# Electroadhesive Auxetics as Programmable Layer Jamming Skins for Formable
Crust Shape Displays

Ahad M. Rauf1, Jack S. Bernardo1, and Sean Follmer1 1Authors are with the
Department of Mechanical Engineering at Stanford University, Stanford, CA
94305, USA<EMAIL_ADDRESS>
###### Abstract

Shape displays are a class of haptic devices that enable whole-hand haptic
exploration of 3D surfaces. However, their scalability is limited by the
mechanical complexity and high cost of traditional actuator arrays. In this
paper, we propose using electroadhesive auxetic skins as a strain-limiting
layer to create programmable shape change in a continuous (“formable crust”)
shape display. Auxetic skins are manufactured as flexible printed circuit
boards with dielectric-laminated electrodes on each auxetic unit cell (AUC),
using monolithic fabrication to lower cost and assembly time. By layering
multiple sheets and applying a voltage between electrodes on subsequent
layers, electroadhesion locks individual AUCs, achieving a maximum in-plane
stiffness variation of 7.6x with a power consumption of 50 µW/AUC. We first
characterize an individual AUC and compare results to a kinematic model. We
then validate the ability of a 5x5 AUC array to actively modify its own axial
and transverse stiffness. Finally, we demonstrate this array in a continuous
shape display as a strain-limiting skin to programmatically modulate the shape
output of an inflatable LDPE pouch. Integrating electroadhesion with auxetics
enables new capabilities for scalable, low-profile, and low-power control of
flexible robotic systems.

##  I Introduction and Related Work

Shape displays allow users to freely explore and manipulate surfaces through
haptic interactions, a concept known as “digital clay” [1, 2]. They
traditionally work using an array of linear actuators that move up and down to
render a discretized 2.5D depth map [3, 4]. Shape displays are powerful tools
for design and architecture, enabling tactile manipulation of CAD models,
physical user interfaces that can adapt to individual users or tasks, and
accessible 3D prototyping tools for blind or visually impaired makers [5, 6].
However, discretized actuator arrays scale poorly to smooth surfaces, where
their mechanical complexity and cost inhibit immersive render resolutions and
display sizes [7, 8].

Formable crust shape displays differ from conventional motor arrays by
embedding the actuation mechanism into the surface itself, reducing assembly
bulk and facilitating curved surface rendering [1, 2]. Using elastomeric
materials as flexible substrates, several actuation methods have been tested
for these programmable surfaces, including pneumatics [9, 10], hydraulics
[11], and shape memory alloys [12]. However, these typically require
complicated and expensive fabrication, long actuation times, or high power
consumption, impeding large-scale continuous shape displays.

Figure 1: Conceptual diagram for a 2-layer auxetic sheet stack. The bottom
sheet’s auxetic unit cells (AUCs, blue) are rotated by 90∘ relative to the top
sheet’s AUCs (orange), causing their squares to rotate in opposite directions
when stretched. (a) Auxetic layer stack stretched axially by force $4f$. The
left AUC is locked in place via electroadhesion, while the right AUC is free
to expand. (b) Locking only part of a formable crust shape display’s surface
results in global shape change. (c) Exploded view of the electroadhesive layer
stack for a single AUC. (d) Electroadhesive normal force $f_{ea}$ is generated
between the metallized electrodes and amplified by the dielectric film when a
voltage differential is applied.

Auxetics present a promising alternative to elastomeric materials when
integrating programmable-stiffness capabilities into flexible surfaces.
Auxetics have a negative Poisson’s ratio ($\nu<0$), allowing them to expand
and conform to high-curvature surfaces without collapsing [13, 14]. They can
be made from traditional materials and plastics using monolithic manufacturing
techniques like flexible printed circuit board (PCB) fabrication and laser
cutting. This lowers assembly complexity and cost while integrating
electronics for programmable actuation [15]. Prior works using 3D printed [16,
17] or laser cut [13, 14, 18] meta-materials have applied these properties to
create passive auxetic skins with pre-programmed stiffness profiles for soft
robots and shape displays, allowing a single device to produce a variety of
output shapes by manually changing the skin wrapped around it at the time.
However, little work has embedded programmable stiffness capabilities into
planar auxetic materials to modulate the shape output at runtime. We propose
that by fabricating auxetic sheets as flexible PCBs, we can integrate planar
electroadhesion actuation with auxetics for scalable, low-cost, and low-power
on-demand stiffness control.

Electroadhesion is the electrostatic attractive force between two metallized
surfaces separated by a dielectric material and held at a potential
difference. It has gained increasing interest for its high force output given
its low mass and power consumption [19], with applications including low-
profile clutches [20, 21, 22], exoskeleton actuation [23, 24], and conformal
grippers [25, 26, 27]. Its integration into volumetric auxetic structures [28]
shows promise for tunable stiffness control, motivating our extension to
planar auxetic surfaces.

Fig. 1 shows the layer stack we used to integrate electroadhesion with auxetic
sheets, for use as a strain limiting layer in a formable crust shape display.
We fabricated auxetic sheets as flexible PCBs with metallized electrodes on
each AUC, and one sheet (colored blue in Fig. 1) is assembled with conductive
adhesive and a dielectric film to amplify the electrostatic attractive force.
When another auxetic sheet (colored orange) with exposed metallized electrodes
is rotated by 90∘ and laid over the assembled sheet, their squares rotate in
opposite directions when stretched. Applying a voltage between the two layers
generates an electroadhesive normal force and thus a frictional shear braking
torque, locking the AUC and preventing it from expanding. This layer stack can
be extended to three or more auxetic sheets by metallizing both sides of the
flexible PCB and alternating high voltage and ground layers. Fig. 1(b)
demonstrates how locally locking selected AUCs results in global shape change
when overlaid as a strain limiting layer on top of an inflatable low-density
polyethylene (LDPE) pouch. Low-stiffness regions inflate more given an
isotropic pressure input, achieving higher curvature than locked surface
regions.

In this work, we motivate the integration of electroadhesion as a programmable
stiffness actuation mechanism for auxetic sheets. We characterize these
stiffening properties across a variety of materials and match the experimental
data to a kinematic model with good conformity. We explore the ability of
locking multiple AUCs in an auxetic array to control the sheets’ axial and
transverse stiffness, achieving global shape change when stretched. Finally,
we demonstrate the ability of this actuator to modulate the shape output of a
continuous, formable crust shape display. A video of the device in action can
be found at https://youtu.be/c7cHUNBqbKU.

##  II Design of an Individual Electroadhesive Auxetic Unit Cell Layer Stack

Figure 2: Dimensional drawing of square AUC when (a) unstretched and (b)
stretched. The metallized pad is offset by 0.444mm to account for space taken
up by electrical wiring.

###  II-A Planar Kinematic Model

We first develop a planar kinematic model for our electroadhesive auxetic
layer stack to understand the relevant design variables. We chose to study
square unit cells for their Poisson’s ratio $\nu=-1$, which allows the sheet
to retain geometric similarity to its original shape even when stretched [29,
15]. A dimensional diagram of our square AUC is shown in Fig. 2. Each square
has side length $a$ and out-of-plane thickness $t$, and the hinge joint
between squares is modeled as a beam with width $\delta$ and length $c$. When
stretched by an applied linear force $f$, each node reaches an angle $\theta$
with respect to the force axis, and each hinge joint reaches a radius of
curvature $\rho$ and bending angle $\phi=2\theta$.

Inspired by [13], we can estimate the AUC’s equilibrium position $\theta$ by
balancing the torques about each individual square’s center from the applied
force $\tau_{f}$, the bending of the hinge joints $\tau_{j}$, and the
frictional torque from electroadhesion $\tau_{ea}$, as shown in Fig. 2(b).
This model relies on three key assumptions: no out-of-plane motion, no
buckling or plastic deformation, and that the hinge joints act as Euler-
Bernoulli beams with no length change along their neutral axes.

The applied force $f$ induces a torque about the center:

$\tau_{f}=fa(\cos(\theta)-\sin(\theta))$ (1)

As derived by [29], we can model the hinge joints as Euler-Bernoulli beams.
The bending radius and resulting torque from all four joints are thus:

$\rho=\frac{c}{\phi}=\frac{c}{2\theta}$ (2)
$\tau_{j}=\frac{4EI}{\rho}=\frac{2E}{3}\left(\frac{\delta^{3}t}{c}\right)\theta$
(3)

where $E$ is the substrate’s Young’s modulus and $I=\frac{\delta^{3}t}{12}$ is
the hinge joint’s area moment of inertia.

Finally, the normal force created from electroadhesion is:

$f_{ea}=\frac{\epsilon_{0}\epsilon_{r}}{2}\left(\frac{V^{2}}{d^{2}}\right)A=:\sigma
A$ (4)

where $\epsilon_{0}=8.854\cdot 10^{-12}N/V^{2}$ is the permittivity of free
space, $\epsilon_{r}$ is the relative permittivity of the dielectric material,
$V$ is the potential difference, $d$ is the dielectric thickness, $A$ is the
overlap area of the charged electrodes, and $\sigma$ serves as the
electrostatic pressure.

