (heterodimensional) cycle if
besides its cyclic intersections involves some heterodimensional tangency. In
order to construct a robust non-transverse heterodimensional cycle one must
construct the tangency involving hyperbolic sets which are not attractors.
This leads to our first question:

######  Question 1.

Is it possible to construct $C^{1}$-robust non-transverse heterodimensional
cycles?

Bearing in mind the classic constructions of robust homoclinic tangencies and
heterodimensional cycles ([19, 7]) via the unfolding of tangencies and cycles
associated with saddles, we ask the following:

######  Question 2.

Can a diffeomorphism $f$ having a non-transverse heterodimensional cycle
associated with saddles $P$ and $Q$ be $C^{r}$-approximated by a
diffeomorphism $g$ with a $C^{r}$-robust non-transverse heterodimensional
cycle associated with hyperbolic sets containing the continuations $P_{g}$ and
$Q_{g}$ of $P$ and $Q$?

On the other hand, we also deal in this paper with the construction of
heterodimensional tangencies with signed co-index $k>0$ of _large_
codimension. Robust tangencies of large codimension were discovered in [3].
Namely, the authors provided a method to construct $C^{2}$-robust _bundle
tangencies_ which are non-trivial intersection between different fiber
bundles. Bundle tangencies include homoclinic, heterodimensional and
equidimensional tangencies. Recently in [4], using similar ideas similar to
this paper, we have constructed new examples of robust homoclinic tangencies
of large codimension. The construction also uses an abstract notion of folding
manifold with respect to a cone-field extending previous approach on robust
homoclinic tangencies in [5]. However, as in the case of this work, the
construction are limited to consider high dimensional manifolds. The lower
possible dimension that allows to have a homoclinic tangency of large
codimension is $d=4$. Similarly, $d=5$ is the lower dimension to construct a
large heterodimensional tangency with signed co-index $k>0$. Thus we address
the following questions:

######  Question 3.

Is it possible to build a robust heterodimensional tangency with signed co-
index $k>0$ (resp. homoclinic tangency) of codimension $c_{T}=2$ in dimension
$d=5$ (resp. $d=4$)?

### Acknowledgements

We are grateful to Artem Raibekas for discussions and helpful suggestions.
During the preparation of this article PB was supported by MTM2017-87697-P
from Ministerio de Economía y Competividad de España and CNPQ-Brasil. SP were
partially supported by CMUP (UID/MAT/00144/2019), which is funded by FCT with
national (MCTES) and European structural funds through the programs FEDER,
under the partnership agreement PT2020. SP also acknowledges financial support
from a postdoctoral grant of the project PTDC/MAT-CAL/3884/2014

