Hölder Shadowing Property

• Hölder shadowing is a quantitative strengthening of the pseudo-orbit tracing property where every δ-pseudo-orbit is tracked by a true orbit within an error bound of Cδ^α.
• In low-dimensional dynamics, a threshold of α > 1/2 forces the elimination of turning points and guarantees conjugacy to a linear expanding endomorphism on the circle.
• Extensions to finite-interval and infinite-dimensional settings show that precise Hölder estimates can lead to structural stability and even imply Lipschitz shadowing in certain contexts.

Searching arXiv for the cited papers and closely related shadowing results to ground the article. Hölder shadowing is a quantitative strengthening of the classical pseudo-orbit tracing property: it requires that every $\delta$-pseudo-orbit be tracked by a true orbit with an error bounded by a power law $C\delta^\alpha$, where $\alpha\in(0,1]$. In the low-dimensional setting studied in “Some Consequences of the Shadowing Property in Low Dimensions” (Koropecki et al., 2011), this additional quantitative control has strong dynamical consequences. For transitive $C^2$ endomorphisms of $S^1$ with finitely many turning points, $\alpha$-Hölder shadowing with $\alpha>1/2$ forces conjugacy to a linear expanding endomorphism; under robust transitivity and orientation preservation, it forces genuine expansion. Other works connect finite-interval Hölder shadowing to structural stability (Tikhomirov, 2011), establish Hölder shadowing on positively invariant neighborhoods for Morse–Smale semigroups in Hilbert spaces (Arrieta et al., 12 Feb 2025), and show that in certain nonautonomous Banach-space settings Hölder regularity of derivatives enters the proof of a Lipschitz shadowing theorem rather than changing the shadowing exponent itself (Backes et al., 2018).

1. Definitions and principal formulations

Let $(X,d)$ be a metric space and $f:X\to X$ a homeomorphism. A bi-infinite $\delta$-pseudo-orbit is a sequence $C\delta^\alpha$0 such that

$C\delta^\alpha$1

A point $C\delta^\alpha$2 $C\delta^\alpha$3-shadows $C\delta^\alpha$4 if

$C\delta^\alpha$5

The map $C\delta^\alpha$6 has the shadowing property if for every $C\delta^\alpha$7 there exists $C\delta^\alpha$8 such that every $C\delta^\alpha$9-pseudo-orbit is $\alpha\in(0,1]$0-shadowed by some orbit (Koropecki et al., 2011).

Given constants $\alpha\in(0,1]$1 and $\alpha\in(0,1]$2, an endomorphism $\alpha\in(0,1]$3 has the $\alpha\in(0,1]$4-Hölder shadowing property if for every $\alpha\in(0,1]$5 and every $\alpha\in(0,1]$6-pseudo-orbit $\alpha\in(0,1]$7 there exists $\alpha\in(0,1]$8 such that

$\alpha\in(0,1]$9

The case $C^2$0 is Lipschitz shadowing. The low-dimensional synthesis explicitly notes that hyperbolic or Axiom A systems with strong transversality satisfy Lipschitz shadowing, and that for diffeomorphisms Lipschitz shadowing is equivalent to hyperbolicity (Koropecki et al., 2011).

For circle endomorphisms, expansion is formulated in either differential or metric form. For a $C^2$1 endomorphism $C^2$2, one equivalent condition is the existence of constants $C^2$3 and $C^2$4 such that

$C^2$5

for all $C^2$6 and all $C^2$7; equivalently, there exists $C^2$8 such that

$C^2$9

for all $S^1$0 and $S^1$1. Conjugacy on $S^1$2 means the existence of a homeomorphism $S^1$3 satisfying

$S^1$4

These formulations are central in the circle theorems of (Koropecki et al., 2011).