Assuming a frictional coefficient $\mu$ and, for simplicity, that the overlap
area is a square of side length $a_{ov}$, we can then compute the frictional
torque as:

$\displaystyle\tau_{ea}$ $\displaystyle=\int r\mu\sigma dA$ (5)
$\displaystyle=\mu\sigma\int_{-a_{ov}/2}^{a_{ov}/2}\int_{-a_{ov}/2}^{a_{ov}/2}\sqrt{x^{2}+y^{2}}\textrm{
}dx\textrm{ }dy$ $\displaystyle=\frac{\mu\sigma
a_{ov}^{3}}{6}\left(\textrm{arsinh}(1)+\sqrt{2}\right)$

By balancing the torque equation

$\tau_{f}-\tau_{j}-\tau_{ea}=0$ (6)

we can solve for the equilibrium angle $\theta$ via a root-finding algorithm
such as Newton’s method.

###  II-B Testing Setup for an Individual Auxetic Unit Cell

In order to test the performance of electroadhesion as a rotary friction
mechanism for auxetic sheets, we patterned an auxetic unit cell into a
flexible printed circuit board (PCB) (PCBWay, Shenzhen, China). The polyimide
substrate is $t=50$ µm thick, and the AUC has dimensions $a=17$ mm, $c=3$ mm,
and $\delta=0.71$ mm. Because of the space taken up by wiring electrical
traces, the total overlap area between copper electrodes on adjacent auxetic
layers is 180.34 mm2, for an average $a_{ov}=13.43$ mm when unstretched.

We consider three dielectric films, which provide different trade-offs in
price, relative permittivity, and thickness:

  * •

Aluminium-sputtered mylar (CS Hyde Company, Illinois, USA), 25 µm thick mylar
with a 25 nm Al layer, $\epsilon_{r}=3$, cost = 0.00065 USD/cm2 in bulk (4.6
m2 rolls)

  * •

Silver-sputtered PVDF-HFP (PolyK Technologies, LLC, Philadelphia, USA), 10 µm
thick biaxially oriented PVDF-HFP (90/10 wt) with 100 nm Ag layer,
$\epsilon_{r}=13$, cost = 0.0059 USD/cm2 in bulk (68 m2 rolls)

  * •

Non-metallized PVDF-TrFE-CFE (PolyK Technologies, LLC), 22 µm thick PVDF-TrFE-
CFE (7 mol% CFE), $\epsilon_{r}=50$, cost = 0.19 USD/cm2 in bulk (1.5 m2
rolls)

Dielectrics and conductive adhesives were cut using a quasi-CW DPSS UV laser
cutter (DPSS Lasers Inc., Series 3500). Dielectric films were cut into 17mm
squares to match the auxetic pattern. For mylar and PVDF-HFP, we etched away
their metallized sputter’s 0.7mm outer ring to prevent arcing and adhered them
to a 14mm square of 3M 1182 double-sided copper tape, before assembling them
onto the flexible PCB as shown in Fig. 1. The non-metallized PVDF-TrFE-CFE was
backed with 3M 9703 z-axis anisotropic conductive transfer tape before being
cut and assembled.

Figure 3: Testing setup for measuring the linear force vs. strain of an
individual auxetic unit cell layer stack.

The testing setup, shown in Fig. 3, measures hysteresis curves for the linear
force required to stretch a multi-layer AUC stack under both locked and
unlocked states. The flexible PCBs have four metallized mounting holes at the
midpoint of their hinge joints. Each mounting hole is clamped between two
threaded standoffs, which are connected to electrical ground and friction fit
into ball bearings for free rotation as the AUC expands. The ball bearings are
held in place vertically by a cover slip and given horizontal travel to expand
when stretched, as expected from the auxetic structure’s negative Poisson’s
ratio. All mounting components for the ball bearings are 3D printed with a
glossy finish from VeroWhitePlus photopolymer using a Strasys Objet24 material
jetting printer, ensuring a low coefficient of friction ($\mu=0.34$). The
hysteresis tests were conducted using an Actuonix S20 linear actuator, and
linear force readings are measured at 4.9 Hz by a Loadstar Sensors RAPG-100G-A
load cell. Because the tight space constraints make it difficult to actively
actuate both the top and bottom AUCs at 90∘ angles relative to each other, we
only actively stretch the bottom layer and let the other lie passively on top.

###  II-C Characterization Results for a Single Auxetic Unit Cell

Fig. 4 shows a sample hysteresis curve from the setup in Sec. II-B. Testing
was restricted to within 5% strain to more accurately assess the effect of
rotational friction without linear sliding, although higher voltages clearly
show the slippage point around 3.5% strain where the AUCs’ locking ability
gets outweighed by the load cell’s spring constant of 1850 N/m. The 2-layer
PVDF-HFP dielectric film achieved the largest locked-over-unlocked stiffness
variation of 7.6x with a locked linear stiffness of 1500 N/m, calculated as
the average ratio of slopes across a 1% strain range. It achieves this with a
maximum power draw of 50 µW/AUC, highlighting the low power consumption of
electroadhesion relative to its locking force. Mylar and PVDF-TrFE-CFE
achieved maximum stiffness variations of 4.6x and 7.0x, respectively. We
proceed with PVDF-HFP for future experiments.

Figure 4: Hysteresis curve for the linear force required to stretch a 2-layer
AUC stack with PVDF-TrFE-CFE dielectric. Voltage was applied only while
stretching. The 95% confidence interval is shown over $N=4$ trials at each
voltage. Figure 5: Additional linear force required to stretch a 2-, 3-, and
4-AUC stack when locked, as compared to when unlocked. Results are averaged
over $N=3$ trials, and the dotted lines show the fitted solid mechanics
models.

Fig. 5 shows the additional linear force required to stretch a locked auxetic
unit cell layer stack at 5% strain, as compared to the unlocked layer stack.
The solid mechanics model in Sec. II-A is plotted for comparison and generally
matches the experimental trends. The solid mechanics model involves two fitted
parameters. The Young’s modulus $E$ is first fit using Brent’s method to the
unlocked AUCs’ experimental data with an average fit of $E$ = 2.1 GPa
($\sigma$ = 0.4 GPa). Second, the effective frictional constant is fit to an
average $\mu_{eff}$ = 0.13 ($\sigma$ = 0.14) to account for non-idealities in
assembly that might introduce air gaps in between the dielectric and
metallized electrodes. This Young’s modulus and fitted efficiency match
closely with values seen in the literature [30, 20, 24]. Across all tests, the
fitted solid mechanics model achieves a Root Mean Squared Error (RMSE) of
0.094 N. As shown by the horizontal predicted and experimental curves at high
voltage, both PVDF-HFP and PVDF-TrFE-CFE achieve near-maximal locking in this
testing apparatus.

##  III Characterization of 5x5 Auxetic Unit Cell Layer Stack

###  III-A Assembly of a 5x5 AUC Layer Stack

In order to explore how the results for an individual AUC extend to an auxetic
array, we patterned a flexible PCB with a 5x5 array of square unit cells with
the same geometry as in Sec. II-B. Towards its functionality as a variable-
stiffness surface for a formable crust shape display, we tested whether
locking local subsets of AUCs can affect the array’s global axial and
transverse stiffness. We assembled the layer stack with PVDF-HFP as the
dielectric, based on results in Sec. II-C. Each flexible PCB includes through
holes around the edges for easy mounting. Pin headers were soldered to the
flexible PCBs and connected to ribbon cables, which were then routed to a
logic PCB using the NMOS source follower circuit shown in Fig. 6 for
programmatic control of each auxetic unit cell via row-column addressing. A
high voltage transistor (STMicroelectronics STB12NM60N) switches a row on one
auxetic layer to either high voltage or ground, while another does the same to
columns of the second layer. A high voltage DC-DC converter (EMCO F50) shifts
applied voltages from 0.51 - 1.29 V to 200 - 600 V. A resistor ($R_{1}=10$
M$\Omega$) is used to limit current draw, and another resistor ($R_{2}=1$
k$\Omega$) limits burden on the Teensy 4.1 microcontroller supplying $V_{in}$.
This high-voltage logic circuitry consumes a maximum of 72.2 µA/AUC at 600 V.
For portability, however, the below results were collected without this logic
PCB by manually powering different electrode pairs.

Figure 6: Logic circuit for driving an individual AUC. High voltage is
supplied from a DC-DC voltage converter, and $V_{in}$ is obtained from a
Teensy 4.1. AUCs can be modeled as capacitors $C\approx 8.3$ nF and two
resistors $R\approx 1.2\Omega$ in series.