## References

  * [1] Abraham, R. and Smale, S.  _Nongenericity of $\Omega$-stability._ Global Analysis Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968, 5–8.
  * [2] Asaoka, M.  _Hyperbolic sets exhibiting $C^{1}$-persistent homoclinic tangency for higher dimensions._ Proc. Amer. Math. Soc., 136, (2008), 677–686.
  * [3] Barrientos, P. G. and Raibekas, A.  _Robust tangencies of large codimension._ Nonlinearity, 30, (2017), 4369–4409.
  * [4] Barrientos, P. G. and Raibekas, A.  _Robust nongeneric unfolding of cycles and tangencies._ preprint (2017).
  * [5] Bonatti, C. and Díaz, L. J.  _Abundance of $C^{1}$-homoclinic tangencies._ Trans. Amer. Math. Soc., 264 (2012), 5111-5148.
  * [6] Bonatti, C. and Díaz, L. J.  _On maximal transitive sets of generic diffeomorphisms._ Publ. Math., Inst. Hautes Étud. Sci., 96 (2003), 171–197.
  * [7] Bonatti, C. and Díaz, L. J.  _Robust heterodimensional cycles and $C^{1}$-generic dynamics._ J. Inst. Math. Jussieu, 7 (2008), 469–525.
  * [8] Díaz, L. J. and Kiriki, S. and Katsutoshi, S.  _Blenders in centre unstable Hénon-like families: with an application to heterodimensional bifurcations._ Nonlinearity, 27 (2014), 353–378.
  * [9] Díaz, L. J., Nogueira, A. and Pujals, E. R.  _Abundance of $C^{1}$-homoclinic tangencies._ Trans. Amer. Math. Soc., 264 (2012), 5111-5148.
  * [10] Díaz, L. J. and Pérez, S. A.  _Hénon-like families and blender-horseshoes at non-transverse heterodimensional cycles._ To appear in: Internat. J. Bifur. Chaos Appl. Sci. Engrg.
  * [11] Hirsch, M. W., Pugh, C. C. and Shub, M.  _Invariant manifolds._ Bull. Amer. Math. Soc., 76 (1970), 1015–1019.
  * [12] Kiriki, S. and Soma, T.  _$C^{2}$ -robust heterodimensional tangencies._ Nonlinearity, 25 (2012), 3277.
  * [13] Palis, J.  _A differentiable invariant of topological conjugacies and moduli of stability._ Dynamical systems, Vol. III—Warsaw, 335–346. Astérisque, No. 51, Soc. Math. France, Paris, 1978.
  * [14] Simon, Carl P.  _A 3-dimensional Abraham-Smale example._ Proc. Amer. Math. Soc., 34 (1972), 629–630.
  * [15] Smale, S.  _Differentiable dynamical systems._ Bull. Amer. Math. Soc., 73 (1967), 747–817.
  * [16] Robinson, C.  _Dynamical systems: stability, symbolic dynamics, and chaos._ CRC press, 1998.
  * [17] Mañé, R.  _A proof of the $C^{1}$ stability conjecture._ Inst. Hautes Études Sci. Publ. Math., 66 (1988), 161–210.
  * [18] Nitecki, Z.  _On semi-stability for diffeomorphisms._ Inventiones mathematicae, 14 (1971), 83–122.
  * [19] Newhouse, S. E.  _The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms._ Inst. Hautes Études Sci. Publ. Math., 50 (1979), 101–151.
  * [20] Williams, R. F.  _The $``{\rm DA}^{\prime\prime}$ maps of Smale and structural stability._ Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968, 329–334.


# Design and Validation of a Metallic Reflectarray for Communications at True
Terahertz Frequencies

Sherif Badran  0000-0002-4345-6329 Northeastern
UniversityBostonMassachusettsUSA<EMAIL_ADDRESS>,  Arjun Singh
0000-0003-0698-6790 SUNY Polytechnic InstituteUticaNew YorkUSA
<EMAIL_ADDRESS>,  Arpit Jaiswal  0000-0002-4421-8816 University at
BuffaloBuffaloNew YorkUSA<EMAIL_ADDRESS>,  Erik Einarsson
0000-0002-3896-2673 University at BuffaloBuffaloNew YorkUSA
<EMAIL_ADDRESS>and  Josep M. Jornet  0000-0001-6351-1754 Northeastern
UniversityBostonMassachusettsUSA<EMAIL_ADDRESS>
(2023)

###### Abstract.

This needs to be adapted to map the new introduction flow. Wireless
communications in the band ($0.110\text{\,}\mathrm{THz}$) is envisioned as a
key wireless technology to fulfill the demand for denser networks and higher
data rates due to the enormous available bandwidth. To overcome the complex
propagation channel, solely engineering the radiation pattern at the
transmitter is not sufficient, wavefront engineering along the propagation
path and at the receiver is also essential. Intelligent reflecting surfaces
(IRSs) can be utilized as a means to enable communication in the presence of
blockage or to aid a resource-limited quasi-omnidirectional transmitter direct
its radiated power. In this paper, we present a novel metallic reflectarray
for THz communications, namely at 1 THz. The fundamental building block of the
reflectarray is designed to have high reflection efficiency, exploiting a gold
patch and gold stubs with incremental lengths for phase manipulation. After
discussing the integration of the multiple reflecting elements in the
reflectarray, the ability to demonstrate controlled reflection or static
beamforming is presented. Finally, we discuss the fabrication procedure of
such reflectarray and validate its working principle through various
experimental measurements using THz Time-Domain Spectroscopy (THz-TDS) and
also using a continuous-wave (CW) terahertz testbed.

Terahertz communications, reflectarrays, graphene plasmonics, intelligent
reflecting surfaces, reconfigurable intelligent surfaces, beamforming.

