Feigenbaum period-doubling map ($g(g(\beta x))=\beta g(x)$)

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.

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.

Odd-address intervals lie in the first image of the central restrictive interval, away from the critical point. The derivative of the Feigenbaum map is bounded above and away from zero there, so the next even interval has comparable length by the mean value theorem. For the wraparound image, $g^{2^n}({\beta }^{n}x)={\beta }^{n}g(x)$ makes its length a fixed fraction $(1-\beta )/2$ of the central interval. Thus the odd and full partition length-power sums differ only by constant factors for each fixed real exponent, and have the same exponential growth rate.

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 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.