###  III-B Axial Stiffness Variation of a 5x5 AUC Layer Stack

In order to measure the ability of an auxetic layer stack to modulate its
axial stiffness, we axially offset the dielectric-assembled auxetic sheet by
half of an AUC relative to the second layer, and clamped the ends of both
sheets at this offset using acrylic sheets and M2 screws through each flexible
PCB’s through holes. This causes the unit cells on subsequent layers to rotate
in opposite direction just like when testing individual AUCs in Sec. II-B,
with the exception of only being able to lock 90% of the electroadhesive pads
at a time.

We tested this multi-layer stack on an Instron 5560 with a 100 N load cell,
with pictorial results shown in Fig. 7 and force vs. strain results shown in
Fig. 8 for the cases when 0%, the bottom 40%, and all of the available 90% of
the rows were locked at 500 V. Tests were conducted up to 5% strain with a
0.1%/s strain rate. Scaled predictions in Fig. 8 were computed by treating the
5x5 cell array as a spring array with five columns in parallel and four
springs in serial in each column (ignoring one AUC/column because of the
attachment offset), and computing the entire array’s axial stiffness based on
the maximum 7.6x locked-over-unlocked stiffness ratio measured in Sec. II-C
for an individual AUC.

As shown in Fig. 8, the unlocked case has the lowest stiffness and matches
well with the predicted global axial stiffness. Unlike with individual AUCs,
which had fixed boundary conditions at each corner, the full sheet is free to
deform out-of-plane. The 90% locked case’s out-of-plane deformation in Fig. 7
highlights the necessity for some compliance when reaching the 5% strain
target, as shown by how its slope gradually decreases in Fig. 8 as AUCs slip
and detach. Notably, both locked cases are much stiffer than predicted, which
we hypothesize is because differences in how the two auxetic sheets deform
out-of-plane actually press individual AUCs together, creating greater locking
forces than the electroadhesion alone could produce.

Figure 7: Example photos of the 5x5 AUC layer stack undergoing linear tests on
an Instron at different locking conditions at 5% strain. Red boxes outline the
locked cells. Figure 8: Linear force vs. strain for a 5x5 AUC layer stack
under different locking conditions. Shaded regions show the 95% confidence
interval, averaged over $N=3$ trials. High-uncertainty regions correspond to
AUC slippage during individual tests. Dotted lines show the expected initial
slope based on individual AUC data in Sec. II-C.

###  III-C Transverse Stiffness Variation of a 5x5 AUC Layer Stack

Unlike traditional uniaxial soft robotic applications, shape displays can make
use of biaxial stiffness control to generate a more diverse range of shapes
[10]. Therefore, we also tested the ability of this multi-layer stack to
modulate its transverse stiffness and bend when given rotational freedom. The
testing method was identical to Sec. III-B’s, except the mounting brackets
were attached to ball bearings for free rotation and we varied the number of
columns locked instead of rows. Results are shown in Fig. 9 and Fig. 10.

Fig. 10 highlights the influence of higher strain, and thus larger out-of-
plane deformation, on the experimental data. Predicted results were obtained
by solving an optimization problem for the minimum spring energy needed to
position the 5x5 AUC spring array at the desired strain, based on the maximum
7.6x locked-over-unlocked stiffness ratio demonstrated in Sec. II-C for an
individual AUC. The experimental data matches the theory well for small
strain, but similar to Sec. III-B the locking forces exceed predictions for
higher strain. Future work should investigate these discrepancies further, but
in Sec. IV we see that in practice mounting auxetic skins onto a soft robotic
body creates boundary constraints that naturally limit out-of-plane
deformation.

Figure 9: Example photos of the 5x5 AUC layer stack undergoing bending angle
tests on an Instron at different locking conditions at 5% strain. The total
angle is computed by adding the rotations of the top and bottom mounts. Figure
10: Bending angle vs. percentage of columns locked in a 5x5 AUC layer stack.
Dotted lines show the expected bending angle based on individual AUC data in
Sec. II-C.

##  IV Integration of Electroadhesive Auxetic Skins into a Formable Crust
Shape Display

In order to test the performance of our electroadhesive auxetic skin for on-
demand stiffness variation in a continuous shape display, we mounted it onto
an inflatable LDPE pouch and measured its ability to modulate the shape
output. Shown in Fig. 11, the inflatable pouch comprises a 50x50 cm2 sheet of
30 µm thick LDPE film crumpled to fit under the 20x20 cm2 auxetic skin. The
LDPE is locked at the edges between two laser cut acrylic sheets, and when
pressurized air is pumped into the interior cavity the pouch balloons outwards
into a uniform dome. The same 5x5 auxetic arrays used in Sec. III are mounted
on top of the LDPE pouch, with the dielectric-assembled bottom sheet rotated
by 90∘ relative to the top sheet. M2 mounting screws are threaded through the
entire height of this assembly stack to fix the edges together.

Fig. 12 shows a side view of the electroadhesive auxetic skins modulating the
output shape’s curvature depending on the number of columns locked. Pressure
is held at 1 kPa, which for the given assembly maximized inflation without
electroadhesive slippage during any of the tested locking conditions. The
auxetic skins enable a 2.3x decrease in global curvature between the unlocked
and 100% locked cases, and they achieve a 2.0x change in output slope between
the display’s left and right sides for the 80% locked case. The LDPE pouch
serves as a conformal substrate, limiting the out-of-plane deformation
observed in Sec. II. Fig. 13 further demonstrates how programmatically locking
different sets of AUCs enables a single auxetic pattern to output a variety of
global output shapes. Depth maps were recorded using a Microsoft Kinect for
Xbox One, and black ripstop nylon squares were taped onto every AUC to limit
reflectivity.

While data resolution still falls behind state-of-the-art [9, 20] due to the
inability to lock AUCs to the base plate, these results are promising for the
application of electroadhesive auxetic skins as a programmable surface for
formable crust shape displays, highlighting the ability of variable stiffness
surfaces to actively control deformation and achieve shape change. The low
cost (0.88 USD/AUC at quantities of 100 flexible PCBs, including high voltage
logic circuitry), low power consumption (50 µW/AUC), low profile ($<$300 µm
thickness), and monolithic manufacturing techniques for lowering assembly
complexity decrease the barrier to entry traditionally required for
distributed tactile displays, opening new options for fast prototyping and
scalable deployment.

Figure 11: Full assembly stack of electroadhesive auxetic sheets onto an
inflatable LDPE pouch to form a shape display. Figure 12: Side view of the
shape display modulating its output shape based on the percentage of columns
locked. Pressure within the LDPE pouch is held at 1 kPa. Figure 13: Elevation
maps and perspective views of the shape display’s output geometry under
different locking conditions: (a) locking every cell, (b) locking no cells,
(c) locking the top right 36%, (d) locking the right 40%, (e) locking the
right 60%, and (f) locking the top 40%. Locked cells, outlined or shaded in
red, expand noticeably less than unlocked cells. Heights are measured as the
difference between a top-down Kinect’s measured depth maps before and after
inflation.

##  V Conclusion and Future Work

In this work, we motivate the application of electroadhesion as a layer
jamming actuation mechanism for auxetic sheets and demonstrate its usage as a
programmable strain limiting skin for a formable crust shape display. By using
monolithic fabrication techniques such as flexible PCB manufacturing and laser
cutting, these actuators present a scalable, low-cost, low-power, and easy to
assemble approach to integrating variable stiffness capabilities into flexible
robot systems. We demonstrated the ability of a multi-layer auxetic stack to
programmatically modulate its axial and transverse stiffness, opening new
possibilities in high degree-of-freedom control of soft robotic systems.

Our current design also has some drawbacks, notably in its simplified planar
kinematic modeling and limited output shape change. Future work should
investigate new analytical 3D models for electroadhesive auxetic skins,
factoring in plastic and out-of-plane deformation as well as fatigue.
Additionally, further work is needed to optimize auxetic patterns supporting
greater strain and thus increased global shape change. Future work should also
consider locking shape display surfaces to the base plate for extra degrees of
freedom [9]. Finally, other soft robotic applications such as vine robots [31]
and haptic clutches [24] which also require low-profile, on-demand stiffness
control should be explored.

## Acknowledgments

This work is supported by the National Science Foundation (NSF) Graduate
Research Fellowship grant no. DGE-1656518, the NSF CAREER award grant no.
2142782, and the Alfred P. Sloan Research Fellowship grant no. FG-2021-15851.
Part of this work was performed at the Stanford Nano Shared Facilities,
supported by the NSF under award ECCS-2026822. We also acknowledge Kai Zhang,
who ideated and prototyped early passive versions of this system.

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$\bigcup_{i=1}^{\ell}f^{-n}[D_{i}^{2}]\subseteq U_{\delta}^{2},$ (64)

where $U^{2}_{\delta}$ is the $\delta$-neighborhood of
$V^{u}((u_{i}^{2})_{i\leq 0})\cap A(u)$ in the leaf $V^{u}((u_{i}^{2})_{i\leq
0})$.