††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: the 7th ACM Workshop on Millimeter-Wave and
Terahertz Networks and Sensing Systems; October 06, 2023; Madrid,
Spain††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Hardware Wireless
devices

##  1\. Introduction

The band has been envisioned as a key asset for the next generation of
wireless communication and sensing systems (akyildiz2022terahertz). The very
large bandwidth available at frequencies (easily tens to hundreds of ) opens
the door to links in front-haul applications, as in wireless immersive , and
ten to hundred in wireless backhaul applications, which can bridge the rural
digital divide. In addition, the smaller wavelength of signals opens the door
to sub-millimetric resolution in radar applications. Moreover, the smaller
wavelength also leads to smaller antennas that can be leveraged in
nanonetworking applications (jornet2023nanonetworking), including wireless
networks on chip or wireless nano-bio sensor and actuator networks. Finally,
the low but non-negligible photon energy of sub-terahertz and terahertz
waves—from $0.41\text{\,}\mathrm{meV}$ up to $41\text{\,}\mathrm{meV}$—leads
to very unique signatures in different elements (from gases in the atmosphere
to nano and bio-materials), which can be leveraged for target classification
purposes, for example.

All these opportunities come with a cost. First, molecular absorption losses
caused by water vapor are often presented as a major problem for systems
operating above $100\text{\,}\mathrm{GHz}$. However, this phenomenon is
present across all frequencies, but exhibits itself in the form of absorption
lines at specific frequencies. As a result, the band is practically divided
into multiple absorption-defined windows, whose usable bandwidth changes with
distance as well as other medium parameters, including pressure, temperature
or relative humidity. Second, the very small size of terahertz antennas leads
to low antenna effective areas. While this is not a problem for nanonetworking
applications, as the expected transmission range is usually under one meter,
the low effective area of the antenna leads to high spreading losses for
macroscale applications of the band. This requires the adoption of high-gain
directional antennas and/or focusing lenses. Finally, the interaction of
radiation with not just gases but effectively most materials can lead to
significant blockage resulting from signal absorption and/or reflection.

Sherif, Arjun, add some meaningful references only To overcome the challenging
propagation of signals, the adoption of has been proposed (di2020smart;
renzo2019smart). can engineer the reflection of signals, introducing, for
example, non-specular reflections, as well as more advanced functionalities
such as polarization switch or, more recently, wavefront engineering including
the transformation of spherical or Gaussian wavefronts into more robust Bessel
beams (singh2022bessel; arjun_wcm_2023). Physically, come mostly in two forms,
namely, programmable reflectarrays and reconfigurable metasurfaces.
Reflectarrays, the reflecting counterpart to conventional transmit and receive
antenna arrays, are integrated by metallic reflecting elements whose size and
spacing is in the order of half a wavelength. The reflection phase or delay
associated with each element can be controlled by means of controllable delay
lines (nayeri2018reflectarray). Metasurfaces, instead, are integrated by
elements that are much smaller than the signal wavelength, known as meta-
atoms. The smaller element size leads to enhanced functionalities but also
results in a much more challenging control of the elements.

Although the end goal is usually to have tunable reflectarrays and
metasurfaces at those frequencies, tunability is not always needed. In many
contexts, having a fixed response is sufficient, such as in the case of an
indoor communication scenario, where there are fixed or stationary blockers
(i.e., a pillar, wall, etc.), in such case, mounting, for instance, a
reflectarray to steer an incoming beam in a fixed direction to another
repeater or is sufficient. Similarly, in nanoscale applications, reflections
from the chip surface can be pre-programmed to provide connectivity with
different cores . Removing the reconfigurability requirement of can
drastically simplify their design, fabrication and control. This is
particularly true at higher operation frequencies, where common phase/delay
control elements are not available (singh2020d).

Towards this goal, in this paper, we design, build and experimentally
characterize a pre-programmed reflectarray that operates in the first
absorption-defined window above 1 THz, i.e., between 1 and 1.05 THz. The
proposed design consists of an array of metallic reflecting patches with
delay-controlling metallic stubs with micrometric dimensions (Section 2). To
realize this structure, we have fabricated patch antennas with a micrometer-
scale dielectric cavity and dimensions tailored to the specific design
criteria (Section 3). We experimentally characterize the structure by
utilizing two complementary approaches, namely, pulse-based broadband time-
domain spectroscopy and narrow-band communication using data-bearing signals
(Section 4). With the results at hand, we then the next steps and potential
future directions (Section 5).