A distinct finite-time variant appears in “Holder Shadowing on Finite Intervals” (Tikhomirov, 2011). For $S^1$5 and $S^1$6, a $S^1$7 diffeomorphism $S^1$8 has $S^1$9 if there exist constants $\alpha$0, $\alpha$1, $\alpha$2 such that, for every $\alpha$3, every $\alpha$4-pseudotrajectory $\alpha$5 on $\alpha$6 is $\alpha$7-shadowed by a true orbit. When the index set is all of $\alpha$8, one recovers the global Hölder shadowing property $\alpha$9.

In infinite-dimensional semigroup settings, the formulation is typically forward in time. For a continuous semigroup $\alpha>1/2$0 on a Hilbert space $\alpha>1/2$1, a forward $\alpha>1/2$2-pseudotrajectory for the time-one map $\alpha>1/2$3 is a sequence $\alpha>1/2$4 with

$\alpha>1/2$5

On a positively invariant bounded neighborhood $\alpha>1/2$6 of a compact global attractor $\alpha>1/2$7, Hölder shadowing means that there exist $\alpha>1/2$8, $\alpha>1/2$9, and $(X,d)$0 such that every forward $(X,d)$1-pseudotrajectory in $(X,d)$2 is shadowed by a true orbit within $(X,d)$3 uniformly in $(X,d)$4 (Arrieta et al., 12 Feb 2025).

2. Low-dimensional context: chain recurrence, surfaces, and the limits of qualitative shadowing

The low-dimensional paper (Koropecki et al., 2011) places Hölder shadowing against a broader topological background built from chain recurrence. The chain recurrent set $(X,d)$5 consists of points $(X,d)$6 such that for every $(X,d)$7 there exists an $(X,d)$8-pseudo-orbit from $(X,d)$9 back to $f:X\to X$0. For fixed $f:X\to X$1, one defines $f:X\to X$2 when there are $f:X\to X$3-pseudo-orbits from $f:X\to X$4 to $f:X\to X$5 and from $f:X\to X$6 to $f:X\to X$7; the equivalence classes are the $f:X\to X$8-transitive components. Conley’s theory supplies a complete Lyapunov function $f:X\to X$9 organizing these classes, and on surfaces each $\delta$0-transitive class is of the form $\delta$1 with $\delta$2 regular values.

For compact orientable surfaces, classical shadowing already has strong consequences. If $\delta$3 is a homeomorphism of a compact orientable surface with the shadowing property, then for any $\delta$4 each $\delta$5-transitive component has a periodic point. Equivalently, $\delta$6 intersects every chain transitive class, and in particular $\delta$7 has a periodic point (Koropecki et al., 2011). The proof uses Conley’s Lyapunov function, an ends compactification of an invariant open set containing the class, a Lefschetz number computation, and Brouwer theory for planar lifts of pseudo-orbits.

That conclusion does not imply that every transitive class is periodic. The same paper constructs, on any compact surface $\delta$8, a $\delta$9 Kupka–Smale diffeomorphism with the shadowing property and with an aperiodic chain transitive component: an invariant circle carrying an irrational rotation with Liouville rotation number. The construction uses a nested sequence of invariant annuli shrinking to the circle, alternating attracting and repelling normal hyperbolicity on boundary components, and “crooked horseshoes” accumulating on the circle. Outside a fixed neighborhood of the circle the dynamics is Axiom A with strong transversality and hence has Lipschitz shadowing; near the circle, the crooked horseshoes provide orbits whose first coordinate $C\delta^\alpha$00-shadows any $C\delta^\alpha$01-pseudo-orbit of the rotation with quantitative control (Koropecki et al., 2011).

This contrast is one of the main motivations for emphasizing Hölder shadowing rather than shadowing alone. On surfaces, classical shadowing forces periodic points in every $C\delta^\alpha$02-transitive class but still admits aperiodic transitive classes. On the circle, by contrast, the quantitative exponent $C\delta^\alpha$03 in the Hölder shadowing estimate has enough rigidity to force hyperbolic behavior.