######  Corollary 7.7.

$\forall\delta\in(0,\delta_{0})$

$\frac{\mathrm{Vol}(U^{2}_{\delta})}{\mathrm{Vol}(U^{1}_{\delta})}\geq
e^{-4\epsilon^{2}(\frac{2d_{C_{0}}(V^{u}((u_{i}^{1})_{i\leq
0}),V^{u}((u_{i}^{2})_{i\leq 0}))}{(p^{s}\wedge
p^{u})^{3}})^{\frac{\beta}{4}}}.$

###### Proof.

Let $n_{\delta}$ as in (64). Then

$\displaystyle\frac{\mathrm{Vol}(U^{2}_{\delta})}{\mathrm{Vol}(U^{1}_{\delta})}=\frac{\mathrm{Vol}(U^{2}_{\delta})}{\mathrm{Vol}(\bigcup_{i=1}^{\ell}f^{-n}[D_{i}^{2}])}\cdot\frac{\mathrm{Vol}(\bigcup_{i=1}^{\ell}f^{-n}[D_{i}^{2}])}{\mathrm{Vol}(\bigcup_{i=1}^{\ell}f^{-n}[D_{i}^{1}])}\cdot\frac{\mathrm{Vol}(\bigcup_{i=1}^{\ell}f^{-n}[D_{i}^{1}])}{\mathrm{Vol}(U^{1}_{\delta})}.$

By (64), the first factor is greater or equal to $1$. By Lemma 7.6, the third
factor is greater or equal to
$e^{-2\epsilon^{2}(\frac{2d_{C_{0}}(V^{u}((u_{i}^{1})_{i\leq
0}),V^{u}((u_{i}^{2})_{i\leq 0}))}{(p^{s}\wedge
p^{u})^{3}})^{\frac{\beta}{4}}}$. By Lemma 7.4 and Corollary 7.4,

$\frac{\mathrm{Vol}(\bigcup_{i=1}^{\ell}f^{-n}[D_{i}^{2}])}{\mathrm{Vol}(\bigcup_{i=1}^{\ell}f^{-n}[D_{i}^{1}])}=\frac{\sum_{i=1}^{\ell}\mathrm{Vol}(f^{-n}[D_{i}^{2}])}{\sum_{i=1}^{\ell}\mathrm{Vol}(f^{-n}[D_{i}^{1}])}=e^{\pm
2\epsilon^{2}(\frac{2d_{C_{0}}(V^{u}((u_{i}^{1})_{i\leq
0}),V^{u}((u_{i}^{2})_{i\leq 0}))}{(p^{s}\wedge
p^{u})^{3}})^{\frac{\beta}{4}}}.$

This completes the proof. ∎

######  Corollary 7.8.

$\mathrm{Vol}(A(u)\cap V^{u}((u_{i}^{2})_{i\leq 0}))>0$.

###### Proof.

$\Sigma$ is locally compact, $\pi$ is uniformly continuous, and so $A(u)$ is
compact. Therefore, if $\mathrm{Vol}(A(u)\cap V^{u}((u_{i}^{2})_{i\leq
0}))=0$, then $\lim_{\delta\to 0}\mathrm{Vol}(U^{2}_{\delta})=0$ as well, but
this is a contradiction to Corollary 7.7. ∎

Now we may apply Lemma 7.6 to the set $A(u)\cap V^{u}((u_{i}^{2})_{\leq 0})$,
and get a $\delta_{1}\in(0,\delta_{0}]$ s.t $\forall\delta\in(0,\delta_{1})$,

$\frac{\mathrm{Vol}(A(u))}{\mathrm{Vol}(U_{\delta}^{2})}\geq
e^{-2\epsilon^{2}(\frac{2d_{C_{0}}(V^{u}((u_{i}^{1})_{i\leq
0}),V^{u}((u_{i}^{2})_{i\leq 0}))}{(p^{s}\wedge
p^{u})^{3}})^{\frac{\beta}{4}}},$

where $\mathrm{Vol}$ is the intrinsic $V^{u}((u_{i}^{2})_{i\leq 0})$-volume.

Therefore, we may use Lemma 7.7 with replacing the roles of $U^{1}_{\delta}$
and $U^{2}_{\delta}$, for all $\delta\in(0,\delta_{1})$,

$\frac{\mathrm{Vol}(U^{1}_{\delta})}{\mathrm{Vol}(U^{2}_{\delta})}=e^{\pm
4\epsilon^{2}(\frac{2d_{C_{0}}(V^{u}((u_{i}^{1})_{i\leq
0}),V^{u}((u_{i}^{2})_{i\leq 0}))}{(p^{s}\wedge
p^{u})^{3}})^{\frac{\beta}{4}}}.$ (65)

It then follows from Lemma 7.6 and Corollary 7.7 that
$\forall\delta\in(0,\delta_{1})$,

$\displaystyle\frac{\mathrm{Vol}(V^{u}((u_{i}^{1})_{i\leq 0})\cap
A(u))}{\mathrm{Vol}(V^{u}((u_{i}^{2})_{i\leq 0})\cap A(u))}=$
$\displaystyle\frac{\mathrm{Vol}(V^{u}((u_{i}^{1})_{i\leq 0})\cap
A(u))}{\mathrm{Vol}(U^{1}_{\delta})}\cdot\frac{\mathrm{Vol}(U^{1}_{\delta})}{\mathrm{Vol}(U^{2}_{\delta})}\cdot\frac{\mathrm{Vol}(U^{2}_{\delta})}{\mathrm{Vol}(V^{u}((u_{i}^{2})_{i\leq
0})\cap A(u))}$ $\displaystyle=$ $\displaystyle e^{\pm
6\epsilon^{2}(\frac{2d_{C_{0}}(V^{u}((u_{i}^{1})_{i\leq
0}),V^{u}((u_{i}^{2})_{i\leq 0}))}{(p^{s}\wedge
p^{u})^{3}})^{\frac{\beta}{4}}}.$ (66)

###  7.3. Construction of strictly weak leaves

In §3 we construct weak manifolds- local stable and unstable leaves for orbits
shadowed by $I$-chains. In particular, this construction does not make use of
an asymptotic exponential contraction on the stable or unstable subspaces of
the shadowed orbit; when exponentially-fast contracting manifolds are the
standard stable and unstable manifolds in Pesin’s stable manifold theorem. A
natural question which rises is whether there are weak manifolds, which cannot
be constructed using Pesin’s stable manifolds theorem. That is, local stable
and unstable leaves which exhibit strictly sub-exponential contraction. Such
manifolds are strictly weak stable and unstable manifolds.

In this chapter we give a general result regarding the existence of strictly
weak leaves. It uses the observation that every hyperbolic periodic orbit, and
every uniformly hyperbolic measure, can be coded by our coding. In §8 we
discuss a general condition for the “codability” of measures- i.e property of
having a lift in our coding space.

Given $x\in\mathrm{RST}$, let $W^{s}(x):=\bigcup_{n\geq
0}f^{-n}[V^{s}(f^{n}(x))]$, $W^{u}(x):=\bigcup_{n\geq
0}f^{n}[V^{s}(f^{-n}(x))]$, which are increasing unions.

######  Definition 7.9 (Ergodic homoclinic class).

Let $p$ be a periodic hyperbolic point. Define

$H(p):=\\{x\in\mathrm{RST}:W^{u}(x)\pitchfork
W^{s}(p)\neq\varnothing,W^{u}(p)\pitchfork W^{s}(x)\neq\varnothing\\}.$

This definition is due to Rodriguez-Hertz, Rodriguez-Hertz, Tahzibi, and Ures
([RHRHTU11]).

######  Definition 7.10 (Maximal irreducible components).

Let $\mathcal{R}$ be partition constructed in §6.2.

  1. (1)

We say that $R\overset{n}{\to}S$ if there exists a word of length $n$,
$(w_{0},w_{1},\ldots,w_{n-2},w_{n-1})$, s.t $w_{i}\in\mathcal{R}$ for all
$i\leq n-1$, $w_{0}=R$ and $w_{n_{1}}=S$, and for all $i\leq n-2$
$f^{-1}[w_{i+1}]\cap w_{i}\neq\varnothing$.

  2. (2)

We write for two partition elements $R,S\in\mathcal{R}$, we write $R\sim S$ if
$\exists n,m\geq 0$ s.t $R\overset{n}{\to}S$ and $S\overset{m}{\to}R$. $\sim$
defines a symmetric and transitive relation on $\mathcal{R}$. When restricting
to $R\in\mathcal{R}$ s.t $R\sim R$, $\sim$ defines an equivalence relation.

  3. (3)

Let $R\in\mathcal{R}$ s.t $R\sim R$, then $\langle R\rangle$ is the
equivalence class of $R$ w.r.t $\sim$.

We call equivalence classes of the equivalence relation $\sim$, maximal
irreducible components.

######  Lemma 7.11 (Transitive coding).

Given an ergodic homoclinc class, there is a maximal irreducible component of
a $\widehat{\Sigma}$, $\widetilde{\Sigma}$, s.t $\widetilde{\Sigma}$ can code
every conservative measure on $H(p)$. In particular, it codes every periodic
orbit of every hyperbolic periodic point $q$ which is homoclinically related
to $p$.

