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E. B. Vul is a coauthor, with Yakov Sinai and Konstantin Khanin, of a 1984 study of Feigenbaum constants using thermodynamic formalism. The work connects the scaling behavior of period doubling with methods developed for statistical mechanics. The initials identify the cited byline; they are retained here without assuming an unverified expansion of the name.^[ref]

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.

Konstantin Khanin coauthored a study connecting Feigenbaum constants with thermodynamic formalism, together with E. B. Vul and Yakov Sinai. This places the repeated scaling of period-doubling dynamics in a framework related to statistical mechanics, connecting detailed orbit geometry with collective scaling descriptions.^[ref]

Yakov Sinai contributed to the study of Feigenbaum constants through thermodynamic formalism, in joint work with E. B. Vul and Konstantin Khanin. Their 1984 paper connects universal period-doubling scaling with methods of statistical mechanics, illustrating how probabilistic and geometric descriptions of a dynamical system can reinforce each other.^[ref]