##  2\. Design and Simulation

In this section, we detail the design of the individual metallic patch, as
well as the additional delay lines that are implemented through the means of a
metallic stub whose length is carefully selected.

###  2.1. Single Element Design and Principle

####  2.1.1. Patch

to change - currently from previous As per (balanis2015antenna), we utilize
the cavity model to derive the required design of the patch :

(1) $W=\frac{c}{2f_{0}\sqrt{\frac{\varepsilon_{r}+1}{2}}},$ (2)
$L=\frac{c}{2f_{0}\sqrt{\varepsilon_{\mathrm{eff}}}}-0.824h\biggl{[}\frac{(\varepsilon_{\mathrm{eff}}+0.3)(\frac{W}{h}+0.264)}{(\varepsilon_{\mathrm{eff}}-0.258)(\frac{W}{h}+0.8)}\biggr{]}.$

Here, for a given design frequency $f_{0}$, the width $W$ and length $L$ of
the patch at a substrate thickness $h$ are found. For optimal efficiency at
the desired frequency, $h$ is required to be in the range of
$0.003-0.05~{}\lambda_{0}$, as per (balanis2016antenna). $\varepsilon_{r}$
represents the dielectric constant of the substrate. Since the electric fields
of the patch antenna undergo fringing at the boundary of the patch edges, the
effective dielectric constant is $\varepsilon_{\mathrm{eff}}$, which is found
from (balanis2016antenna) as:

(3)
$\varepsilon_{\mathrm{eff}}=\frac{\varepsilon_{r}+1}{2}+\frac{\varepsilon_{r}-1}{2}[1+12h/W]^{-1/2}.$

The patch acts as a good reflectarray element through the reciprocity
principle: The patch is designed in transmission mode, with resonance at the
desired frequency verified through the S11 parameter, or the reflection
coefficient (balanis2016antenna). A good transmitter is also a good receiver,
and since the reflectarray element works by first receiving and then re-
transmitting the signal, the patch is a resonant reflectarray element
(nayeri2018reflectarray). gold only, no graphene

####  2.1.2. Delay lines

to do

###  2.2. Reflectarray Integration

to do

If there is nothing to write here, just make the previous subsubsections into
subsections and remove this heading

##  3\. Fabrication

(a) Cross-sectional schematic of element structure  (b) Optical micrograph of
IRS (scale is $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$)  (c) Photograph of
IRS

Figure 1. Schematic and optical images of the IRS. The size of the patterned
area in (c) is $12.75\times$12.75\text{\,}\mathrm{mm}$$.

We fabricated the IRS on substrates consisting of a $300\text{\,}\mathrm{nm}$
($20\text{\,}\mathrm{nm}$ Ti + $260\text{\,}\mathrm{nm}$ Ag +
$20\text{\,}\mathrm{nm}$ Au) metallic ground plane deposited atop a Si/SiO2
wafer. $300\text{\,}\mathrm{nm}$ was chosen to ensure that the metal is
sufficiently thicker than the skin depth for THz waves incident on Ag to
ensure high reflectivity (abohmra2020terahertz). A
$2.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ SU8 layer was spin coated on the
ground plane and the reflectarray designs were patterned on the SU8 dielectric
layer by photolithography.

The fabrication steps for this reflectarray are as follows. A Si/SiO2
substrate was cleaned by sonication in acetone, isopropyl alcohol, and
deionized water for $5\text{\,}\mathrm{min}$ each, followed by N2 blow dry and
heating for $5\text{\,}\mathrm{min}$ at
$150\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The substrate was then treated
with O2 plasma at $65\text{\,}\mathrm{W}$ power for $120\text{\,}\mathrm{s}$.
The O2 flow rate was set to
$98\text{\,}\mathrm{s}\mathrm{c}\mathrm{c}\mathrm{m}$ and the pressure in the
chamber was $300\text{\,}\mathrm{mTorr}$. This helps in removing organic
contaminants from the surface of the substrate. The cleaned substrate was then
inserted into an electron beam evaporation system at
${10}^{-6}\text{\,}\mathrm{Torr}$ for deposition of the ground plane metal.
The structure of the deposited metal was: $20\text{\,}\mathrm{nm}$ Ti,
$260\text{\,}\mathrm{nm}$ Ag, and $20\text{\,}\mathrm{nm}$ Ti. Silver was
chosen as the desired ground plane metal because it shows the highest
reflectivitys. The $20\text{\,}\mathrm{nm}$ Ti layers on top and bottom of Ag
served as adhesion layers. $0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ SiO2
was then deposited on top of the metal at ${10}^{-6}\text{\,}\mathrm{Torr}$. A
schematic of the final structure is shown in Fig. 1a