This lemma is due to Buzzi, Crovisier, and Sarig [BCS]. This formulation of
the lemma appears in [BO21, § 5] (with another set taking the role of
$\mathrm{RST}$ in the definition of ergodic homoclinic classes, but both sets
carry all uniformly hyperbolic measures and it is sufficient for the proof).

######  Definition 7.12.

Given an ergodic invariant hyperbolic measure $\mu$, we write
$\chi^{u}(\mu):=\sum_{i=1}^{d}\max(\chi_{i}(\mu),0)$ and
$\chi^{s}(\mu):=\sum_{i=1}^{d}\max(-\chi_{i}(\mu),0)$, where
$\chi_{1}(\mu),\ldots,\chi_{d}(\mu)$ are the Lyapunov exponents of $\mu$ (with
multiplicity). If $p$ is a hyperbolic periodic point, $\chi^{s/u}(p)$ means
$\chi^{s/u}(\mu_{p})$, where $\mu_{p}$ is the invariant measure supported on
the orbit of $p$.131313The Lyapunov exponents of a periodic orbit are well-
defined, as working in coordinates, the differential can be put in a Jordan-
block form, with respective eigenvalues. See [BO21, Lemma 5.4] for more
details.

######  Proposition 7.13 (Existence of strictly weak foliations).

Let $H(p)$ s.t $\inf\\{\chi^{s}(q):q\in H(p)\text{ periodic}\\}=0$. Then
$H(p)$ admits strictly weak stable manifolds. A similar statement applies to
strictly weak unstable manifolds.

###### Proof.

By Lemma 7.11 we have a maximal irreducible component $\widetilde{\Sigma}$
which codes all $q\in H(p)$ which are periodic. Therefore, there are periodic
words in $\widetilde{\Sigma}$, $\\{\underline{w}^{(i)}\\}_{i\geq 0}$,
$\underline{w}^{(i)}=(w^{(i)}_{0},\ldots,w^{(i)}_{l_{i}-1})$ s.t the Lyapunov
exponents $\chi_{i}:=\chi^{s}(\underline{w}^{(i)})\to 0$.

Fix $w_{0}^{(0)}$ and denote it by $R$. Let $n_{i}:=\min\\{n\geq
0:R\overset{n}{\to}w_{0}^{(i)}\\}$ (which is well-defined since
$\widetilde{\Sigma}$ is an irreducible component); and let
$\underline{c}^{(i)}$ be a connecting word of such length. Similarly, let
$m_{i}:=\min\\{m\geq 0:w_{0}^{(i)}\overset{m}{\to}R\\}$; and let
$\underline{d}^{(i)}$ be a connecting word of such length.

Define for all $i\geq 1$,

$p_{i}:=\min\\{p\geq 0:p\cdot l_{i}\geq\frac{1}{\chi_{i}}\cdot d\cdot\log
M_{f}\cdot(m_{i}+p_{i-1}+n_{i+1}+l_{i+1})\\}.$ (67)

Consider the chain

$\underline{R}:=(\underline{w}^{(0)})\cdot\underline{c}^{(1)}\cdot(\underline{w}^{(1)})^{p_{1}}\cdot\underline{d}^{(1)}\cdots(\underline{w}^{(n)})\cdot\underline{c}^{(n+1)}\cdot(\underline{w}^{(n+1)})^{p_{n+1}}\cdot\underline{d}^{(n+1)}\cdots,$

where $\cdot$ means a concatenation, and $(\cdot)^{\ell}$ means a
concatenation of a word to itself $\ell$ times. By construction, this is as an
admissible chain, which returns to $R$ infinitely often. Take any recurrent
extension of $\underline{R}$ to the negative coordinates,
$\underline{R}^{\pm}\in\widehat{\Sigma}^{\\#}$, and write
$x:=\widehat{\pi}(\underline{R}^{\pm})$.

For simplicity of presentation, we first prove the proposition under the
assumption that $\mathrm{dim}H^{s}(x)=1$, and then show how it extends to
higher dimensions. When $\mathrm{dim}H^{s}=1$, there is only one normalized
stable tangent vector at the point (up to a sign), denoted by
$\omega^{s}(\cdot)$. Then the quantity $S^{2}(\cdot)$ is well-defined.
Consider the chain $\underline{w}^{(i,\pm)}$ which is the periodic extension
of $\underline{w}^{(i)}$ to $\widehat{\Sigma}$. Write
$w_{j}^{(i)}=\psi_{x_{j}^{(i)}}^{p^{s,(i)}_{j},p^{u,(i)}_{j}}$. Then (3) tells
us that

$\displaystyle\log|d_{\widehat{\pi}(\underline{w}^{(i,\pm)})}f^{l_{i}}\omega^{s}(\widehat{\pi}(\underline{w}^{(i,\pm)}))|=$
$\displaystyle\sum_{j=0}^{l_{i}-1}\log\sqrt{1-\frac{2}{S^{2}(f^{j}(\widehat{\pi}(\underline{w}^{(i,\pm)})))}}+\log
S(\widehat{\pi}(\underline{w}^{(i,\pm)}))-\log
S(\widehat{\pi}(\underline{w}^{(i,\pm)})),$
$\displaystyle\log|d_{f^{t_{i}}(x)}f^{l_{i}}\omega^{s}(f^{t_{i}(x)})|=$
$\displaystyle\sum_{j=0}^{l_{i}-1}\log\sqrt{1-\frac{2}{S^{2}(f^{j+t_{i}}(x))}}+\log
S(f^{t_{i}}(x))-\log S(f^{t_{i}+l_{i}}(x)).$ (68)

The proof of Lemma 4.7 allows to compare multiplicatively the quantities of
both equations of (7.3), through the centers of charts
$\\{x^{(i)}_{j}\\}_{j=t_{i}}^{t_{i}+l_{i}}$ (notice
$S^{2}(\cdot)\geq\frac{2}{1-M_{f}^{-2}}$). The choice of $p_{i}$ in (67)
guarantees, together with the fact that $\chi_{i}\to 0$, that $\chi^{s}(x)=0$.

Now, in order to see the proof when $\mathrm{dim}H^{s}(x)>1$, the role of
$\omega^{s}(\cdot)$ is replaced by the unique (up to sign) normalized volume
form on $H^{s}(\cdot)$. One can define

$\widehat{S}^{2}(x):=2\sum_{m\geq 0}|d_{x}f^{m}\omega^{s}(x)|^{2},$

which is finite by the Hadamard inequality and the fact that $x\in 0$-summ.
(3) derives similarly
$|d_{x}f\omega^{s}(x)|=\frac{\widehat{S}(x)}{\widehat{S}(f(x))}\sqrt{1-\frac{2}{\widehat{S}^{2}(x)}}$,
and so (7.3) follows. The comparison of the two quantities in (7.3) through
the center of charts is done similarly by an analogous version of Lemma 4.7,
which follows from Lemma 4.7: the map $\pi_{x}^{i,j}:H^{s}(f^{t_{i}+j}(x))\to
H^{s}(x_{j}^{(i)})$ satisfies $\frac{|\pi_{x}^{i,j}\xi|}{|\xi|}=e^{\pm
Q_{\epsilon}(x_{j})^{\frac{\beta}{2}-\frac{1}{2\gamma}}}$ (Lemma 4.6).
Therefore $\mathrm{Jac}(\pi_{x}^{i,j})=e^{\pm
dQ_{\epsilon}(x_{j})^{\frac{\beta}{2}-\frac{1}{2\gamma}}}$; whence the vector
estimates of Lemma 4.7 pull to volume form estimates as well. Strictly sub-
exponential contraction of the volume form on $V^{s}(\underline{R})$ implies
no tangent vector at $x$ which contracts exponentially fast. ∎

##  8\. Codability

In §6.2 we saw that $\widehat{\pi}[\widehat{\Sigma}^{\\#}]=\mathrm{RST}$, so
in particular we can code any measure which is carried by $\mathrm{RST}$. It
is clear that every hyperbolic measure is carried by $0$-summ, but it is not
clear for which ergodic measures almost every orbit is strongly temperable
(recall Definition 2.8).

First, we should notice that every periodic hyperbolic orbit, and every
uniformly hyperbolic measure, are carried by $\mathrm{RST}$, since
$\|C_{0}^{-1}(\cdot)\|$ is bounded on every orbit. Being able to code
uniformly hyperbolic measures is important, since it allows to “lift” the
pressure of Hölder continuous potentials (as well as the geometric potential)
over hyperbolic measures. That is,

$\sup\\{h_{\mu}(f)+\int\phi d\mu:\mu\text{
hyperbolic}\\}=\sup\\{h_{\mu}(f)+\int\phi d\mu:\mu\text{ uniformly
hyperbolic}\\},$

for a Hölder continuous $\phi:M\to\mathbb{R}$. This is true since every
hyperbolic measure can be coded by a TMS which lifts the potential [BO18], and
[Sar09, Theorem 4.3] shows that it can be approximated by the pressure over
compact subshifts (which project to uniformly hyperbolic horseshoes).

