= Feigenbaum geometric potential
{c}
{title2=$-B\leq U\leq-b<0$}
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,
$$
-B\le U\le-b<0,
$$
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 satisfies
$$
\left|\log|\Delta_i^{(n)}|-S_n^*U(i)\right|\le C,
$$
with 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>.<Feigenbaum universality and the thermodynamic formalism by E. B. Vul and others>[Thermodynamic description of the period-doubling geometry]{ref}<Multifractality of the Feigenbaum attractor and fractional derivatives by Uriel Frisch and others>[Section 2.1, equations (8)–(14): bounded potential, decaying variations and interval-length comparisons]{ref}
The bounded logarithmic error is important. For fixed real $t$, exponentiation gives
$$
e^{-|t|C}e^{tS_n^*U(i)}\le |\Delta_i^{(n)}|^t\le e^{|t|C}e^{tS_n^*U(i)}.
$$
After 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 $g'(0)=0$, and partition depth is distinct from elapsed orbit time.
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