Dimension of the accumulation set of any hair for the exponential map

Abstract: We study the dynamics of the exponential map on the complex plane. The set $Λ<em>{\mathbf{c}}$ of all points sharing a given itinerary $\mathbf{c}$ is non-empty if and only if $\mathbf{c}$ is an exponentially bounded itinerary. For such itineraries, $Λ</em>{\mathbf{c}}$ also contains a curve of escaping points, and hence its Hausdorff dimension is at least~$1$. We prove that for every exponentially bounded itinerary this dimension is in fact equal to~$1$. In comparison, for certain itineraries, the set $Λ_{\mathbf{c}}$ exhibits highly complicated topological structures, such as indecomposable continua.

• The paper proves that every non-empty itinerary set Λ_c for fλ(z) of the exponential map e^z under the condition that “c” is exponentially bounded has a Hausdorff dimension of 1.
• The proof defines three unique regions with a common proof strategy and requires stratify the points in each by a complex estimation of escape time
• This key fact grows from enhanced techniques including derivative growth lemmas and adjustments for the excecution

The problem and its context

The paper studies the Hausdorff dimension of the sets $\Lambda_c$ of points sharing a prescribed itinerary under the exponential map $f_\lambda(z)=\lambda e^z$. Partitioning the plane into horizontal strips $$P_k=\{z:(2k-1)\pi<\operatorname{Im}z\leqslant(2k+1)\pi\}$$, each point $z$ is assigned an itinerary $c=c_0c_1\ldots$ recording the strip visited by each iterate. Devaney and Krych established that $\Lambda_c\neq\emptyset$ if and only if $c$ is exponentially bounded, and Schleicher and Zimmer later showed (with a parameter-independent definition of exponential boundedness) that for such itineraries $\Lambda_c^\infty$ contains an escaping curve — the tail of a hair — which Viana proved to be $C^\infty$-smooth.

The topological structure of $\Lambda_c$ can be far more complicated than a single curve: Devaney and Jarque showed that for $f_\lambda(z)=\lambda e^z$0 and bounded block-form itineraries with sufficiently long zero blocks, $f_\lambda(z)=\lambda e^z$1 is an indecomposable continuum; related phenomena occur at Misiurewicz parameters such as $f_\lambda(z)=\lambda e^z$2, and bounded indecomposable continua disjoint from the hair itself are also known. Since every non-empty $f_\lambda(z)=\lambda e^z$3 contains a smooth curve, $f_\lambda(z)=\lambda e^z$4 always holds. Zdunik and Pawelec had previously shown that for bounded itineraries this lower bound is sharp: $f_\lambda(z)=\lambda e^z$5. The present work extends this to all exponentially bounded itineraries — including unbounded ones — showing that topological complexity never raises the Hausdorff dimension above one.

Main result

For $f_\lambda(z)=\lambda e^z$6 and any itinerary $f_\lambda(z)=\lambda e^z$7, the authors prove

$f_\lambda(z)=\lambda e^z$8

which, combined with the existence of an escaping curve in every non-empty $f_\lambda(z)=\lambda e^z$9, yields the corollary that $$P_k=\{z:(2k-1)\pi<\operatorname{Im}z\leqslant(2k+1)\pi\}$$0 whenever $$P_k=\{z:(2k-1)\pi<\operatorname{Im}z\leqslant(2k+1)\pi\}$$1 (equivalently, whenever $$P_k=\{z:(2k-1)\pi<\operatorname{Im}z\leqslant(2k+1)\pi\}$$2 is exponentially bounded). This is a strong rigidity statement: even when $$P_k=\{z:(2k-1)\pi<\operatorname{Im}z\leqslant(2k+1)\pi\}$$3 is an indecomposable continuum accumulating everywhere on itself, its Hausdorff dimension equals exactly one. The restriction to $$P_k=\{z:(2k-1)\pi<\operatorname{Im}z\leqslant(2k+1)\pi\}$$4 is a deliberate simplification; the authors state that the techniques appear to extend to all real $$P_k=\{z:(2k-1)\pi<\operatorname{Im}z\leqslant(2k+1)\pi\}$$5, but not to non-real parameters such as $$P_k=\{z:(2k-1)\pi<\operatorname{Im}z\leqslant(2k+1)\pi\}$$6.