In particular, since our coding lifts all hyperbolic periodic orbits, one can
always construct invariant probability measures on the shift space. Such
measures can even be Markov measures and fully supported on the shift space.
They project to invariant probability measures on the manifold; and as long as
the dynamics we code are not uniformly hyperbolic, these measures will be non-
uniformly hyperbolic, and carried by $\mathrm{RST}$.

One can nonetheless ask, in what generality can measures lift to
$\widehat{\Sigma}$? Below we give a general condition which is sufficient
(though may be hard to test), and in a closing discussion later in this
chapter, we review in what sense this condition is optimal.

######  Proposition 8.1.

Let $x\in 0\mathrm{-summ}$. If $\|C_{0}^{-1}(f^{n}(x))\|^{2}=o(|n|)$,
$n\to\pm\infty$, then $x\in\mathrm{ST}$.

In particular, this condition does not depend on the calibrating parameters
$\gamma,\Gamma$, nor $\beta$, nor $\epsilon$. So an invariant probability
measure carried by such orbits will be carried by any Markov partition
constructed per this paper.

###### Proof.

Let $\delta>0$ to be determined later. Let $n_{\delta}\in\mathbb{N}$ s.t
$\forall n\geq n_{\delta}$, $\|C_{0}^{-1}(f^{n}(x))\|^{2}\leq\delta n$. Set
$c_{\delta}:=\min\\{\frac{C_{\beta,\epsilon}}{\|C_{0}^{-1}(f^{n}(x))\|^{2\gamma}}:0\leq
n\leq n_{\delta}\\}$.

For every $n\geq 0$, define
$q^{+}(f^{n}(x)):=\min\\{c_{\delta},\frac{C_{\beta,\epsilon}}{(\delta
n)^{\gamma}}\\}\leq\frac{C_{\beta,\epsilon}}{\|C_{0}^{-1}(f^{n}(x))\|^{2\gamma}}$.
In order to bound the variation of $q^{+}$, it is enough to bound the
variation of $\frac{C_{\beta,\epsilon}}{(\delta n)^{\gamma}}$ for $n\geq
n_{\delta}+1$. Then for any $n\geq n_{\delta}+1$, (w.l.o.g $n_{\delta}$ is big
enough so $(1-\frac{1}{n})\geq e^{-2\frac{1}{n}}$)

$\displaystyle\frac{\frac{C_{\beta,\epsilon}}{(\delta
n)^{\gamma}}}{\frac{C_{\beta,\epsilon}}{(\delta(n-1))^{\gamma}}}\geq(\frac{n-1}{n})^{\gamma}\geq
e^{-2\frac{\gamma}{n}}=e^{-2\gamma\cdot\frac{\delta}{(C_{\beta,\epsilon})^{\frac{1}{\gamma}}}\cdot(\frac{C_{\beta,\epsilon}}{(\delta
n)^{\gamma}})^{\frac{1}{\gamma}}}\geq
e^{-\Gamma\cdot(\frac{C_{\beta,\epsilon}}{(\delta
n)^{\gamma}})^{\frac{1}{\gamma}}},$

where the last inequality is when we choose $\delta>0$ sufficiently small.
Then we get,

$I(\frac{C_{\beta,\epsilon}}{(\delta
n)^{\gamma}})=\frac{C_{\beta,\epsilon}}{(\delta
n)^{\gamma}}e^{\Gamma\cdot(\frac{C_{\beta,\epsilon}}{(\delta
n)^{\gamma}})^{\frac{1}{\gamma}}}\geq\frac{C_{\beta,\epsilon}}{(\delta(n-1))^{\gamma}},\text{
whence }\frac{C_{\beta,\epsilon}}{(\delta n)^{\gamma}}\geq
I^{-1}(\frac{C_{\beta,\epsilon}}{(\delta(n-1))^{\gamma}}).$

This construction can be done for $f^{k}(x)$, $k\in\mathbb{Z}$, the same way.
Therefore the Ledrappier formula is well-defined for every $k\in\mathbb{Z}$,

$p^{s}_{k}:=\max\\{t\in\mathcal{I}:I^{-n}(t)\leq\frac{C_{\beta,\epsilon}}{\|C_{0}^{-1}(f^{n}(f^{k}(x)))\|^{2\gamma}}\text{
for all }n\geq 0\\}.$

Similarly, the following is well-defined as well:

$p^{u}_{k}:=\max\\{t\in\mathcal{I}:I^{-n}(t)\leq\frac{C_{\beta,\epsilon}}{\|C_{0}^{-1}(f^{-n}(f^{k}(x)))\|^{2\gamma}}\text{
for all }n\geq 0\\}.$

It follows from the Ledrappier formula that for all $k\in\mathbb{Z}$,

  1. (1)

$p^{s}_{k},p^{u}_{k}\leq\frac{C_{\beta,\epsilon}}{\|C_{0}^{-1}(f^{k}(x))\|^{2\gamma}}$,

  2. (2)

$p^{u}_{k+1}=\min\\{I(p_{k}^{u}),\frac{C_{\beta,\epsilon}}{\|C_{0}^{-1}(f^{k+1}(x))\|^{2\gamma}}\\}$,
$p^{s}_{k-1}=\min\\{I(p_{k}^{s}),\frac{C_{\beta,\epsilon}}{\|C_{0}^{-1}(f^{k-1}(x))\|^{2\gamma}}\\}$.

Therefore $q(f^{k}(x)):=\min\\{p^{s}_{k},p^{u}_{k}\\}$ is a strongly tempered
kernel for $x$, as in Lemma 2.24. ∎

###  8.1. Examples and discussion

In this section we consider a family of systems which is called almost Anosov,
introduced by Hu and Young in [HY95], and we mention a few results regarding
our coding of these systems. The full details appear in an upcoming work. We
then have a short discussion regarding optimality of our construction (i.e the
notions of $0$-summability and strong temperability, and the condition in
Proposition 8.1 for codability).

######  Definition 8.2 (Almost-Anosov).

Let $f\in\mathrm{Diff}^{r}(\mathbb{T}^{2})$, $r\geq 2$, be a topologically
transitive diffeomorphism of the two dimensional torus. Assume that $f$ admits
an invariant splitting of the tangent bundle everywhere $T_{x}M=E^{s}(x)\oplus
E^{u}(x)$ and an indifferent fixed point $p$, s.t

  1. (1)

$\exists K^{s}<1$, $\forall x$ $|d_{x}f|_{E^{s}}|\leq K^{s}$

  2. (2)

$\exists K^{u}(\cdot)\geq 1$ continuous on $M$, s.t $\forall x$
$|d_{x}f|_{E^{u}}|\geq K^{u}(x)$,

  3. (3)

For any $x\neq p$, $K^{u}(x)>1$, and $|d_{p}f|_{E^{u}}|=K^{u}(p)=1$.

Hu and Young introduce this class of systems to show that while these systems
are very nicely behaving (“almost” uniform hyperbolicity), the admit no SRB
measures.

The following result appears with details in an upcoming work, and for this
paper we assume it.

######  Theorem 8.3.

Given an almost Anosov diffeomorphism, every $x\neq p$ is $0$-summable.

######  Corollary 8.4.

Given an almost Anosov system, every invariant measure ($\neq\delta_{p}$) is
carried by $\mathrm{RST}$ (for any choice of the calibrating parameters), and
hence codable.

###### Proof.

Let $\mu$ be an invariant probability measure. W.l.o.g $\mu$ is ergodic. If
$\mu(p)=0$, then $\mu$-a.e point is summable by Theorem 8.3. In addition,
$|d_{x}f|_{H^{u}(x)}|\geq 1$ everywhere, therefore

$\displaystyle U^{2}(f(x))=$ $\displaystyle 2\sum_{m\geq
0}|d_{f(x)}f^{-m}|_{H^{u}(f(x))}|^{2}=2+|d_{f(x)}f^{-1}|_{H^{u}(f(x))}|^{2}\cdot
2\sum_{m\geq 0}|d_{x}f^{-m}|_{H^{u}(x)}|^{2}$ $\displaystyle=$ $\displaystyle
2+|d_{f(x)}f^{-1}|_{H^{u}(f(x))}|^{2}\cdot U^{2}(x)\leq 2+U^{2}(x).$ (69)

Thus, $U^{2}(f^{n}(x))\leq U^{2}(x)+2n$ for all $n\geq 0$. Then $(U^{2}\circ
f-U^{2})^{+}\in L^{1}(\mu)$. One can then show that necessarily $U^{2}\circ
f-U^{2}\in L^{1}(\mu)$ and $\int(U^{2}\circ f-U^{2})d\mu=0$ (full details
appear in the upcoming work where we show Theroem 8.3). Then by the point-wise
ergodic theorem, $\mu$-a.e,

$\frac{U^{2}\circ f^{n}}{n}=\frac{1}{n}\sum_{k=0}^{n-1}(U^{2}\circ
f-U^{2})\circ f^{k}+\frac{U^{2}}{n}\to 0.$

Similarly $U^{2}\circ f^{-n}=o(n)$, $n\to\infty$, $\mu$-a.e.