To create a $2.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ dielectric layer, SU8
2002 (dielectric constant, $\kappa=2.75$) was mixed with SU8 2010 in a 5:1
ratio to get the desired viscosity. This mixture was spin-coated on the
sputtered SiO2 at 3000 rpm for $30\text{\,}\mathrm{s}$ followed by post-coat
bake at $95\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for
$1\text{\,}\mathrm{min}$. The baked SU8 was then exposed to UV light for
$40\text{\,}\mathrm{s}$ using a mask aligner followed by a post-exposure bake
at $150\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $4\text{\,}\mathrm{min}$.
This hardens the coated layer of SU8 making it adhere to the underlying SiO2.
As the adhesive strength between metal and SU8 is weak, another
$0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ layer of SiO2 was deposited on top
of this exposed $2.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ SU8 in order to
ensure the adhesion. We have found that in the absence of these sputtered SiO2
layers, the SU8 fails to stick to the ground plane and to the array and is
removed during the final lift-off process.

Photolithography was used to write the array pattern onto the substrate. In
order to get better lift-off results, a bilayer photoresist coating approach
was used. First, LOR3B photoresist was spin coated on the substrate (Si/SiO2
\+ ground plane + dielectric separation layer) at
$2500\text{\,}\mathrm{r}\mathrm{p}\mathrm{m}$ for $45\text{\,}\mathrm{s}$
followed by post-coat bake at $190\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for
$4\text{\,}\mathrm{min}$. A second photoresist (Microposit S1813) was then
spin coated at 5000 rpm for $45\text{\,}\mathrm{s}$ on top of the LOR3B
followed by post-coat bake at $115\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for
$1\text{\,}\mathrm{min}$. The substrate was subsequently placed in contact
with the designed photomask using the mask aligner and UV exposed (i-line,
$350\text{\,}\mathrm{W}$ Hg lamp) for $50\text{\,}\mathrm{s}$
($4.5\text{\,}\mathrm{A}$).

After the photolithography, the photoresists were developed in Microposit
MF-319 solution for $45\text{\,}\mathrm{s}$ followed by rinsing in DI water
for $30\text{\,}\mathrm{s}$. A $300\text{\,}\mathrm{nm}$ layer of metal
($20\text{\,}\mathrm{nm}$ Ti, $260\text{\,}\mathrm{nm}$ Ag, and
$20\text{\,}\mathrm{nm}$ Au) was deposited to form the metal arrays. The Au
layer on top was deposited in order to prevent the Ag surface from oxidation.
The sample was then left in Microposit Remover 1165 solution overnight at
$70\text{\,}\mathrm{\SIUnitSymbolCelsius}$. After the lift-off, the sample was
rinsed in IPA and DI water for $5\text{\,}\mathrm{min}$ each, followed by N2
blow dry.

##  4\. Experimental Validation

In this section, we show various measurements of the reflectarray using two
platforms, i.e., terahertz time-domain spectroscopy (THz-TDS) and a
continuous-wave terahertz testbed, to demonstrate controlled reflection and
validate its working principal.

###  4.1. Terahertz Time-Domain Spectroscopy

The advantage of THz-TDS (Advantest TAS7500TS) is rapid, broadband
measurements in reflection or transmission (see e.g., (Grischkowsky:90;
Dai:04)). A schematic of the measurement setup (Vandrevala:19) is shown in
Fig. 2

Figure 2. Schematic of the THz-TDS measurement setup in reflection geometry.

