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The dyadic adding machine is addition by one on the space of infinite binary digit sequences, interpreted with the least significant digit first. Carries propagate toward later digits: $000\dots \mapsto 100\dots \mapsto 010\dots \mapsto 110\dots$. The all-ones sequence maps to the all-zeros sequence.

Equivalently, it is the compatible collection of cyclic maps $i\mapsto i+1(\mathrm{mod}{2}^n)$ for all $n$, or translation by one on the 2-adic integers. Every orbit visits every finite binary cylinder and is therefore dense. At the Feigenbaum attractor, this describes the order of visits to nested intervals; it is not the digit-discarding evolution of a full shift.

For the normalized Feigenbaum map, put $\gamma =-g(1)$ and defineThe Feigenbaum attractor is the nested intersection $A={\bigcap }_n{\bigcup }_i{\Delta }_i^{(n)}$. Each level interval has two children, and the largest interval length tends to zero. Consequently every compatible infinite binary address selects one point of this Cantor set.^[ref]

At level $n$, the intervals are visited cyclically: the label changes from $i$ to $i+1$ modulo $2^n$. Expressing $i$ in binary therefore turns iteration on $A$ into the dyadic adding machine. One adds one with carries, rather than discarding the first symbol as in a full shift. For example, addresses written with the least significant digit first begin $000\dots \mapsto 100\dots \mapsto 010\dots \mapsto 110\dots$.

There are $2^n$ geometric pieces at depth $n$, but those pieces are visited in a cycle. Counting them does not make the time evolution a two-symbol full shift. The Feigenbaum geometric potential uses this depth to study spatial contraction and Hausdorff dimension.

The Feigenbaum geometric potential is a bounded negative symbolic potential $U$ encoding the lengths of nested intervals of the Feigenbaum attractor. In an appropriate binary coding,and changing only the distant tail of an address changes $U$ by an exponentially small amount. For odd interval labels, a sum along reversed address prefixes satisfieswith a bound independent of the depth and address. Even-label intervals have lengths comparable to neighboring odd-label intervals. These uniform estimates connect geometric lengths with thermodynamic formalism.^[ref][ref]

The bounded logarithmic error is important. For fixed real $t$, exponentiation givesAfter summing and taking $n^{-1}\log$, the error is at most $|t|C/n$, which vanishes. Thus geometric partition sums and symbolic partition sums give the same limiting topological pressure. The almost-additive partition-function limit supplies existence under the bounded-distortion hypotheses; bounded negative potentials give a strictly decreasing pressure.

This potential measures geometric contraction across partition levels. It is not an unbounded derivative potential evaluated at the critical point, where $g^{\prime }(0)=0$, and partition depth is distinct from elapsed orbit time.

The Feigenbaum period-doubling map is the universal, even, quadratic-critical fixed map associated with repeated period-doubling bifurcations. One normalization isComposing twice and rescaling therefore reproduces the same graph. This self-similarity is the mechanism behind a common spatial scaling in otherwise different one-dimensional maps with a quadratic maximum.^[ref]

The quadratic maximum means $g^{\prime }(0)=0$ and $g^{\prime \prime }(0)\ne 0$. Twice applying the chain rule at zero givesThe term involving $g^{\prime }(0{)}^2$ vanishes, and cancellation is legitimate because the maximum is nondegenerate. Thus the endpoint derivative is determined by the spatial scaling.

Writing $\gamma =-\beta >0$ makes the successive central intervals $[-{\gamma }^n,{\gamma }^n]$ shrink by the factor $\gamma$. Their iterates generate the Feigenbaum attractor; nonuniform interval lengths are encoded by the Feigenbaum geometric potential. The constant $\beta$ here describes spatial scaling, rather than the spacing of successive bifurcation parameter values.

Use zero-based indices $0\le i<2^n$; the two index conventions in the printed question enumerate the same partition. Let $O_n(\gamma )$ be the sum of length powers over odd intervals and $H_n(\gamma )$ the sum over all intervals. For odd-address words, the given symbolic potential estimate has $c\le |{\Delta }_i^{(n)}|e^{-H_n^{\mathrm{pot}}(i)}\le C$, with constants independent of $n,i$. Raising these bounds to a fixed real power $\gamma$ and reversing the order when necessary gives finite positive constants $c_{\gamma },C_{\gamma }$ such thatTherefore $n^{-1}\log O_n(\gamma )\to f(\gamma )$. It remains to justify the even intervals, since the supplied estimate covers only half of the partition.

In the standard period-doubling partition, ${\Delta }_0^{(n)}={\beta }^n[-1,1]$ and ${\Delta }_i^{(n)}=g^i({\Delta }_0^{(n)})$. All odd-address intervals lie in the first image restrictive interval, a compact set away from the unique critical point zero. Hence there are uniform constants $0<d\le |g^{\prime }|\le D_1$ there. For an odd $i<2^n-1$, its image is the next even interval, and the mean value theorem gives $d|{\Delta }_i^{(n)}|\le |{\Delta }_{i+1}^{(n)}|\le D_1|{\Delta }_i^{(n)}|$.

For the wraparound pair the image of the last interval is a subinterval of ${\Delta }_0^{(n)}$, not necessarily the whole of it. Iterating the renormalization identity givesThus $|g({\Delta }_{2^n-1}^{(n)})|=|\beta {|}^n(1-\beta )$, while $|{\Delta }_0^{(n)}|=2|\beta {|}^n$. The image is the fixed fraction $(1-\beta )/2$ of the central interval. Combining this with the derivative bounds proves uniform comparability of the last odd interval and the central even interval as well. This establishes the odd-even comparison for Feigenbaum partition lengths for every pair, including wraparound.

Consequently, for each fixed real $\gamma$, the even sum is bounded above and below by constant multiples of $O_n(\gamma )$. The same is true of $H_n(\gamma )$, soFinally take $\gamma >{\gamma }_{\ast }$. Strict decrease gives $f(\gamma )<0$, so this length-power sum tends to zero exponentially. The cover mesh also tends to zero: negative bounded $U$ and the odd-interval estimate give $|{\Delta }_i^{(n)}|\le C{e}^{-bn}$ on odd intervals, and the odd-even comparison transfers this bound to all intervals. Since the intervals cover the Feigenbaum attractor, they are admissible arbitrarily fine covers in the definition of Hausdorff measure, and ${\mathcal{H}}^{\gamma }(F)=0$. Therefore the pressure upper bound for Hausdorff dimension yieldsThe covering argument supplies the requested upper bound; it does not assume a dimension equality or a lower bound not proved here.

Takeshi Matsumoto studied the multifractal measure of the Feigenbaum attractor with Uriel Frisch and Konstantin Khanin. Their analysis related power-law tails of fractional derivatives of an integrated invariant measure to its spectrum of local singularities.^[ref]

This supplies a way to detect nonuniform scaling from a transformed observable, rather than assigning the entire attractor one local exponent. The paper distinguishes its heuristic theoretical steps from high-accuracy numerical checks.

Uriel Frisch studied nonuniform scaling of invariant measures with Konstantin Khanin and Takeshi Matsumoto. Their work on the Feigenbaum attractor related the statistics of fractional derivatives to the local singularity structure of a multifractal measure.^[ref]

A single fractal dimension need not describe how probability concentrates differently from point to point. The derivative-based test offers another way to probe that spatially varying scaling, with the paper explicitly comparing heuristic arguments and precise numerical calculations.