Finally, notice that the continuous splitting on a compact manifold implies
that $\sphericalangle(H^{s}(x),H^{u}(x))$ is bounded from below everywhere. In
addition, $S^{2}\leq 2\sum_{m\geq
0}(K^{s})^{2m}=\frac{2}{1-(K^{s})^{2}}<\infty$. Then
$\|C_{0}^{-1}(x)\|^{2}\leq C_{M}\cdot U^{2}(x)$ for all $x\neq p$, where
$C_{M}$ is some global constant. Thus $\|C_{0}^{-1}\circ f^{n}\|^{2}=o(|n|)$,
$n\to\pm\infty$, $\mu$-a.e. By Proposition 8.1, $\mu$ is carried by
$\mathrm{RST}$ and is codable. ∎

Remark: We saw that $\widehat{\pi}[\widehat{\Sigma}^{\\#}]=\mathrm{RST}$
carries every invariant probability measure in the almost Anosov setup.
Furthermore, we can see that in some cases we must have measures with
arbitrarily small Lyapunov exponents, while we can code all of them
simultaneously in a finite-to-one almost everywhere fashion. This follows from
the fact that almost Anosov systems can be achieved as a perturbation (or a
“slow-down”) of a mixing hyperbolic toral automorphism (which admit a mixing
finite Markov partition). One can make the element of the partition which
contains $p$ arbitrarily small, call it $a$. Fix another element $b$, and
construct a word from $b$ to $b$ with an arbitrarily long segment $a\ldots a$
(admissible since $p$ is a fixed point). Then This codes a periodic (hence
hyperbolic) orbit, with arbitrarily small Lyapunov exponent. One then gets the
existence of strictly weak manifolds by Proposition 7.13.

After establishing the relationship between being codable and a sub-linear
growth-rate of $\|C_{0}^{-1}(f^{n}(\cdot))\|^{2}$, one can ask the following
three questions:

  1. (1)

Can the definition of $0$-summable orbits be changed to get stronger
contractions in charts? (Thus allowing us to require weaker properties than
strong temperability).

  2. (2)

Can the definition of strong temperability be weakened and still allow for a
shadowing theory?

  3. (3)

Is there a weaker condition than sub-linear growth-rate of
$\|C_{0}^{-1}(f^{n}(\cdot))\|^{2}$ to guarantee codability?

We will give non-formal heuristic answers to these questions, which justify
the “optimality” of the notions in this paper.

  1. (1)

The answer to this question has three parts. First, one can ask wether we can
consider orbits which converge in a power higher than $2$. First notice that
for every hyperbolic orbit, the sum converges regardless of the power, and
that every periodic orbit which converges in some power must be hyperbolic.
Furthermore, every measure which is uniformly summable in some power must be
hyperbolic. Therefore one does not lose much by choosing the power $2$, but
gains a lot by the fact that the parallelogram law dictates that only for the
power $2$, the scaling functions will constitute a norm on the tangent space
which is induced by an inner-product. This bilinear form which induces the
scaling functions (i.e the Lyapunov inner-product) is paramount to our
construction of the Lyapunov change of coordinates.

One can also ask if perhaps adding an exponential term in the definition of
summability, which depends on the Lyapunov exponent of the orbit, can yield a
simpler construction while still coding all “sufficiently nice” hyperbolic
orbits. The heuristic answer is that this is conceptually wrong. A Lyapunov
exponent is not a property of a point (resp. symbol), it is a property of the
orbit (resp. chain). Imposing an edge condition which requires some
correspondence between the Lyapunov exponents of neighboring charts, goes
against the Markov structure which we sought to construct. Without imposing
some correspondence on the exponents through an edge condition, one cannot use
a composition of Pesin charts
$d_{x}f^{n}=C_{0}(f^{n}(x))\circ(C_{0}^{-1}(f^{n}(x))d_{f^{n-1}(x)}fC_{0}(f^{n-1}x))\circ\cdots\circ(C_{0}^{-1}(f(x))d_{x}fC_{0}(x))\circ
C_{0}^{-1}(x)$ without accumulating errors.

Lastly, once can ask if some weights can be added in the definition of
summability to aid/hinder convergence. A natural choice of such weights which
has been studied before is exponential weights. Their short-comes are clear,
as they restrict us to $\chi$-hyperbolic orbits, where $\chi$ depends on the
exponent in the weights. If one chooses rates where the ratio between them
does not go to a constant number, then shifting the sequence does not change
it by a factor, and again, creates discrepancy. For these heuristic reason,
the notion of $0$-summability is quite natural.

  2. (2)

(8.1) shows us that even for a family of systems which are very nicely
behaving, one gets naturally the relation $\frac{1}{U^{2\gamma}\circ
f}\geq\frac{1}{U^{2\gamma}}e^{-2(\frac{1}{U^{2\gamma}})^{\frac{1}{\gamma}}}$.
Therefore, heuristically, when we wish to give a general description to a
phenomenon which has a relatively easily understood example, we cannot expect
behavior nicer than that of the example generally. It turns out that strong
temperability, which utilizes exactly the relation of the form
$\frac{C_{\beta,\epsilon}}{\|C_{0}^{-1}\|^{2\gamma}\circ
f}\geq\frac{C_{\beta,\epsilon}}{\|C_{0}^{-1}\|^{2\gamma}}e^{-\Gamma(\frac{C_{\beta,\epsilon}}{\|C_{0}^{-1}\|^{2\gamma}\circ
f})^{\frac{1}{\gamma}}}$, is sufficient for a shadowing theory in the general
case.

  3. (3)

In Lemma 2.6 we saw that being strongly temperable implies that
$\|C_{0}^{-1})\|^{2\gamma}\circ f^{n}=O(n^{\gamma})$, where the big $O$
constant depends on $\gamma$ and $\Gamma$. Therefore an orbit which is coded
by our coding or is strongly temperable for any choice of the calibrating
parameters, will have to satisfy $\|C_{0}^{-1})\|^{2}\circ f^{n}=o(n)$. This
implies that this condition captures a significant portion of the orbits we
wish to code.

##  9\. Temperability and rate of contraction

In this section we present a relationship between two properties which are
studied in this paper: Temperability and the rate of contraction on stable and
unstable manifolds.

We begin by showing that as long as an orbit visits a level set of $S^{2}$
with some positive frequency, it must contract tangent vectors on the stable
direction exponentially fast asymptotically (similarly for $U^{2}$ and the
unstable direction). This shows us that hyperbolicity emerges as a phenomenon
of positive recurrence to level sets of the scaling function. In particular,
every invariant probability measure carried by $0$-summ must be hyperbolic.
However, the lack of positive recurrence does not imply the lack of
recurrence.

We therefore continue to study the rate of contraction of tangent vectors in
the stable or unstable subspaces, without assuming positive recurrence. We
show that the temperability properties induce a rate of contraction. By
temperability, we mean asymptotic growth-rate bounds for $\|C_{0}^{-1}\|^{2}$
(or $S^{2}/U^{2}$). We saw in Proposition 8.1 that a sub-linear growth-rate
implies being strongly temperable. Proposition 9.2 considers more rates.

###  9.1. Hyperbolicity as positive recurrence of the scaling functions

######  Proposition 9.1.

Let $x\in 0\mathrm{-summ}$, and let $\xi\in H^{s}(x)$ s.t $|\xi|=1$. Write
$\xi_{k}:=\frac{d_{x}f^{k}\xi}{|d_{x}f^{k}\xi|}$. Then if $\lim\frac{1}{n}\log
S(f^{n}(x))=0$,

  1. (1)

$\limsup\limits_{n\to\infty}\frac{1}{n}\log|d_{x}f^{n}\xi|<0$ if and only if
$\exists C>0$ s.t $\liminf\limits_{n\to\infty}\frac{1}{n}\\#\\{0\leq i\leq
n:S^{2}(f^{k}(x),\xi_{k})\leq C\\}>0$.

  2. (2)

$\liminf\limits_{n\to\infty}\frac{1}{n}\log|d_{x}f^{n}\xi|<0$ if and only if
$\exists C>0$ s.t $\limsup\limits_{n\to\infty}\frac{1}{n}\\#\\{0\leq i\leq
n:S^{2}(f^{k}(x),\xi_{k})\leq C\\}>0$.

A similar statement holds for the backwards orbit and $U^{2}$.

