During the recession of 2007−2009, the u. s. federal government's tax collections fell from about $2.6 trillion down to about $2.1 trillion while gdp declined by about 4 percent. does the u. s. tax system appear to have built-in stabilizers?

Yes. The U.S. tax system has a built-in stabilizers.

These built-in stabilizers are called automatic stabilizers. Automatic stabilizers are defined as the features of tax and transfer system that lends stability of the economy without direct intervention from the policy makers.

These stabilizers tempers the economy when it overheats and provides economic stimulus when it slumps. 

When:                           Automatic Stabilizers:
Incomes are high          tax liabilities rise and eligibility for government benefits falls
Incomes are low            tax liabilities drop and more families become eligible for government                                         transfer programs (food stamps, unemployment insurance)

I'm not 100% sure but i think it is service producing industries