3. Circle endomorphisms, the threshold $C\delta^\alpha$04, and expanding dynamics

The central one-dimensional theorem in (Koropecki et al., 2011) states: let $C\delta^\alpha$05 be a $C\delta^\alpha$06 endomorphism of the circle with finitely many turning points. If $C\delta^\alpha$07 is transitive and satisfies the $C\delta^\alpha$08-Hölder shadowing property with $C\delta^\alpha$09, then $C\delta^\alpha$10 is conjugate to a linear expanding endomorphism of the circle. A companion theorem strengthens the conclusion under robust transitivity: if $C\delta^\alpha$11 is a $C\delta^\alpha$12 orientation-preserving endomorphism of the circle with finitely many turning points, satisfies $C\delta^\alpha$13-Hölder shadowing with $C\delta^\alpha$14, and is $C\delta^\alpha$15-robustly transitive for some $C\delta^\alpha$16, then $C\delta^\alpha$17 is an expanding endomorphism (Koropecki et al., 2011).

The threshold $C\delta^\alpha$18 is explicit in the argument. Near a turning point $C\delta^\alpha$19, $C\delta^\alpha$20 regularity implies quadratic flattening:

$C\delta^\alpha$21

for $C\delta^\alpha$22 sufficiently close to $C\delta^\alpha$23. If $C\delta^\alpha$24, a pseudo-orbit can be formed by taking a single jump from $C\delta^\alpha$25 to $C\delta^\alpha$26 and then following the true orbit of $C\delta^\alpha$27. The jump size is $C\delta^\alpha$28. Hölder shadowing then gives a shadowing error of size $C\delta^\alpha$29. To make this error much smaller than $C\delta^\alpha$30, one needs $C\delta^\alpha$31, i.e. $C\delta^\alpha$32 (Koropecki et al., 2011).

The paper explicitly records the boundary of this mechanism. If $C\delta^\alpha$33, then the shadowing error can be comparable to $C\delta^\alpha$34, and the turning-point elimination argument breaks down. It does not construct counterexamples for $C\delta^\alpha$35, and the optimality of the threshold beyond this mechanism remains open in that text. A plausible implication is that the exponent $C\delta^\alpha$36 is intrinsic to the quadratic local geometry of turning points rather than to circle transitivity alone.

The transitivity hypothesis is also essential in the stated circle theorems. Once turning points are excluded, the map is a local homeomorphism, and the paper proves that a transitive non-invertible local homeomorphism of degree $C\delta^\alpha$37 with $C\delta^\alpha$38 is conjugate to the linear expanding map $C\delta^\alpha$39. Robust transitivity then rules out the remaining nonexpanding alternatives through perturbative arguments and a one-dimensional dichotomy attributed there to Mañé (Koropecki et al., 2011).

4. Turning-point exclusion and the structure of the proof

The proof strategy in the circle case begins by assuming the existence of a turning point $C\delta^\alpha$40 and producing a contradiction with transitivity. One chooses $C\delta^\alpha$41 near $C\delta^\alpha$42, builds the single-jump pseudo-orbit from $C\delta^\alpha$43 to $C\delta^\alpha$44, and invokes $C\delta^\alpha$45-Hölder shadowing to obtain an orbit $C\delta^\alpha$46 with shadowing accuracy of order $C\delta^\alpha$47. The local monotonicity of $C\delta^\alpha$48 on either side of $C\delta^\alpha$49 allows the construction of a small interval $C\delta^\alpha$50 around $C\delta^\alpha$51, bounded by the two preimages of $C\delta^\alpha$52 inside a neighborhood $C\delta^\alpha$53 on which $C\delta^\alpha$54 is strictly monotone on each side (Koropecki et al., 2011).

Because $C\delta^\alpha$55 when $C\delta^\alpha$56, one has $C\delta^\alpha$57 for $C\delta^\alpha$58 sufficiently small. This implies that the forward iterates $C\delta^\alpha$59 remain small enough to contain another turning point $C\delta^\alpha$60 after finitely many iterates. Repeating the argument along the finite set of turning points yields a finite cycle of intervals $C\delta^\alpha$61 and an iterate $C\delta^\alpha$62 with

$C\delta^\alpha$63

that is, a compact inclusion. Such a trapping interval contradicts transitivity. Therefore the map has no turning points and is consequently a local homeomorphism (Koropecki et al., 2011).