In this THz-TDS system, shown schematically in Fig. 2, a $\mathrm{fs}$ pulse
from an IR laser is incident on a photoconductive antenna, which generates a
broadband THz pulse. The nominal range is either $0.34\text{\,}\mathrm{THz}$
or $0.57\text{\,}\mathrm{THz}$, depending on the emitter. After interacting
with the sample, the THz pulse is detected at a second photoconductive antenna
by mixing with a delayed IR pulse.

The non-specular reflection by the IRS should direct the component of the
broadband THz pulse at the resonance frequency _away_ from the detector,
resulting in a clear dip in reflectance. This can be clearly seen in the
reflectance spectrum plotted in Fig. 3, where the blue curve shows a clear dip
in reflectance centered at $0.9\text{\,}\mathrm{THz}$. The dashed red curve is
the reflection of a uniform metal film of equal thickness, and both are
normalized to the reflection from the bare substrate.

(a) THz reflectance vs. metal film  (b) Reflectance at $None$ and $None$ to
THz polarization

Figure 3. THz-TDS reflectance measurements of IRS normalized to substrate.

The dip at $0.9\text{\,}\mathrm{THz}$ is eliminated when the sample is rotated
$None$ to the incident THz polarization (Fig. 3b), indicating the response is
indeed due to the element geometry and not some other factor.

###  4.2. Terahertz Testbed

We can cut this a lot… we don’t to explain again what teranova is. We don’t
need subsubections for hardware and software, just one paragraph for each at
most

Figure 4. The terahertz testbed depicting its various modular components.

The reflectarray is further validated and characterized using a continuous-
wave terahertz platform (SEN2020107370) depicted. We discuss the various
aspects of the platform and demonstrate controlled reflection at
$1\text{\,}\mathrm{THz}$.

####  4.2.1. Hardware Setup

Figure 5. A block diagram depicting the interconnection of the various modular
components of the terahertz testbed.

The testbed consists of a transmitter with a high-performance analog
programmable signal generator (PSG) and an arbitrary waveform generator (AWG)
from Keysight Technologies, an upconverter frontend, along with a directional
high-gain horn antenna, encompassing true terahertz frequencies (1–1.05 THz).
The PSG is used to generate the local oscillator (LO) signal, it can generate
signals with frequencies up to 50 GHz. The AWG is utilized to generate the
intermediate frequency (IF) signal, which will be then mixed with the LO
signal, and thus, upconverted to a higher radio frequency (RF) signal. The
arbitrary waveform generator has a high sampling rate of up to 92 GSa/s. The
upconverter is manufactured by Virginia Diodes, Inc. (VDI) and operates in the
1–1.05 THz band. It consists of a frequency multiplier chain of $\times 24$, a
frequency mixer with a double sideband (DSB) conversion loss of 15 dB, and a
rectangular waveguide to facilitate connecting the antenna. The transmit power
at RF before feeding the antenna is about -15 dBm
($31.6\text{\,}\mathrm{\SIUnitSymbolMicro W}$) and the horn antenna gain is 26
dBi with a half-power beamwidth (HPBW) of $None$.

The testbed receiver consists of a similar PSG and a high-performance digital
storage oscilloscope (DSO) from Keysight Technologies, a downconverter
frontend, along with a similar directional high-gain horn antenna. The
receiver PSG is used to generate the LO signal at the receiver side. The DSO
has a very high sampling rate of up to 160 GSa/s and a bandwidth of up to 63
GHz. The downconverter is also manufactured by VDI and operates in the same
1–1.05 THz band. It has the same architecture as the upconverter (i.e.,
frequency multiplier chain and mixer), additionally, it is equipped with a
high-gain IF low-noise amplifier (LNA). Fig. 5 depicts how the different
transmitter and receiver components are connected to one another. The 10 MHz
reference cable is utilized to synchronize the transmitter and receiver PSGs
internal oscillators to compensate for the carrier frequency and phase offsets
(CFO and CPO), ensuring the specific hardware response is not affecting the
actual measurements.

