Douglas Lind, Klaus Schmidt, Evgeny Verbitskiy
Entropy and Growth Rate of Periodic Points of Algebraic Z$^d$--Actions
Preprint series: ESI preprints

MSC:

28D20 Entropy and other invariants

47A35 Ergodic theory, See also {28Dxx}

58F11 Ergodic theory; invariant measures, See also {28Dxx}

54C70 Entropy

54H20 Topological dynamics, See also {28Dxx, 34C35, 58Fxx}

28D99 None of the above, but in this section

11K99 None of the above but in this section

13F20 Polynomial rings and ideals, See also {11C08}

Abstract: Expansive algebraic $\zd$-actions corresponding to ideals are
characterized by the property that the complex variety of the ideal
is disjoint from the multiplicative unit torus. For such actions
it is known that the limit for the growth rate of periodic points
exists and equals the entropy of the action. We extend this result to
actions for which the complex variety intersects the multiplicative
torus in a finite set. The main technical tool is the use of
homoclinic points which decay rapidly enough to be summable.