Hausdorff Dimension and Topological Entropies of a Solenoid

Abstract

The purpose of this paper is to elucidate the interrelations between three essentially different concepts: solenoids, topological entropy, and Hausdorff dimension. For this purpose, we describe the dynamics of a solenoid by topological entropy-like quantities and investigate the relations between them. For L-Lipschitz solenoids and locally λ — expanding solenoids, we show that the topological entropy and fractal dimensions are closely related. For a locally λ — expanding solenoid, we prove that its topological entropy is lower estimated by the Hausdorff dimension of X multiplied by the logarithm of λ .

Correction: Biś, A., et al. Hausdorff Dimension and Topological Entropies of a Solenoid. Entropy 2020, 22, 506

“…The authors wish to make the following corrections to this paper [1]: On page 1, the paper erroneously states that the solenoid X ∞ is uniquely determined by the sequence of epimorphisms f ∞ . Consequently, the results of the paper do not apply to solenoids, understood as geometric objects that arose due to an inverse limit construction.…”