Feigenbaum geometric potential ($-B\le U\le -b<0$)

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.