The next step is topological. For a transitive non-invertible local homeomorphism of the circle, the paper constructs a semiconjugacy $C\delta^\alpha$64 to the linear expanding endomorphism $C\delta^\alpha$65. Transitivity excludes nontrivial interval fibers of $C\delta^\alpha$66, because such fibers would force periodic dynamics on the image. Hence the semiconjugacy is actually a homeomorphism, and the original map is conjugate to $C\delta^\alpha$67 (Koropecki et al., 2011).

The robustly transitive case adds a perturbative layer. Using a family of lifts $C\delta^\alpha$68 on the universal cover and associated circle maps $C\delta^\alpha$69, recurrent points can be closed to periodic ones under small translations. This is used to show that critical points cannot persist under robust transitivity. The final step invokes a one-dimensional dichotomy: a $C\delta^\alpha$70 transitive endomorphism of $C\delta^\alpha$71 without critical points is either conjugate to a rotation, has a non-hyperbolic periodic point, or is expanding. Non-invertibility excludes rotation, and robust transitivity excludes non-hyperbolic periodic points because small perturbations would create sinks; therefore the map is expanding (Koropecki et al., 2011).

5. Finite-interval Hölder shadowing and structural stability

“Holder Shadowing on Finite Intervals” (Tikhomirov, 2011) studies a weaker, finite-time version of Hölder shadowing. The property $C\delta^\alpha$72 requires that every $C\delta^\alpha$73-pseudotrajectory of length at most $C\delta^\alpha$74 be shadowed with accuracy $C\delta^\alpha$75. The main theorem states that if a $C\delta^\alpha$76 diffeomorphism satisfies $C\delta^\alpha$77 with

$C\delta^\alpha$78

then it is structurally stable. A direct corollary is that if a $C\delta^\alpha$79 diffeomorphism satisfies the global Hölder shadowing property $C\delta^\alpha$80 with $C\delta^\alpha$81, then it is structurally stable, because global Hölder shadowing implies $C\delta^\alpha$82 for every $C\delta^\alpha$83 (Tikhomirov, 2011).

The proof is organized through linearized dynamics. Along any true trajectory $C\delta^\alpha$84 one considers the derivative cocycle $C\delta^\alpha$85 and the inhomogeneous linear system

$C\delta^\alpha$86

Finite-interval Hölder shadowing implies a sublinear growth property $C\delta^\alpha$87 for this linear system, for some $C\delta^\alpha$88. The sublinear growth property means that for every bounded inhomogeneity, solutions exist on finite intervals with norm bounded by $C\delta^\alpha$89, where $C\delta^\alpha$90 is the interval length (Tikhomirov, 2011).

The paper then proves that $C\delta^\alpha$91 with $C\delta^\alpha$92 implies exponential dichotomy on $C\delta^\alpha$93 and $C\delta^\alpha$94 together with a transversality condition. Through results quoted there from Todorov, exponential dichotomy plus transversality is equivalent to the bounded solution property for the linearized inhomogeneous system. This, in turn, yields Mañé’s criterion for structural stability:

$C\delta^\alpha$95

where

$C\delta^\alpha$96

and

$C\delta^\alpha$97

Thus finite-time Hölder shadowing above the threshold forces global structural stability (Tikhomirov, 2011).

The paper also identifies sharpness phenomena. The identity map satisfies $C\delta^\alpha$98 whenever $C\delta^\alpha$99, yet it is not structurally stable. In addition, there exists a non-structurally stable $\alpha\in(0,1]$00 diffeomorphism of $\alpha\in(0,1]$01 such that

$\alpha\in(0,1]$02

These examples show that equality at the threshold does not suffice in general, and that the conditions $\alpha\in(0,1]$03 and $\alpha\in(0,1]$04 are close to optimal (Tikhomirov, 2011).