Proof strategy

The set $$P_k=\{z:(2k-1)\pi<\operatorname{Im}z\leqslant(2k+1)\pi\}$$7 of points with bounded orbit consists of at most two points, so it suffices to cover the unbounded part. Fixing $$P_k=\{z:(2k-1)\pi<\operatorname{Im}z\leqslant(2k+1)\pi\}$$8, the authors construct coverings $$P_k=\{z:(2k-1)\pi<\operatorname{Im}z\leqslant(2k+1)\pi\}$$9 with mesh tending to zero and total $z$0-weight uniformly bounded by $z$1, giving $z$2; letting $z$3 concludes the argument. A key choice is to take the cutoff $z$4 on the forward orbit of the singular value $z$5, so that $z$6, ensuring $z$7.

The plane is split into three regions treated by distinct subfamilies:

• Right half-plane ($z$8): pull back rectangles from level zero via inverse branches; counting arguments give at most $z$9 pieces per rectangle $c=c_0c_1\ldots$0 with diameter $c=c_0c_1\ldots$1, yielding weight $c=c_0c_1\ldots$2 per rectangle.
• Left half-plane ($c=c_0c_1\ldots$3): orbits starting far left shadow the orbit of $c=c_0c_1\ldots$4 until reaching roughly $c=c_0c_1\ldots$5 (Lemma on escape from $c=c_0c_1\ldots$6), so one iterates until the image has diameter at least $c=c_0c_1\ldots$7 and pulls back elements of $c=c_0c_1\ldots$8 using a branch of $c=c_0c_1\ldots$9. Bounded distortion of $\Lambda_c\neq\emptyset$0 follows from the Koebe Distortion Theorem, with distortion constant $\Lambda_c\neq\emptyset$1 arbitrarily close to $\Lambda_c\neq\emptyset$2 as $\Lambda_c\neq\emptyset$3.
• Middle strip ($\Lambda_c\neq\emptyset$4): this requires entirely new techniques relative to [PZ]. Points are stratified by escape time $\Lambda_c\neq\emptyset$5 into sets $\Lambda_c\neq\emptyset$6, further split by sign ($\Lambda_c\neq\emptyset$7) according to whether some iterate crosses the negative real axis — possible only once per piece, and then along a line segment interior to the piece. A derivative-growth lemma shows that for each point there exists $\Lambda_c\neq\emptyset$8 with $\Lambda_c\neq\emptyset$9 (with $c$0), obtained by partitioning trajectories into sectors where the cumulative derivative grows geometrically, using explicit estimates near $c$1 and distortion control further out.

Each subfamily contributes at most $c$2 to the first-level sum, and an induction over levels transfers these bounds: the right-family estimate $c$3 per rectangle is preserved, while left and middle families inherit their weights from pulled-back right-families at shifted itineraries, with the geometric factor $c$4 controlling the middle sum. The authors emphasize careful tracking of constants throughout, noting the proof is more technical than the underlying idea warrants.

Limitations and open questions

The method is confined to real parameters: the authors explicitly note it does not work for $c$5, notably $c$6, where hairs accumulate on themselves through two distinct mechanisms. Although the authors expect the argument extends to all real $c$7, this is asserted rather than carried out, and the extension would require re-verifying the delicate constant-dependent estimates. The full catalogue of topological types realized by $c$8 across all parameters and itineraries also remains open, though the dimension result shows this classification cannot be detected by Hausdorff dimension.

Conclusion

The paper establishes that for the exponential map $c$9, every non-empty itinerary set $\Lambda_c^\infty$0 has Hausdorff dimension exactly one, regardless of whether it is a simple hair or an indecomposable continuum. The proof combines classical covering techniques for the outer half-planes with new escape-time stratification and derivative-growth estimates in the middle region, extending the earlier bounded-itinerary result of Zdunik and Pawelec to all exponentially bounded itineraries.