###### Proof.

Write
$\zeta^{u}_{n}:=\log\frac{1}{\sqrt{1-\frac{2}{S^{2}(f^{n}(x),\xi_{n})}}}\in(0,\log
M_{f}]$. By (3), $\forall n\geq 0$,

$\frac{1}{n}\log|d_{x}f^{n}\xi|=-\frac{1}{n}\sum_{k=0}^{n-1}\zeta^{u}_{k}+\frac{1}{n}\log
S(x,\xi)-\frac{1}{n}\log
S(f^{n}(x),\xi_{n})\leq-\frac{1}{n}\sum_{k=0}^{n-1}\zeta^{u}_{k}+\frac{1}{n}\log
S(x,\xi).$ (70)

Then the following claim implies the proposition.

Claim:

  1. (1)

$\limsup\frac{1}{n}\sum_{i=0}^{n-1}\zeta^{u}_{i}>0$ if and only if
$\exists\delta>0$ s.t $\limsup\frac{1}{n}\\#\\{0\leq i\leq
n:S^{2}(f^{i}(x),\xi_{i})\leq\frac{1}{\delta}\\}>\delta$,

  2. (2)

$\liminf\frac{1}{n}\sum_{i=0}^{n-1}\zeta^{u}_{i}>0$ if and only if
$\exists\delta>0$ s.t $\liminf\frac{1}{n}\\#\\{0\leq i\leq
n:S^{2}(f^{i}(x),\xi_{i})\leq\frac{1}{\delta}\\}>\delta$.

Proof: We start by showing item (1). Let $n_{k}$ be a subsequence.

$\Rightarrow$: Assume for contradiction that for any $\delta>0$ there exists
$k_{\delta}\geq 0$ s.t $\forall k\geq k_{\delta}$,

$\frac{1}{n_{k}}\\#\\{i\leq
n_{k}:S^{2}(f^{i}(x),\xi_{i})\leq\frac{1}{\delta}\\}\leq\delta.$

Notice, if $S^{2}(f^{i}(x),\xi_{i})\geq\frac{1}{\delta}$, then
$\zeta^{u}_{i}\leq\log\frac{1}{\sqrt{1-2\delta}}\leq 2\delta$ (for all
sufficiently small $\delta>0$). Therefore,

$\frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1}\zeta^{u}_{i}=\frac{1}{n_{k}}\sum_{i\leq
n_{k}:S^{2}(f^{i}(x),\xi_{i})<\frac{1}{\delta}}\zeta^{u}_{i}+\frac{1}{n_{k}}\sum_{i\leq
n_{k}:S^{2}(f^{i}(x),\xi_{i})\geq\frac{1}{\delta}}\zeta^{u}_{i}\leq\log
M_{f}\cdot\delta+2\delta\leq(\log M_{f}+2)\cdot\delta.$ (71)

This is a contradiction when
$0<\delta<\frac{1}{2}\frac{\limsup\frac{1}{n}\sum_{i=0}^{n-1}\zeta^{u}_{i}}{\log
M_{f}+2}=\frac{1}{2}\frac{\lim\frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1}\zeta^{u}_{i}}{\log
M_{f}+2}$, for some subsequence $n_{k}$.

$\Leftarrow$: Assume that $\exists\delta>0$ s.t $\limsup\frac{1}{n}\\#\\{0\leq
i\leq n:S^{2}(f^{i}(x),\xi_{i})\leq\frac{1}{\delta}\\}>\delta$. Notice that if
$S^{2}(f^{i}(x),\xi)\leq\frac{1}{\delta}$ then
$\zeta^{u}_{i}\geq\log\frac{1}{\sqrt{1-2\delta}}\geq\delta$. Then by (71), for
$n_{k}$ s.t $\frac{1}{n_{k}}\\#\\{0\leq i\leq
n_{k}:S^{2}(f^{i}(x),\xi_{i})\leq\frac{1}{\delta}\\}>\frac{\delta}{2}$,

$\frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1}\zeta^{u}_{i}>\frac{1}{n_{k}}\sum_{i\leq
n_{k}:S^{2}(f^{i}(x),\xi_{i})<\frac{1}{\delta}}\zeta^{u}_{i}\geq\frac{\delta}{2}\cdot\delta>0.$

This concludes item (1). The proof of item (2) is similar, and we leave it to
the reader. ∎

Remark: (70) shows that for the “if” direction, the assumption
$\lim\frac{1}{n}\log S\circ f^{n}=0$ is not needed. This is in fact the
direction which implies how hyperbolicity emerges from positive recurrence.

###  9.2. Temperability and rate of contraction

Let $F(x):=e^{\alpha x}$, where $\alpha,x>0$. We get that
$\frac{F(x)}{F^{\prime}(x)}=\frac{1}{\alpha}<\infty$. That is, for an
exponential $F$, this quantity is bounded away from infinity uniformly in $x$
(however big the bound may be).

We wish to consider all rates which are strictly weaker than exponential, in
the following sense: Let $F:\mathbb{R}^{+}\to\mathbb{R}^{+}$ be a
differentiable function s.t

$\lim_{x\to\infty}\frac{F(x)}{F^{\prime}(x)}=\infty,$

and assume that $\frac{F(x)}{F^{\prime}(x)}$ is non-decreasing. So, as long as
$\frac{F(x)}{F^{\prime}(x)}$ grows and is unbounded (no matter how slowly it
grows), we consider it. The following proposition tells us that for any rate
which is weaker than exponential in the sense above, there is a temperability
rate which implies such contraction on the stable or unstable subspaces.

######  Proposition 9.2.

Let $F:\mathbb{R}^{+}\to\mathbb{R}^{+}$ differentiable s.t
$\frac{F(x)}{F^{\prime}(x)}\uparrow\infty$. Let $x\in 0\mathrm{-summ}$,
$\xi\in H^{s}(x)$ s.t $|\xi|=1$. Write
$\xi_{n}:=\frac{d_{x}f^{n}\xi}{|d_{x}f^{n}\xi|}$, and assume that $\exists
C>0$ s.t $\forall n\geq 0$, $\frac{1}{2}S^{2}(f^{n}(x),\xi_{n})\leq
C\frac{F(n)}{F^{\prime}(n)}$. Then
$|d_{x}f^{n}\xi|=O(\frac{1}{(F(n))^{\frac{1-\tau}{C}}})$, $\forall\tau>0$. A
similar statement holds for contraction on $H^{u}(x)$.

###### Proof.

We start by proving an inequality. Let $n_{1}>n_{0}\geq 1$. Write
$G(t):=\frac{F^{\prime}(t)}{F(t)}$, which is monotone, then,

$|\int_{n_{0}}^{n_{1}}G(t)dt-\sum_{k=n_{0}}^{n_{1}-1}G(k)|=\sum_{k=n_{0}}^{n_{1}-1}G(k)-\int_{k}^{k+1}G(t)dt\leq\sum_{k=n_{0}}^{n_{1}-1}G(k)-G(k+1)=G(n_{0})-G(n_{1})\leq
G(1).$ (72)

We now may continue with proving the proof. Write
$\zeta_{n}^{u}:=\log\frac{1}{\sqrt{1-\frac{2}{S^{2}(f^{n}(x),\xi_{n})}}}$. Let
$\tau>0$, then for all $n\geq n_{x}$, where $n_{x}$ is large enough so
$\frac{F^{\prime}(n_{x})}{CF(n_{x})}$ is sufficiently small,
$\zeta^{u}_{n}\geq\frac{(1-\tau)}{C}\frac{F^{\prime}(n)}{F(n)}$. Then, for all
$n\geq n_{x}$, by (70),

$\displaystyle|d_{x}f^{n}\xi|\leq$ $\displaystyle
S(x,\xi)\cdot\exp\left(-\sum_{k=n_{x}}^{n-1}\frac{(1-\tau)}{C}\frac{F^{\prime}(k)}{F(k)}\right)\leq
S(x)\cdot\exp\left(-\frac{(1-\tau)}{C}\sum_{k=n_{x}}^{n-1}\frac{F^{\prime}(k)}{F(k)}\right)$
$\displaystyle\leq$ $\displaystyle
S(x)\cdot\exp\left(\frac{F^{\prime}(1)}{F(1)C}-\frac{(1-\tau)}{C}\int_{n_{x}}^{n}\frac{F^{\prime}(t)}{F(t)}dt\right)=C_{x}\cdot\exp\left(-\frac{(1-\tau)}{C}\int_{n_{x}}^{n}(\log
F(t))^{\prime}dt\right)\text{ }\because\eqref{finallyIwannaGoHome}$
$\displaystyle=$ $\displaystyle C_{x}\cdot\exp\left(-\frac{(1-\tau)}{C}(\log
F(n)-\log
F(n_{x}))dt\right)=C_{x}^{\prime}\cdot\frac{1}{(F(n))^{\frac{1-\tau}{C}}}.$

In particular, when $\|C_{0}^{-1}(f^{n}(x))\|^{2}=o(n)$ then
$\|d_{x}f^{n}|_{H^{s}(x)}\|$ decreases super-polynomially. Unfortunately, this
relationship is not easily reversible. That is, super-polynomial contraction
does not necessarily imply strong temperability, since we cannot commute a
bound on the sum in the exponent, into a bound onto its terms (while the other
way around works).

∎

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