6. Infinite-dimensional and nonautonomous generalizations

In “Shadowing for Infinite Dimensional Dynamical Systems” (Arrieta et al., 12 Feb 2025), the setting is a continuous semigroup $\alpha\in(0,1]$05 on a separable Hilbert space $\alpha\in(0,1]$06, assumed dissipative and endowed with a compact global attractor $\alpha\in(0,1]$07. Under Morse–Smale hypotheses—hyperbolicity of invariant objects, transversality of stable and unstable manifolds, finiteness of equilibria and periodic orbits in $\alpha\in(0,1]$08, compactness of $\alpha\in(0,1]$09, and sufficient regularity and smoothing—the time-one map $\alpha\in(0,1]$10 has Lipschitz shadowing, while for every bounded positively invariant neighborhood $\alpha\in(0,1]$11 the restriction $\alpha\in(0,1]$12 has Hölder shadowing with some exponent $\alpha\in(0,1]$13 (Arrieta et al., 12 Feb 2025).

The loss from Lipschitz to Hölder occurs away from the attractor. On $\alpha\in(0,1]$14, the dynamics is confined to a compact, hyperbolic, finite-dimensional network with uniform spectral gaps and uniform local invariant manifold sizes, and the derivative of $\alpha\in(0,1]$15 is uniformly bounded; this yields linear dependence of the shadowing error on $\alpha\in(0,1]$16. On $\alpha\in(0,1]$17, the time-one map is typically compact or noninvertible, the derivative may only be Hölder continuous with exponent $\alpha\in(0,1]$18, and uniform hyperbolicity constants are only available on $\alpha\in(0,1]$19. The paper describes a representative mechanism in which a Lyapunov–Perron argument yields

$\alpha\in(0,1]$20

leading to

$\alpha\in(0,1]$21

with the explicit caveat that the precise exponent depends on the balance of constants (Arrieta et al., 12 Feb 2025). The same work derives applications to structural stability on the attractor and Hölder continuity of global attractors under perturbation, including estimates of Hausdorff distance of the form

$\alpha\in(0,1]$22

A different Banach-space generalization appears in “Shadowing for nonautonomous dynamics” (Backes et al., 2018). There the nonautonomous system is

$\alpha\in(0,1]$23

where $\alpha\in(0,1]$24 is a sequence of invertible bounded operators with an exponential dichotomy, and the maps $\alpha\in(0,1]$25 are differentiable with uniformly bounded derivatives and Hölder-continuous derivatives:

$\alpha\in(0,1]$26

For admissible Banach sequence spaces $\alpha\in(0,1]$27, the paper defines $\alpha\in(0,1]$28-pseudotrajectories and proves a $\alpha\in(0,1]$29-Lipschitz shadowing theorem with uniqueness. In that proof, Hölder regularity enters through the estimate

$\alpha\in(0,1]$30

which yields a contraction condition

$\alpha\in(0,1]$31

The resulting shadowing exponent is nevertheless $\alpha\in(0,1]$32, not a subunit Hölder exponent; the shadowing error scales linearly with $\alpha\in(0,1]$33, while the Hölder exponent $\alpha\in(0,1]$34 controls the admissible radius of the fixed point argument (Backes et al., 2018). This separates two roles that can be played by Hölder regularity: it may define the shadowing modulus itself, as in (Koropecki et al., 2011, Tikhomirov, 2011), and (Arrieta et al., 12 Feb 2025), or it may act as an analytical hypothesis used to prove a Lipschitz shadowing statement, as in (Backes et al., 2018).

Across these settings, the Hölder shadowing property functions as a quantitative bridge between approximate and exact dynamics. In circle dynamics it rules out turning points above the threshold $\alpha\in(0,1]$35; on finite intervals it forces structural stability when $\alpha\in(0,1]$36 cross the corresponding threshold; in infinite-dimensional semigroups it captures the loss of regularity away from the attractor; and in nonautonomous Banach-space dynamics Hölder continuity of derivatives governs the fixed-point estimates underlying global Lipschitz shadowing.