####  4.2.2. Digital Signal Processing Backend

The digital signal processing (DSP) backend of the testbed comprises of
designing discrete-time pulse-shaped modulated waveforms in MATLAB and
uploading them to the AWG. The AWG then generates the modulated IF signals,
which are upconverted to RF by the VDI upconverters, transmitted over the air,
received by the VDI downconverter frontends, get downconverted back to IF, and
finally captured and stored via the DSO. Post-processing and DSP techniques
can be then applied to the captured signals in MATLAB. With this flexibility,
we can experimentally test any arbitrary modulation scheme/technique. The AWG
indefinitely sends the same waveform (i.e., frame) repeatedly. The frame
structure we design consists of header bits, which comprise of a high-
autocorrelation sequence/code (e.g., Barker codes, maximal merit factor codes,
etc.) known to both the transmitter and receiver, which are used for
time/frame synchronization at the receiver, silence (i.e., zeros) which is
used to aid in visualizing the looping frames from the AWG at the receiver,
and thus facilitates capturing by visualizing the number of incoming frames,
pilot bits which are known to both the transmitter and receiver and are used
for channel estimation/training, payload bits which are the actual
information-bearing/data bits, and zero padding (i.e., zeros) which designates
the end of the frame and to comply with the AWG input format, as the number of
samples in the waveform file must be a multiple of 128. For channel
equalization, we implement a minimum mean square error (MMSE) equalizer with
optimization of the number of taps.

####  4.2.3. Controlled Reflection Validation

Figure 6. Experimental validation setup of the fabricated metallic
reflectarray using the $11.05\text{\,}\mathrm{THz}$ up and downconverter
frontends depicting a controlled reflection scenario at
$\theta=$22.5\text{\,}\mathrm{\SIUnitSymbolDegree}$$.

(a) Metallic Sheet (b) Reflectarray

Figure 7. Power spectrum ($50\text{\,}\mathrm{\SIUnitSymbolOhm}$) at IF for
the received reflected signal at $\theta=$$$ for a normally-incident signal.

(a) Metallic Sheet (b) Reflectarray

Figure 8. QPSK constellation diagram (Bandwidth = $500\text{\,}\mathrm{MHz}$,
Bitrate = $500\text{\,}\mathrm{M}\mathrm{b}\mathrm{p}\mathrm{s}$) for the
received reflected signal at $\theta=$$$ for a normally incident signal.

The transmitter is placed such that the incident beam is normal to the
reflectarray plane of incidence, and the receiver is placed in the direction
of the controlled reflection ($\theta=$$$), as depicted in Fig. 6. First, we
send, in intermediate frequency, a signal comprising of the sum of five
sinusoidal signals at 1, 2, 3, 4, and $5\text{\,}\mathrm{GHz}$. They were
successfully received with high signal-to-noise ratio (SNR) as shown in Fig. 7
in the case of the reflectarray, but nothing was observed when replacing the
reflectarray with a metallic sheet. Second, we send, in intermediate
frequency, a QPSK modulated data signal with $f_{c}=$1\text{\,}\mathrm{GHz}$$
and $BW=$500\text{\,}\mathrm{MHz}$$, and comprising of 200 pilot bits and 2000
data bits. The constellation diagram for the transmitted, normalized received,
and equalized signals is shown in Fig. 8. The effective received data rate was
495.05 Mbps, with a measured bit error rate (BER) of 0.0335 (67 error bits out
of 2000 data bits) without any error correction. The measured BER when the
reflectarray was replaced by a metallic sheet was 0.499, which indicates
failed reception. Finally, the SNR was measured at 1 GHz IF to be 32.9194 dB
in the controlled reflection scenario ($\theta=$$$) and 1.3899 dB in the
specular reflection scenario (i.e. equal incident and reflected angles from
the normal). The previous results state the clear difference between the
reflectarray and, for example, a metallic sheet, which acts as a specular
reflector and cannot achieve controlled reflection.

##  5\. Conclusion

In this paper, we demonstrated the first fabricated reflectarray achieving
static controlled reflection at true terahertz frequencies. We first discussed
the various aspects and challenges relating to the design and fabrication,
then we experimentally validated the working principal of the reflectarray
using two systems, i.e., THz–TDS and a CW–THz testbed. For future work, we
plan to fabricate another prototype achieving a higher reflection efficiency,
thus allowing us to utilize the reflectarray for multi-Gbps PSK and QAM
modulated data signals. Eventually, the end goal is to fabricate a prototype
of a fully tunable graphene-based reflectarray.

###### Acknowledgements.

This work was funded by the Air Force Research Laboratory award
FA8750-20-1-0500 and the National Science Foundation awards CNS-1955004 and
CNS-2011411.